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class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Units"> <div class="vector-toc-text"> 7 Units </div> </a> <ul id="toc-Units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Molecular_origins" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Molecular_origins"> <div class="vector-toc-text"> 8 Molecular origins </div> </a> <button aria-controls="toc-Molecular_origins-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Molecular origins subsection </button> <ul id="toc-Molecular_origins-sublist" class="vector-toc-list"> <li id="toc-Pure_gases" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pure_gases"> <div class="vector-toc-text"> 8.1 Pure gases </div> </a> <ul id="toc-Pure_gases-sublist" class="vector-toc-list"> <li id="toc-Chapman–Enskog_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Chapman–Enskog_theory"> <div class="vector-toc-text"> 8.1.1 Chapman–Enskog theory </div> </a> <ul id="toc-Chapman–Enskog_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bulk_viscosity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bulk_viscosity"> <div class="vector-toc-text"> 8.1.2 Bulk viscosity </div> </a> <ul id="toc-Bulk_viscosity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pure_liquids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pure_liquids"> <div class="vector-toc-text"> 8.2 Pure liquids </div> </a> <ul id="toc-Pure_liquids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mixtures_and_blends" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mixtures_and_blends"> <div class="vector-toc-text"> 8.3 Mixtures and blends </div> </a> <ul id="toc-Mixtures_and_blends-sublist" class="vector-toc-list"> <li id="toc-Gaseous_mixtures" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Gaseous_mixtures"> <div class="vector-toc-text"> 8.3.1 Gaseous mixtures </div> </a> <ul id="toc-Gaseous_mixtures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Blends_of_liquids" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Blends_of_liquids"> <div class="vector-toc-text"> 8.3.2 Blends of liquids </div> </a> <ul id="toc-Blends_of_liquids-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Solutions_and_suspensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solutions_and_suspensions"> <div class="vector-toc-text"> 8.4 Solutions and suspensions </div> </a> <ul id="toc-Solutions_and_suspensions-sublist" class="vector-toc-list"> <li id="toc-Aqueous_solutions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Aqueous_solutions"> <div class="vector-toc-text"> 8.4.1 Aqueous solutions </div> </a> <ul id="toc-Aqueous_solutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Suspensions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Suspensions"> <div class="vector-toc-text"> 8.4.2 Suspensions </div> </a> <ul id="toc-Suspensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Amorphous_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Amorphous_materials"> <div class="vector-toc-text"> 8.5 Amorphous materials </div> </a> <ul id="toc-Amorphous_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eddy_viscosity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eddy_viscosity"> <div class="vector-toc-text"> 8.6 Eddy viscosity </div> </a> <ul id="toc-Eddy_viscosity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Prediction" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Prediction"> <div class="vector-toc-text"> 9 Prediction </div> </a> <ul id="toc-Prediction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Selected_substances" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Selected_substances"> <div class="vector-toc-text"> 10 Selected substances </div> </a> <button aria-controls="toc-Selected_substances-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Selected substances subsection </button> <ul id="toc-Selected_substances-sublist" class="vector-toc-list"> <li id="toc-Water" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Water"> <div class="vector-toc-text"> 10.1 Water </div> </a> <ul id="toc-Water-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Air" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Air"> <div class="vector-toc-text"> 10.2 Air </div> </a> <ul id="toc-Air-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_common_substances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_common_substances"> <div class="vector-toc-text"> 10.3 Other common substances </div> </a> <ul id="toc-Other_common_substances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Order_of_magnitude_estimates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Order_of_magnitude_estimates"> <div class="vector-toc-text"> 10.4 Order of magnitude estimates </div> </a> <ul id="toc-Order_of_magnitude_estimates-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 11 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 12 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> 12.1 Footnotes </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 12.2 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 12.3 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 13 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Viscosity</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 78 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-78" aria-hidden="true" > 78 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Viskositeit" title="Viskositeit – Afrikaans" lang="af" hreflang="af" data-title="Viskositeit" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D8%B2%D9%88%D8%AC%D8%A9" title="لزوجة – Arabic" lang="ar" hreflang="ar" data-title="لزوجة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Mafa" title="Mafa – Asturian" lang="ast" hreflang="ast" data-title="Mafa" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%96zl%C3%BCl%C3%BCk" title="Özlülük – Azerbaijani" lang="az" hreflang="az" data-title="Özlülük" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A2%D8%BA%D8%AF%D8%A7%D9%84%DB%8C%DA%A9" title="آغدالیک – South Azerbaijani" lang="azb" hreflang="azb" data-title="آغدالیک" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%A6%E0%A7%8D%E0%A6%B0%E0%A6%A4%E0%A6%BE" title="সান্দ্রতা – Bangla" lang="bn" hreflang="bn" data-title="সান্দ্রতা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Li%C3%A2m-t%C5%8D%CD%98" title="Liâm-tō͘ – Minnan" lang="nan" hreflang="nan" data-title="Liâm-tō͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D1%8F%D0%B7%D0%BA%D0%B0%D1%81%D1%86%D1%8C" title="Вязкасць – Belarusian" lang="be" hreflang="be" data-title="Вязкасць" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%B8%D1%82%D0%B5%D1%82" title="Вискозитет – Bulgarian" lang="bg" hreflang="bg" data-title="Вискозитет" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Viskoznost" title="Viskoznost – Bosnian" lang="bs" hreflang="bs" data-title="Viskoznost" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Viscositat" title="Viscositat – Catalan" lang="ca" hreflang="ca" data-title="Viscositat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D3%B3%D0%BD%D0%B5%D0%BA" title="Нӳнек – Chuvash" lang="cv" hreflang="cv" data-title="Нӳнек" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Viskozita" title="Viskozita – Czech" lang="cs" hreflang="cs" data-title="Viskozita" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gludedd" title="Gludedd – Welsh" lang="cy" hreflang="cy" data-title="Gludedd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Viskositet" title="Viskositet – Danish" lang="da" hreflang="da" data-title="Viskositet" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Viskosit%C3%A4t" title="Viskosität – German" lang="de" hreflang="de" data-title="Viskosität" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Viskoossus" title="Viskoossus – Estonian" lang="et" hreflang="et" data-title="Viskoossus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%99%CE%BE%CF%8E%CE%B4%CE%B5%CF%82" title="Ιξώδες – Greek" lang="el" hreflang="el" data-title="Ιξώδες" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%9F%D0%B5%D0%B7%D0%BD%D0%B0%D0%BC%D0%BE" title="Пезнамо – Erzya" lang="myv" hreflang="myv" data-title="Пезнамо" data-language-autonym="Эрзянь" data-language-local-name="Erzya" class="interlanguage-link-target">Эрзянь</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Viscosidad" title="Viscosidad – Spanish" lang="es" hreflang="es" data-title="Viscosidad" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Viskozeco" title="Viskozeco – Esperanto" lang="eo" hreflang="eo" data-title="Viskozeco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Biskositate" title="Biskositate – Basque" lang="eu" hreflang="eu" data-title="Biskositate" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D8%A7%D9%86%E2%80%8C%D8%B1%D9%88%DB%8C" title="گران‌روی – Persian" lang="fa" hreflang="fa" data-title="گران‌روی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Viscosit%C3%A9" title="Viscosité – French" lang="fr" hreflang="fr" data-title="Viscosité" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Slaodacht" title="Slaodacht – Irish" lang="ga" hreflang="ga" data-title="Slaodacht" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Viscosidade" title="Viscosidade – Galician" lang="gl" hreflang="gl" data-title="Viscosidade" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%90%EC%84%B1%EB%8F%84" title="점성도 – Korean" lang="ko" hreflang="ko" data-title="점성도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%AE%D5%B8%D6%82%D6%81%D5%AB%D5%AF%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Մածուցիկություն – Armenian" lang="hy" hreflang="hy" data-title="Մածուցիկություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B6%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%A8%E0%A4%A4%E0%A4%BE" title="श्यानता – Hindi" lang="hi" hreflang="hi" data-title="श्यानता" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Viskoznost" title="Viskoznost – Croatian" lang="hr" hreflang="hr" data-title="Viskoznost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kekentalan" title="Kekentalan – Indonesian" lang="id" hreflang="id" data-title="Kekentalan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Seigja" title="Seigja – Icelandic" lang="is" hreflang="is" data-title="Seigja" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Viscosit%C3%A0" title="Viscosità – Italian" lang="it" hreflang="it" data-title="Viscosità" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A6%D7%9E%D7%99%D7%92%D7%95%D7%AA" title="צמיגות – Hebrew" lang="he" hreflang="he" data-title="צמיגות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Viskositas" title="Viskositas – Javanese" lang="jv" hreflang="jv" data-title="Viskositas" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B8%E0%B3%8D%E0%B2%A8%E0%B2%BF%E0%B2%97%E0%B3%8D%E0%B2%A7%E0%B2%A4%E0%B3%86" title="ಸ್ನಿಗ್ಧತೆ – Kannada" lang="kn" hreflang="kn" data-title="ಸ್ನಿಗ್ಧತೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%91%E1%83%9A%E1%83%90%E1%83%9C%E1%83%A2%E1%83%94" title="სიბლანტე – Georgian" lang="ka" hreflang="ka" data-title="სიბლანტე" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D2%B1%D1%82%D2%9B%D1%8B%D1%80%D0%BB%D1%8B%D2%9B" title="Тұтқырлық – Kazakh" lang="kk" hreflang="kk" data-title="Тұтқырлық" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Viskozite" title="Viskozite – Haitian Creole" lang="ht" hreflang="ht" data-title="Viskozite" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Viskozit%C4%81te" title="Viskozitāte – Latvian" lang="lv" hreflang="lv" data-title="Viskozitāte" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Viskosit%C3%A9it" title="Viskositéit – Luxembourgish" lang="lb" hreflang="lb" data-title="Viskositéit" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Klampumas" title="Klampumas – Lithuanian" lang="lt" hreflang="lt" data-title="Klampumas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Viszkozit%C3%A1s" title="Viszkozitás – Hungarian" lang="hu" hreflang="hu" data-title="Viszkozitás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82" title="Вискозност – Macedonian" lang="mk" hreflang="mk" data-title="Вискозност" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Fitsiloly" title="Fitsiloly – Malagasy" lang="mg" hreflang="mg" data-title="Fitsiloly" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B6%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%A8%E0%B4%A4" title="ശ്യാനത – Malayalam" lang="ml" hreflang="ml" data-title="ശ്യാനത" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kelikatan" title="Kelikatan – Malay" lang="ms" hreflang="ms" data-title="Kelikatan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%97%D1%83%D0%BD%D0%B3%D0%B0%D0%B0%D1%80%D0%B0%D0%BB%D1%82" title="Зунгааралт – Mongolian" lang="mn" hreflang="mn" data-title="Зунгааралт" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Viscositeit" title="Viscositeit – Dutch" lang="nl" hreflang="nl" data-title="Viscositeit" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AD%E0%A4%BF%E0%A4%B8%E0%A5%8D%E0%A4%95%E0%A5%8B%E0%A4%B8%E0%A4%BF%E0%A4%9F%E0%A5%80" title="भिस्कोसिटी – Nepali" lang="ne" hreflang="ne" data-title="भिस्कोसिटी" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B2%98%E5%BA%A6" title="粘度 – Japanese" lang="ja" hreflang="ja" data-title="粘度" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Viskositet" title="Viskositet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Viskositet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Viskositet" title="Viskositet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Viskositet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Yopishqoqlik" title="Yopishqoqlik – Uzbek" lang="uz" hreflang="uz" data-title="Yopishqoqlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A9%87%E0%A8%B8" title="ਲੇਸ – Punjabi" lang="pa" hreflang="pa" data-title="ਲੇਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AE%D9%BC%D9%86%D9%87" title="خټنه – Pashto" lang="ps" hreflang="ps" data-title="خټنه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Lepko%C5%9B%C4%87" title="Lepkość – Polish" lang="pl" hreflang="pl" data-title="Lepkość" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Viscosidade" title="Viscosidade – Portuguese" lang="pt" hreflang="pt" data-title="Viscosidade" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Viscozitate" title="Viscozitate – Romanian" lang="ro" hreflang="ro" data-title="Viscozitate" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D1%8F%D0%B7%D0%BA%D0%BE%D1%81%D1%82%D1%8C" title="Вязкость – Russian" lang="ru" hreflang="ru" data-title="Вязкость" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Viskoziteti" title="Viskoziteti – Albanian" lang="sq" hreflang="sq" data-title="Viskoziteti" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Viscosity" title="Viscosity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Viscosity" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Viskozita" title="Viskozita – Slovak" lang="sk" hreflang="sk" data-title="Viskozita" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Viskoznost" title="Viskoznost – Slovenian" lang="sl" hreflang="sl" data-title="Viskoznost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82" title="Вискозност – Serbian" lang="sr" hreflang="sr" data-title="Вискозност" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Viskoznost" title="Viskoznost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Viskoznost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Viskositeetti" title="Viskositeetti – Finnish" lang="fi" hreflang="fi" data-title="Viskositeetti" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Viskositet" title="Viskositet – Swedish" lang="sv" hreflang="sv" data-title="Viskositet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%BF%E0%AE%9A%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%81%E0%AE%AE%E0%AF%88" title="பிசுக்குமை – Tamil" lang="ta" hreflang="ta" data-title="பிசுக்குமை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B1%8D%E0%B0%A8%E0%B0%BF%E0%B0%97%E0%B1%8D%E0%B0%A5%E0%B0%A4" title="స్నిగ్థత – Telugu" lang="te" hreflang="te" data-title="స్నిగ్థత" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%AB%E0%B8%99%E0%B8%B7%E0%B8%94" title="ความหนืด – Thai" lang="th" hreflang="th" data-title="ความหนืด" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Viskozite" title="Viskozite – Turkish" lang="tr" hreflang="tr" data-title="Viskozite" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%27%D1%8F%D0%B7%D0%BA%D1%96%D1%81%D1%82%D1%8C" title="В'язкість – Ukrainian" lang="uk" hreflang="uk" data-title="В'язкість" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%84%D8%B2%D9%88%D8%AC%D8%AA" title="لزوجت – Urdu" lang="ur" hreflang="ur" data-title="لزوجت" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%99_nh%E1%BB%9Bt" title="Độ nhớt – Vietnamese" lang="vi" hreflang="vi" data-title="Độ nhớt" data-language-autonym="Tiếng Việt" 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.mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Viscosity</th></tr><tr><td colspan="2" class="infobox-image"><a href="/wiki/File:Viscosities.gif" class="mw-file-description" title="Viscosities"><img alt="Viscosities" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Viscosities.gif/330px-Viscosities.gif" decoding="async" width="300" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Viscosities.gif/500px-Viscosities.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/4/44/Viscosities.gif 2x" data-file-width="600" data-file-height="463" /></a><div class="infobox-caption">A simulation of liquids with different viscosities. The liquid on the left has lower viscosity than the liquid on the right.</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><a href="/wiki/Eta_(letter)" class="mw-redirect" title="Eta (letter)">η</a>, <a href="/wiki/Mu_(letter)" title="Mu (letter)">μ</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from other quantities</div></th><td class="infobox-data">μ = <a href="/wiki/Shear_modulus" title="Shear modulus">G</a>·<a href="/wiki/Time" title="Time">t</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {M}}{\mathsf {L}}^{-1}{\mathsf {T}}^{-1}}"> <semantics> <mrow 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href="/wiki/Strain_(mechanics)" title="Strain (mechanics)">Strain</a> <ul><li><a href="/wiki/Finite_strain_theory" title="Finite strain theory">Finite strain</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain</a></li></ul></li> <li><a href="/wiki/Compatibility_(mechanics)" title="Compatibility (mechanics)">Compatibility</a></li> <li><a href="/wiki/Bending" title="Bending">Bending</a></li> <li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Contact mechanics</a> <ul><li><a href="/wiki/Frictional_contact_mechanics" title="Frictional contact mechanics">frictional</a></li></ul></li> <li><a href="/wiki/Material_failure_theory" title="Material failure theory">Material failure theory</a></li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Fluid" title="Fluid">Fluids</a></th></tr><tr><td class="sidebar-content"> <div class="wraplinks"> <ul><li><a href="/wiki/Hydrostatics" title="Hydrostatics">Statics</a> · <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Dynamics</a></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a> · <a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li> <li><a href="/wiki/Hagen%E2%80%93Poiseuille_equation" title="Hagen–Poiseuille equation">Poiseuille equation</a> · <a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a></li> <li><a class="mw-selflink selflink">Viscosity</a> <ul><li>(<a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a> · <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian</a>)</li></ul></li> <li><a href="/wiki/Buoyancy" title="Buoyancy">Buoyancy</a> · <a href="/wiki/Mixing_(process_engineering)" title="Mixing (process engineering)">Mixing</a> · <a href="/wiki/Pressure" title="Pressure">Pressure</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Liquid" title="Liquid">Liquids</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Adhesion" title="Adhesion">Adhesion</a></li> <li><a href="/wiki/Capillary_action" title="Capillary action">Capillary action</a></li> <li><a href="/wiki/Chromatography" title="Chromatography">Chromatography</a></li> <li><a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">Cohesion (chemistry)</a></li> <li><a href="/wiki/Surface_tension" title="Surface tension">Surface tension</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Gas" title="Gas">Gases</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Atmosphere" title="Atmosphere">Atmosphere</a></li> <li><a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's law</a></li> <li><a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a></li> <li><a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">Combined gas law</a></li> <li><a href="/wiki/Fick%27s_law" class="mw-redirect" title="Fick's law">Fick's law</a></li> <li><a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></li> <li><a href="/wiki/Graham%27s_law" title="Graham's law">Graham's law</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></th></tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Rheology" title="Rheology">Rheology</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a></li> <li><a href="/wiki/Rheometry" title="Rheometry">Rheometry</a></li> <li><a href="/wiki/Rheometer" title="Rheometer">Rheometer</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Smart_fluid" title="Smart fluid">Smart fluids</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electrorheological_fluid" title="Electrorheological fluid">Electrorheological</a></li> <li><a href="/wiki/Magnetorheological_fluid" title="Magnetorheological fluid">Magnetorheological</a></li> <li><a href="/wiki/Ferrofluid" title="Ferrofluid">Ferrofluids</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Robert_Boyle" title="Robert Boyle">Boyle</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Jacques_Charles" title="Jacques Charles">Charles</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Adolf_Eugen_Fick" title="Adolf Eugen Fick">Fick</a></li> <li><a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Gay-Lussac</a></li> <li><a href="/wiki/Thomas_Graham_(chemist)" title="Thomas Graham (chemist)">Graham</a></li> <li><a href="/wiki/Robert_Hooke" title="Robert Hooke">Hooke</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Navier</a></li> <li><a href="/wiki/Walter_Noll" title="Walter Noll">Noll</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st Baronet">Stokes</a></li> <li><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Continuum_mechanics" title="Template:Continuum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Continuum_mechanics" title="Template talk:Continuum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Continuum_mechanics" title="Special:EditPage/Template:Continuum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Viscosity is a measure of a <a href="/wiki/Fluid" title="Fluid">fluid</a>'s rate-dependent <a href="/wiki/Drag_(physics)" title="Drag (physics)">resistance</a> to a change in shape or to movement of its neighboring portions relative to one another.<a href="#cite_note-Britanica-1">[1]</a> For liquids, it corresponds to the informal concept of thickness; for example, <a href="/wiki/Syrup" title="Syrup">syrup</a> has a higher viscosity than <a href="/wiki/Water" title="Water">water</a>.<a href="#cite_note-2">[2]</a> Viscosity is defined scientifically as a force multiplied by a time divided by an area. Thus its <a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI units</a> are newton-seconds per square meter, or pascal-seconds.<a href="#cite_note-Britanica-1">[1]</a> Viscosity quantifies the internal <a href="/wiki/Friction" title="Friction">frictional force</a> between adjacent layers of fluid that are in relative motion.<a href="#cite_note-Britanica-1">[1]</a> For instance, when a viscous fluid is forced through a tube, it flows more quickly near the tube's center line than near its walls.<a href="#cite_note-3">[3]</a> Experiments show that some <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a> (such as a <a href="/wiki/Pressure" title="Pressure">pressure</a> difference between the two ends of the tube) is needed to sustain the flow. This is because a force is required to overcome the friction between the layers of the fluid which are in <a href="/wiki/Relative_motion" class="mw-redirect" title="Relative motion">relative motion</a>. For a tube with a constant rate of flow, the strength of the compensating force is proportional to the fluid's viscosity. In general, viscosity depends on a fluid's state, such as its temperature, pressure, and rate of deformation. However, the dependence on some of these properties is negligible in certain cases. For example, the viscosity of a <a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a> does not vary significantly with the rate of deformation. Zero viscosity (no resistance to <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a>) is observed only at <a href="/wiki/Cryogenics" title="Cryogenics">very low temperatures</a> in <a href="/wiki/Superfluidity" title="Superfluidity">superfluids</a>; otherwise, the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> requires all fluids to have positive viscosity.<a href="#cite_note-FOOTNOTEBalescu1975428–429-4">[4]</a><a href="#cite_note-FOOTNOTELandauLifshitz1987-5">[5]</a> A fluid that has zero viscosity (non-viscous) is called ideal or inviscid. For <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian fluid</a>'s viscosity, there are <a href="/wiki/Pseudoplastic" class="mw-redirect" title="Pseudoplastic">pseudoplastic</a>, <a href="/wiki/Plastic_flow" class="mw-redirect" title="Plastic flow">plastic</a>, and <a href="/wiki/Dilatant" title="Dilatant">dilatant</a> flows that are time-independent, and there are <a href="/wiki/Thixotropic" class="mw-redirect" title="Thixotropic">thixotropic</a> and <a href="/wiki/Rheopectic" class="mw-redirect" title="Rheopectic">rheopectic</a> flows that are time-dependent. <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=1" title="Edit section: Etymology">edit</a>]</div> The word "viscosity" is derived from the <a href="/wiki/Latin" title="Latin">Latin</a> viscum ("<a href="/wiki/Mistletoe" title="Mistletoe">mistletoe</a>"). Viscum also referred to a viscous <a href="/wiki/Glue" class="mw-redirect" title="Glue">glue</a> derived from mistletoe berries.<a href="#cite_note-etym_visc-6">[6]</a> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=2" title="Edit section: Definitions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Dynamic_viscosity">Dynamic viscosity</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=3" title="Edit section: Dynamic viscosity">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Laminar_shear.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Laminar_shear.svg/220px-Laminar_shear.svg.png" decoding="async" width="220" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Laminar_shear.svg/330px-Laminar_shear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Laminar_shear.svg/440px-Laminar_shear.svg.png 2x" data-file-width="350" data-file-height="250" /></a><figcaption>Illustration of a planar <a href="/wiki/Couette_flow" title="Couette flow">Couette flow</a>. Since the shearing flow is opposed by friction between adjacent layers of fluid (which are in relative motion), a force is required to sustain the motion of the upper plate. The relative strength of this force is a measure of the fluid's viscosity.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Laminar_shear_flow.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Laminar_shear_flow.svg/220px-Laminar_shear_flow.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Laminar_shear_flow.svg/330px-Laminar_shear_flow.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Laminar_shear_flow.svg/440px-Laminar_shear_flow.svg.png 2x" data-file-width="250" data-file-height="250" /></a><figcaption>In a general parallel flow, the shear stress is proportional to the gradient of the velocity.</figcaption></figure> In <a href="/wiki/Materials_science" title="Materials science">materials science</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>, there is often interest in understanding the forces or <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stresses</a> involved in the <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">deformation</a> of a material. For instance, if the material were a simple spring, the answer would be given by <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> stresses. In other materials, stresses are present which can be attributed to the <a href="/wiki/Strain_rate" title="Strain rate">deformation rate over time</a>. These are called viscous stresses. For instance, in a fluid such as water the stresses which arise from shearing the fluid do not depend on the distance the fluid has been sheared; rather, they depend on how quickly the shearing occurs. Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar <a href="/wiki/Couette_flow" title="Couette flow">Couette flow</a>. In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /> (see illustration to the right). If the speed of the top plate is low enough (to avoid turbulence), then in steady state the fluid particles move parallel to it, and their speed varies from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /> at the bottom to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /> at the top.<a href="#cite_note-FOOTNOTEMewisWagner201219-7">[7]</a> Each layer of fluid moves faster than the one just below it, and friction between them gives rise to a <a href="/wiki/Force_(physics)" class="mw-redirect" title="Force (physics)">force</a> resisting their relative motion. In particular, the fluid applies on the top plate a force in the direction opposite to its motion, and an equal but opposite force on the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed. In many fluids, the flow velocity is observed to vary linearly from zero at the bottom to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /> at the top. Moreover, the magnitude of the force, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" />, acting on the top plate is found to be proportional to the speed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /> and the area <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> of each plate, and inversely proportional to their separation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" />: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\mu A{\frac {u}{y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>μ</mi> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mi>y</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\mu A{\frac {u}{y}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/413ef0745d061f367dd3cb768a833a47c5e5bae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.797ex; height:5.176ex;" alt="{\displaystyle F=\mu A{\frac {u}{y}}.}" /></dd></dl> The proportionality factor is the dynamic viscosity of the fluid, often simply referred to as the viscosity. It is denoted by the <a href="/wiki/Mu_(letter)" title="Mu (letter)">Greek letter mu</a> (μ). The dynamic viscosity has the <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {(mass/length)/time} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {(mass/length)/time} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1643916c6922137cf5d421fa9036a112e3752fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.916ex; height:2.843ex;" alt="{\displaystyle \mathrm {(mass/length)/time} }" />, therefore resulting in the <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a> and the <a href="/wiki/SI_derived_unit" title="SI derived unit">derived units</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mu ]={\frac {\rm {kg}}{\rm {m{\cdot }s}}}={\frac {\rm {N}}{\rm {m^{2}}}}{\cdot }{\rm {s}}={\rm {Pa{\cdot }s}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>μ</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi mathvariant="normal">s</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi mathvariant="normal">s</mi> </mrow> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mu ]={\frac {\rm {kg}}{\rm {m{\cdot }s}}}={\frac {\rm {N}}{\rm {m^{2}}}}{\cdot }{\rm {s}}={\rm {Pa{\cdot }s}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0505c4de127e4d762cc7174f9e205606cbef004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.477ex; height:5.509ex;" alt="{\displaystyle [\mu ]={\frac {\rm {kg}}{\rm {m{\cdot }s}}}={\frac {\rm {N}}{\rm {m^{2}}}}{\cdot }{\rm {s}}={\rm {Pa{\cdot }s}}=}" /> <a href="/wiki/Pressure" title="Pressure">pressure</a> multiplied by <a href="/wiki/Time" title="Time">time</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" /> energy per unit volume multiplied by time.</dd></dl> The aforementioned ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u/y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u/y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43abd4cf5e9cc7266564a98c97d9437dd2d6d89e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.648ex; height:2.843ex;" alt="{\displaystyle u/y}" /> is called the rate of shear deformation or <a href="/wiki/Shear_velocity" title="Shear velocity">shear velocity</a>, and is the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the fluid speed in the direction <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> to the normal vector of the plates (see illustrations to the right). If the velocity does not vary linearly with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" />, then the appropriate generalization is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =\mu {\frac {\partial u}{\partial y}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =\mu {\frac {\partial u}{\partial y}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36db791abdbffb5b7822b4d2ecda9706c70f4777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.833ex; height:6.009ex;" alt="{\displaystyle \tau =\mu {\frac {\partial u}{\partial y}},}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =F/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =F/A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f373a47fc8548cf14fe51a206b2056c520c0a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.947ex; height:2.843ex;" alt="{\displaystyle \tau =F/A}" />, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial u/\partial y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial u/\partial y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90ebe5203a48f9eb132e7136331dd1155c1e4d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.284ex; height:2.843ex;" alt="{\displaystyle \partial u/\partial y}" /> is the local shear velocity. This expression is referred to as <a href="/wiki/Newton%27s_law_of_viscosity" class="mw-redirect" title="Newton's law of viscosity">Newton's law of viscosity</a>. In shearing flows with planar symmetry, it is what defines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" />. It is a special case of the general definition of viscosity (see below), which can be expressed in coordinate-free form. Use of the <a href="/wiki/Mu_(letter)" title="Mu (letter)">Greek letter mu</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" />) for the dynamic viscosity (sometimes also called the absolute viscosity) is common among <a href="/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical</a> and <a href="/wiki/Chemical_engineer" title="Chemical engineer">chemical engineers</a>, as well as mathematicians and physicists.<a href="#cite_note-FOOTNOTEStreeterWylieBedford1998-8">[8]</a><a href="#cite_note-FOOTNOTEHolman2002-9">[9]</a><a href="#cite_note-FOOTNOTEIncroperaDeWittBergmanLavine2007-10">[10]</a> However, the <a href="/wiki/Eta_(letter)" class="mw-redirect" title="Eta (letter)">Greek letter eta</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" />) is also used by chemists, physicists, and the <a href="/wiki/IUPAC" class="mw-redirect" title="IUPAC">IUPAC</a>.<a href="#cite_note-FOOTNOTENičJirátKošataJenkins1997-11">[11]</a> The viscosity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> is sometimes also called the shear viscosity. However, at least one author discourages the use of this terminology, noting that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> can appear in non-shearing flows in addition to shearing flows.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200719-12">[12]</a> <div class="mw-heading mw-heading3"><h3 id="Kinematic_viscosity">Kinematic viscosity</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=4" title="Edit section: Kinematic viscosity">edit</a>]</div> In fluid dynamics, it is sometimes more appropriate to work in terms of kinematic viscosity (sometimes also called the momentum diffusivity), defined as the ratio of the dynamic viscosity (μ) over the <a href="/wiki/Density" title="Density">density</a> of the fluid (ρ). It is usually denoted by the <a href="/wiki/Nu_(letter)" title="Nu (letter)">Greek letter nu</a> (ν): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu ={\frac {\mu }{\rho }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mi>ρ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ={\frac {\mu }{\rho }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9360cb86665dfec20c9b49252d65e6f2c210a3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.215ex; height:5.343ex;" alt="{\displaystyle \nu ={\frac {\mu }{\rho }},}" /></dd></dl> and has the <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {(length)^{2}/time} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">h</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {(length)^{2}/time} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84464328f5df854f423ac3e64d08472e06160c45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.877ex; height:3.176ex;" alt="{\displaystyle \mathrm {(length)^{2}/time} }" />, therefore resulting in the <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a> and the <a href="/wiki/SI_derived_unit" title="SI derived unit">derived units</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\nu ]=\mathrm {\frac {m^{2}}{s}} =\mathrm {{\frac {N{\cdot }m}{kg}}{\cdot }s} =\mathrm {{\frac {J}{kg}}{\cdot }s} =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>ν</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">s</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi mathvariant="normal">m</mi> </mrow> <mrow> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi mathvariant="normal">s</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">J</mi> <mrow> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">g</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi mathvariant="normal">s</mi> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\nu ]=\mathrm {\frac {m^{2}}{s}} =\mathrm {{\frac {N{\cdot }m}{kg}}{\cdot }s} =\mathrm {{\frac {J}{kg}}{\cdot }s} =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abefedddb99ec0896354cfcfcdffd26d00903265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.615ex; height:6.176ex;" alt="{\displaystyle [\nu ]=\mathrm {\frac {m^{2}}{s}} =\mathrm {{\frac {N{\cdot }m}{kg}}{\cdot }s} =\mathrm {{\frac {J}{kg}}{\cdot }s} =}" /> <a href="/wiki/Specific_energy" title="Specific energy">specific energy</a> multiplied by <a href="/wiki/Time" title="Time">time</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" /> energy per unit mass multiplied by time.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="General_definition">General definition</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=5" title="Edit section: General definition">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Viscous_stress_tensor" title="Viscous stress tensor">Viscous stress tensor</a> and <a href="/wiki/Volume_viscosity" title="Volume viscosity">Volume viscosity</a></div> In very general terms, the viscous stresses in a fluid are defined as those resulting from the relative velocity of different fluid particles. As such, the viscous stresses must depend on spatial gradients of the flow velocity. If the velocity gradients are small, then to a first approximation the viscous stresses depend only on the first derivatives of the velocity.<a href="#cite_note-FOOTNOTELandauLifshitz198744–45-13">[13]</a> (For Newtonian fluids, this is also a linear dependence.) In Cartesian coordinates, the general relationship can then be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{ij}=\sum _{k}\sum _{\ell }\mu _{ijk\ell }{\frac {\partial v_{k}}{\partial r_{\ell }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </munder> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{ij}=\sum _{k}\sum _{\ell }\mu _{ijk\ell }{\frac {\partial v_{k}}{\partial r_{\ell }}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225b6f2a22322d9c692d855546851a6250ff33df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.514ex; height:6.676ex;" alt="{\displaystyle \tau _{ij}=\sum _{k}\sum _{\ell }\mu _{ijk\ell }{\frac {\partial v_{k}}{\partial r_{\ell }}},}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{ijk\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{ijk\ell }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3252a506287883a348729441bcd21cafe3010f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.421ex; height:2.343ex;" alt="{\displaystyle \mu _{ijk\ell }}" /> is a viscosity tensor that maps the <a href="/wiki/Velocity_gradient" class="mw-redirect" title="Velocity gradient">velocity gradient</a> tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial v_{k}/\partial r_{\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial v_{k}/\partial r_{\ell }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ec861059d90d942d8d26b2ebf79f4ef7f0df23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.982ex; height:2.843ex;" alt="{\displaystyle \partial v_{k}/\partial r_{\ell }}" /> onto the viscous stress tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97d0190e7517254a655e8033badbc01c8590381c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.493ex; height:2.343ex;" alt="{\displaystyle \tau _{ij}}" />.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200718-14">[14]</a> Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{ijkl}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{ijkl}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ccc7586bc977f4d32cd07ebe15e44c6bd85a983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.226ex; height:2.343ex;" alt="{\displaystyle \mu _{ijkl}}" /> in total. However, assuming that the viscosity rank-2 tensor is <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> reduces these 81 coefficients to three independent parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" />: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{ijk\ell }=\alpha \delta _{ij}\delta _{k\ell }+\beta \delta _{ik}\delta _{j\ell }+\gamma \delta _{i\ell }\delta _{jk},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> <mo>+</mo> <mi>β</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ℓ</mi> </mrow> </msub> <mo>+</mo> <mi>γ</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ℓ</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{ijk\ell }=\alpha \delta _{ij}\delta _{k\ell }+\beta \delta _{ik}\delta _{j\ell }+\gamma \delta _{i\ell }\delta _{jk},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f572060e79931c80f6ca66f4a2a04d49ef4c784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.878ex; height:3.009ex;" alt="{\displaystyle \mu _{ijk\ell }=\alpha \delta _{ij}\delta _{k\ell }+\beta \delta _{ik}\delta _{j\ell }+\gamma \delta _{i\ell }\delta _{jk},}" /></dd></dl> and furthermore, it is assumed that no viscous forces may arise when the fluid is undergoing simple rigid-body rotation, thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =\gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b82201fad2d4f0a3b374681cf6963f7b85cd3687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.693ex; height:2.676ex;" alt="{\displaystyle \beta =\gamma }" />, leaving only two independent parameters.<a href="#cite_note-FOOTNOTELandauLifshitz198744–45-13">[13]</a> The most usual decomposition is in terms of the standard (scalar) viscosity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> and the <a href="/wiki/Bulk_viscosity" class="mw-redirect" title="Bulk viscosity">bulk viscosity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" /> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\kappa -{\tfrac {2}{3}}\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>κ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\kappa -{\tfrac {2}{3}}\mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2c110353da78f2f1aa9c1bbedcffd9cdf5a6acf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.825ex; height:3.676ex;" alt="{\displaystyle \alpha =\kappa -{\tfrac {2}{3}}\mu }" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =\gamma =\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =\gamma =\mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e8190aa7e1ca1496ba6f5b54f1303e0f536144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.193ex; height:2.676ex;" alt="{\displaystyle \beta =\gamma =\mu }" />. In vector notation this appears as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {v} +(\nabla \mathbf {v} )^{\mathrm {T} }\right]-\left({\frac {2}{3}}\mu -\kappa \right)(\nabla \cdot \mathbf {v} )\mathbf {\delta } ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo>=</mo> <mi>μ</mi> <mrow> <mo>[</mo> <mrow> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>μ</mi> <mo>−</mo> <mi>κ</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {v} +(\nabla \mathbf {v} )^{\mathrm {T} }\right]-\left({\frac {2}{3}}\mu -\kappa \right)(\nabla \cdot \mathbf {v} )\mathbf {\delta } ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c881ede5c0e043dbe36b7b5a30b4c6bf92204e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.766ex; height:6.176ex;" alt="{\displaystyle {\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {v} +(\nabla \mathbf {v} )^{\mathrm {T} }\right]-\left({\frac {2}{3}}\mu -\kappa \right)(\nabla \cdot \mathbf {v} )\mathbf {\delta } ,}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\delta } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\delta } }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32cabc0eaa140779f6e02929b1d7e7e3537e88b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \mathbf {\delta } }" /> is the unit tensor.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200719-12">[12]</a><a href="#cite_note-FOOTNOTELandauLifshitz198745-15">[15]</a> This equation can be thought of as a generalized form of Newton's law of viscosity. The bulk viscosity (also called volume viscosity) expresses a type of internal friction that resists the shearless compression or expansion of a fluid. Knowledge of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" /> is frequently not necessary in fluid dynamics problems. For example, an incompressible fluid satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {v} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {v} =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/760025de0202db29f86a1c2c4bef18d54ccc0f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.287ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {v} =0}" /> and so the term containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" /> drops out. Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" /> is often assumed to be negligible for gases since it is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /> in a <a href="/wiki/Monatomic" class="mw-redirect" title="Monatomic">monatomic</a> <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a>.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200719-12">[12]</a> One situation in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" /> can be important is the calculation of energy loss in <a href="/wiki/Sound" title="Sound">sound</a> and <a href="/wiki/Shock_wave" title="Shock wave">shock waves</a>, described by <a href="/wiki/Stokes%27_law_(sound_attenuation)" class="mw-redirect" title="Stokes' law (sound attenuation)">Stokes' law of sound attenuation</a>, since these phenomena involve rapid expansions and compressions. The defining equations for viscosity are not fundamental laws of nature, so their usefulness, as well as methods for measuring or calculating the viscosity, must be established using separate means. A potential issue is that viscosity depends, in principle, on the full microscopic state of the fluid, which encompasses the positions and momenta of every particle in the system.<a href="#cite_note-FOOTNOTEBalescu1975-16">[16]</a> Such highly detailed information is typically not available in realistic systems. However, under certain conditions most of this information can be shown to be negligible. In particular, for Newtonian fluids near equilibrium and far from boundaries (bulk state), the viscosity depends only space- and time-dependent macroscopic fields (such as temperature and density) defining local equilibrium.<a href="#cite_note-FOOTNOTEBalescu1975-16">[16]</a><a href="#cite_note-FOOTNOTEChapmanCowling1970-17">[17]</a> Nevertheless, viscosity may still carry a non-negligible dependence on several system properties, such as temperature, pressure, and the amplitude and frequency of any external forcing. Therefore, precision measurements of viscosity are only defined with respect to a specific fluid state.<a href="#cite_note-FOOTNOTEMillat1996-18">[18]</a> To standardize comparisons among experiments and theoretical models, viscosity data is sometimes extrapolated to ideal limiting cases, such as the zero shear limit, or (for gases) the zero density limit. <div class="mw-heading mw-heading2"><h2 id="Momentum_transport">Momentum transport</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=6" title="Edit section: Momentum transport">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Transport_phenomena" title="Transport phenomena">Transport phenomena</a></div> Transport theory provides an alternative interpretation of viscosity in terms of momentum transport: viscosity is the material property which characterizes momentum transport within a fluid, just as <a href="/wiki/Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a> characterizes <a href="/wiki/Heat" title="Heat">heat</a> transport, and (mass) <a href="/wiki/Mass_diffusivity" title="Mass diffusivity">diffusivity</a> characterizes mass transport.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot2007-19">[19]</a> This perspective is implicit in Newton's law of viscosity, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =\mu (\partial u/\partial y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =\mu (\partial u/\partial y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477b9bd2cefa37576eb367b39b8cab4365f4f525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.795ex; height:2.843ex;" alt="{\displaystyle \tau =\mu (\partial u/\partial y)}" />, because the shear stress <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /> has units equivalent to a momentum <a href="/wiki/Flux" title="Flux">flux</a>, i.e., momentum per unit time per unit area. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /> can be interpreted as specifying the flow of momentum in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /> direction from one fluid layer to the next. Per Newton's law of viscosity, this momentum flow occurs across a velocity gradient, and the magnitude of the corresponding momentum flux is determined by the viscosity. The analogy with heat and mass transfer can be made explicit. Just as heat flows from high temperature to low temperature and mass flows from high density to low density, momentum flows from high velocity to low velocity. These behaviors are all described by compact expressions, called <a href="/wiki/Constitutive_relations" class="mw-redirect" title="Constitutive relations">constitutive relations</a>, whose one-dimensional forms are given here: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {J} &=-D{\frac {\partial \rho }{\partial x}}&&{\text{(Fick's law of diffusion)}}\\[5pt]\mathbf {q} &=-k_{t}{\frac {\partial T}{\partial x}}&&{\text{(Fourier's law of heat conduction)}}\\[5pt]\tau &=\mu {\frac {\partial u}{\partial y}}&&{\text{(Newton's law of viscosity)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Fick's law of diffusion)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Fourier's law of heat conduction)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>τ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Newton's law of viscosity)</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {J} &=-D{\frac {\partial \rho }{\partial x}}&&{\text{(Fick's law of diffusion)}}\\[5pt]\mathbf {q} &=-k_{t}{\frac {\partial T}{\partial x}}&&{\text{(Fourier's law of heat conduction)}}\\[5pt]\tau &=\mu {\frac {\partial u}{\partial y}}&&{\text{(Newton's law of viscosity)}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6380b89b0d24d9c9deb9ef04f333430b073c45cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:51.582ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {J} &=-D{\frac {\partial \rho }{\partial x}}&&{\text{(Fick's law of diffusion)}}\\[5pt]\mathbf {q} &=-k_{t}{\frac {\partial T}{\partial x}}&&{\text{(Fourier's law of heat conduction)}}\\[5pt]\tau &=\mu {\frac {\partial u}{\partial y}}&&{\text{(Newton's law of viscosity)}}\end{aligned}}}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" /> is the density, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7686846b1a6b756cb514954000004ab5e7b2a5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.381ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} }" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7be005a326b7ac3fe4c24bca391369f44c4c2876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }" /> are the mass and heat fluxes, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352386366dcb1dcc184d82669513975561f2d2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.037ex; height:2.509ex;" alt="{\displaystyle k_{t}}" /> are the mass diffusivity and thermal conductivity.<a href="#cite_note-FOOTNOTESchroeder1999-20">[20]</a> The fact that mass, momentum, and energy (heat) transport are among the most relevant processes in continuum mechanics is not a coincidence: these are among the few physical quantities that are conserved at the microscopic level in interparticle collisions. Thus, rather than being dictated by the fast and complex microscopic interaction timescale, their dynamics occurs on macroscopic timescales, as described by the various equations of transport theory and hydrodynamics. <div class="mw-heading mw-heading2"><h2 id="Newtonian_and_non-Newtonian_fluids">Newtonian and non-Newtonian fluids</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=7" title="Edit section: Newtonian and non-Newtonian fluids">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Viscous_regimes_chart.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Viscous_regimes_chart.png/250px-Viscous_regimes_chart.png" decoding="async" width="250" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Viscous_regimes_chart.png/500px-Viscous_regimes_chart.png 1.5x" data-file-width="821" data-file-height="835" /></a><figcaption>Viscosity, the slope of each line, varies among materials.</figcaption></figure> Newton's law of viscosity is not a fundamental law of nature, but rather a <a href="/wiki/Constitutive_equation" title="Constitutive equation">constitutive equation</a> (like <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>, <a href="/wiki/Fick%27s_law" class="mw-redirect" title="Fick's law">Fick's law</a>, and <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a>) which serves to define the viscosity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" />. Its form is motivated by experiments which show that for a wide range of fluids, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> is independent of strain rate. Such fluids are called <a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a>. <a href="/wiki/Gas" title="Gas">Gases</a>, <a href="/wiki/Water" title="Water">water</a>, and many common liquids can be considered Newtonian in ordinary conditions and contexts. However, there are many <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian fluids</a> that significantly deviate from this behavior. For example: <ul><li><a href="/wiki/Shear_thickening" class="mw-redirect" title="Shear thickening">Shear-thickening</a> (dilatant) liquids, whose viscosity increases with the rate of shear strain.</li> <li><a href="/wiki/Shear_thinning" title="Shear thinning">Shear-thinning</a> liquids, whose viscosity decreases with the rate of shear strain.</li> <li><a href="/wiki/Thixotropic" class="mw-redirect" title="Thixotropic">Thixotropic</a> liquids, that become less viscous over time when shaken, agitated, or otherwise stressed.</li> <li><a href="/wiki/Rheopectic" class="mw-redirect" title="Rheopectic">Rheopectic</a> liquids, that become more viscous over time when shaken, agitated, or otherwise stressed.</li> <li><a href="/wiki/Bingham_plastic" title="Bingham plastic">Bingham plastics</a> that behave as a solid at low stresses but flow as a viscous fluid at high stresses.</li></ul> <a href="/wiki/Frederick_Thomas_Trouton" title="Frederick Thomas Trouton">Trouton</a>'s ratio is the ratio of <a href="/wiki/Extensional_viscosity" title="Extensional viscosity">extensional viscosity</a> to <a href="/wiki/Shear_viscosity" class="mw-redirect" title="Shear viscosity">shear viscosity</a>. For a Newtonian fluid, the Trouton ratio is 3.<a href="#cite_note-FOOTNOTERóżańskaRóżańskiOchowiakMitkowski201447–55-21">[21]</a><a href="#cite_note-FOOTNOTETrouton1906426–440-22">[22]</a> Shear-thinning liquids are very commonly, but misleadingly, described as thixotropic.<a href="#cite_note-FOOTNOTEMewisWagner2012228–230-23">[23]</a> Viscosity may also depend on the fluid's physical state (temperature and pressure) and other, external, factors. For gases and other <a href="/wiki/Compressible_fluid" class="mw-redirect" title="Compressible fluid">compressible fluids</a>, it depends on temperature and varies very slowly with pressure. The viscosity of some fluids may depend on other factors. A <a href="/wiki/Magnetorheological_fluid" title="Magnetorheological fluid">magnetorheological fluid</a>, for example, becomes thicker when subjected to a <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>, possibly to the point of behaving like a solid. <div class="mw-heading mw-heading2"><h2 id="In_solids">In solids</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=8" title="Edit section: In solids">edit</a>]</div> The viscous forces that arise during fluid flow are distinct from the <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> forces that occur in a solid in response to shear, compression, or extension stresses. While in the latter the stress is proportional to the amount of shear deformation, in a fluid it is proportional to the rate of deformation over time. For this reason, <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> used the term fugitive elasticity for fluid viscosity. However, many liquids (including water) will briefly react like elastic solids when subjected to sudden stress. Conversely, many "solids" (even <a href="/wiki/Granite" title="Granite">granite</a>) will flow like liquids, albeit very slowly, even under arbitrarily small stress.<a href="#cite_note-FOOTNOTEKumagaiSasajimaIto1978157–161-24">[24]</a> Such materials are best described as <a href="/wiki/Viscoelasticity" title="Viscoelasticity">viscoelastic</a>—that is, possessing both elasticity (reaction to deformation) and viscosity (reaction to rate of deformation). Viscoelastic solids may exhibit both shear viscosity and bulk viscosity. The <a href="/wiki/Extensional_viscosity" title="Extensional viscosity">extensional viscosity</a> is a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of the shear and bulk viscosities that describes the reaction of a solid elastic material to elongation. It is widely used for characterizing polymers. In <a href="/wiki/Geology" title="Geology">geology</a>, earth materials that exhibit viscous deformation at least three <a href="/wiki/Order_of_magnitude" title="Order of magnitude">orders of magnitude</a> greater than their elastic deformation are sometimes called <a href="/wiki/Rheid" title="Rheid">rheids</a>.<a href="#cite_note-FOOTNOTESchererPardenekSwiatek198814-25">[25]</a> <div class="mw-heading mw-heading2"><h2 id="Measurement">Measurement</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=9" title="Edit section: Measurement">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Viscometer" title="Viscometer">Viscometer</a></div> Viscosity is measured with various types of <a href="/wiki/Viscometer" title="Viscometer">viscometers</a> and <a href="/wiki/Rheometer" title="Rheometer">rheometers</a>. Close temperature control of the fluid is essential to obtain accurate measurements, particularly in materials like lubricants, whose viscosity can double with a change of only 5 °C. A rheometer is used for fluids that cannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than is the case for a viscometer.<a href="#cite_note-FOOTNOTEHannan2007-26">[26]</a> For some fluids, the viscosity is constant over a wide range of shear rates (<a href="/wiki/Newtonian_fluids" class="mw-redirect" title="Newtonian fluids">Newtonian fluids</a>). The fluids without a constant viscosity (<a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian fluids</a>) cannot be described by a single number. Non-Newtonian fluids exhibit a variety of different correlations between shear stress and shear rate. One of the most common instruments for measuring kinematic viscosity is the glass capillary viscometer. In <a href="/wiki/Coating" title="Coating">coating</a> industries, viscosity may be measured with a cup in which the <a href="/wiki/Efflux_time" class="mw-redirect" title="Efflux time">efflux time</a> is measured. There are several sorts of cup—such as the <a href="/wiki/Zahn_cup" title="Zahn cup">Zahn cup</a> and the <a href="/wiki/Ford_viscosity_cup" title="Ford viscosity cup">Ford viscosity cup</a>—with the usage of each type varying mainly according to the industry. Also used in coatings, a Stormer viscometer employs load-based rotation to determine viscosity. The viscosity is reported in Krebs units (KU), which are unique to Stormer viscometers. Vibrating viscometers can also be used to measure viscosity. Resonant, or vibrational viscometers work by creating shear waves within the liquid. In this method, the sensor is submerged in the fluid and is made to resonate at a specific frequency. As the surface of the sensor shears through the liquid, energy is lost due to its viscosity. This dissipated energy is then measured and converted into a viscosity reading. A higher viscosity causes a greater loss of energy.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Extensional viscosity can be measured with various <a href="/wiki/Rheometer" title="Rheometer">rheometers</a> that apply <a href="/wiki/Extensional_stress" class="mw-redirect" title="Extensional stress">extensional stress</a>. <a href="/wiki/Volume_viscosity" title="Volume viscosity">Volume viscosity</a> can be measured with an <a href="/wiki/Acoustic_rheometer" title="Acoustic rheometer">acoustic rheometer</a>. <a href="/wiki/Apparent_viscosity" title="Apparent viscosity">Apparent viscosity</a> is a calculation derived from tests performed on <a href="/wiki/Drilling_fluid" title="Drilling fluid">drilling fluid</a> used in oil or gas well development. These calculations and tests help engineers develop and maintain the properties of the drilling fluid to the specifications required. Nanoviscosity (viscosity sensed by nanoprobes) can be measured by <a href="/wiki/Fluorescence_correlation_spectroscopy" title="Fluorescence correlation spectroscopy">fluorescence correlation spectroscopy</a>.<a href="#cite_note-FOOTNOTEKwapiszewskaSzczepańskiKalwarczykMichalska2020-27">[27]</a> <div class="mw-heading mw-heading2"><h2 id="Units">Units</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=10" title="Edit section: Units">edit</a>]</div> The <a href="/wiki/International_System_of_Units" title="International System of Units">SI</a> unit of dynamic viscosity is the <a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a>-second per square meter (N·s/m2), also frequently expressed in the equivalent forms <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascal</a>-<a href="/wiki/Second" title="Second">second</a> (Pa·s), <a href="/wiki/Kilogram" title="Kilogram">kilogram</a> per meter per second (kg·m−1·s−1) and <a href="/wiki/Poiseuille" title="Poiseuille">poiseuille</a> (Pl). The <a href="/wiki/CGS_system" class="mw-redirect" title="CGS system">CGS</a> unit is the <a href="/wiki/Poise_(unit)" title="Poise (unit)">poise</a> (P, or g·cm−1·s−1 = 0.1 Pa·s),<a href="#cite_note-FOOTNOTEMcNaughtWilkinson1997poise-28">[28]</a> named after <a href="/wiki/Jean_L%C3%A9onard_Marie_Poiseuille" title="Jean Léonard Marie Poiseuille">Jean Léonard Marie Poiseuille</a>. It is commonly expressed, particularly in <a href="/wiki/ASTM" class="mw-redirect" title="ASTM">ASTM</a> standards, as centipoise (cP). The centipoise is convenient because the viscosity of water at 20 °C is about 1 cP, and one centipoise is equal to the SI millipascal second (mPa·s). The SI unit of kinematic viscosity is square meter per second (m2/s), whereas the CGS unit for kinematic viscosity is the stokes (St, or cm2·s−1 = 0.0001 m2·s−1), named after Sir <a href="/wiki/George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">George Gabriel Stokes</a>.<a href="#cite_note-FOOTNOTEGyllenbok2018213-29">[29]</a> In U.S. usage, stoke is sometimes used as the singular form. The <a href="/wiki/Submultiple" class="mw-redirect" title="Submultiple">submultiple</a> centistokes (cSt) is often used instead, 1 cSt = 1 mm2·s−1 = 10−6 m2·s−1. 1 cSt is 1 cP divided by 1000 kg/m^3, close to the density of water. The kinematic viscosity of water at 20 °C is about 1 cSt. The most frequently used systems of <a href="/wiki/Imperial_and_US_customary_measurement_systems" title="Imperial and US customary measurement systems">US customary, or Imperial</a>, units are the <a href="/wiki/Foot-pound-second_system" class="mw-redirect" title="Foot-pound-second system">British Gravitational</a> (BG) and <a href="/wiki/English_Engineering_units" class="mw-redirect" title="English Engineering units">English Engineering</a> (EE). In the BG system, dynamic viscosity has units of <a href="/wiki/Pound_(force)" title="Pound (force)">pound</a>-seconds per square <a href="/wiki/Foot" title="Foot">foot</a> (lb·s/ft2), and in the EE system it has units of <a href="/wiki/Pound_(force)" title="Pound (force)">pound-force</a>-seconds per square foot (lbf·s/ft2). The pound and pound-force are equivalent; the two systems differ only in how force and mass are defined. In the BG system the pound is a basic unit from which the unit of mass (the <a href="/wiki/Slug_(unit)" title="Slug (unit)">slug</a>) is defined by <a href="/wiki/Newton%27s_Second_Law" class="mw-redirect" title="Newton's Second Law">Newton's Second Law</a>, whereas in the EE system the units of force and mass (the pound-force and <a href="/wiki/Pound_(mass)" title="Pound (mass)">pound-mass</a> respectively) are defined independently through the Second Law using the <a href="/wiki/Gc_(engineering)" title="Gc (engineering)">proportionality constant gc</a>. Kinematic viscosity has units of square feet per second (ft2/s) in both the BG and EE systems. Nonstandard units include the <a href="/wiki/Reyn" title="Reyn">reyn</a> (lbf·s/in2), a British unit of dynamic viscosity.<a href="#cite_note-30">[30]</a> In the automotive industry the <a href="/wiki/Viscosity_index" title="Viscosity index">viscosity index</a> is used to describe the change of viscosity with temperature. The <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of viscosity is fluidity, usually symbolized by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =1/\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =1/\mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c07421ec9c5c44bc833e20f6dd55b54f18a8d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.21ex; height:2.843ex;" alt="{\displaystyle \phi =1/\mu }" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=1/\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=1/\mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca1bd8b8dfd6d80de780c2d3b005aa70a507128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.566ex; height:2.843ex;" alt="{\displaystyle F=1/\mu }" />, depending on the convention used, measured in reciprocal poise (P−1, or <a href="/wiki/Centimetre" title="Centimetre">cm</a>·<a href="/wiki/Second" title="Second">s</a>·<a href="/wiki/Gram" title="Gram">g</a>−1), sometimes called the rhe. Fluidity is seldom used in <a href="/wiki/Engineering" title="Engineering">engineering</a> practice.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] At one time the petroleum industry relied on measuring kinematic viscosity by means of the <a href="/wiki/Saybolt_viscometer" class="mw-redirect" title="Saybolt viscometer">Saybolt viscometer</a>, and expressing kinematic viscosity in units of <a href="/wiki/Saybolt_universal_second" class="mw-redirect" title="Saybolt universal second">Saybolt universal seconds</a> (SUS).<a href="#cite_note-ASTM_D2161-31">[31]</a> Other abbreviations such as SSU (Saybolt seconds universal) or SUV (Saybolt universal viscosity) are sometimes used. Kinematic viscosity in centistokes can be converted from SUS according to the arithmetic and the reference table provided in <a href="/wiki/ASTM" class="mw-redirect" title="ASTM">ASTM</a> D 2161. <div class="mw-heading mw-heading2"><h2 id="Molecular_origins">Molecular origins</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=11" title="Edit section: Molecular origins">edit</a>]</div> Momentum transport in gases is mediated by discrete molecular collisions, and in liquids by attractive forces that bind molecules close together.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot2007-19">[19]</a> Because of this, the dynamic viscosities of liquids are typically much larger than those of gases. In addition, viscosity tends to increase with temperature in gases and decrease with temperature in liquids. Above the liquid-gas <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical point</a>, the liquid and gas phases are replaced by a single <a href="/wiki/Supercritical_fluid" title="Supercritical fluid">supercritical phase</a>. In this regime, the mechanisms of momentum transport interpolate between liquid-like and gas-like behavior. For example, along a supercritical <a href="/wiki/Isobaric_process" title="Isobaric process">isobar</a> (constant-pressure surface), the kinematic viscosity decreases at low temperature and increases at high temperature, with a minimum in between.<a href="#cite_note-FOOTNOTETrachenkoBrazhkin2020-32">[32]</a><a href="#cite_note-FOOTNOTETrachenkoBrazhkin2021-33">[33]</a> A rough estimate for the value at the minimum is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{\text{min}}={\frac {1}{4\pi }}{\frac {\hbar }{\sqrt {m_{\text{e}}m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <msqrt> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mi>m</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{\text{min}}={\frac {1}{4\pi }}{\frac {\hbar }{\sqrt {m_{\text{e}}m}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d401b274dd0eb77450983de8424eb8c1ee6e367c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:18.365ex; height:6.343ex;" alt="{\displaystyle \nu _{\text{min}}={\frac {1}{4\pi }}{\frac {\hbar }{\sqrt {m_{\text{e}}m}}}}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" /> is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\text{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\text{e}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c45a5eb082ea4dafd3cb43ca39e033989e4a52eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.003ex; height:2.009ex;" alt="{\displaystyle m_{\text{e}}}" /> is the <a href="/wiki/Electron_mass" title="Electron mass">electron mass</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /> is the molecular mass.<a href="#cite_note-FOOTNOTETrachenkoBrazhkin2021-33">[33]</a> In general, however, the viscosity of a system depends in detail on how the molecules constituting the system interact, and there are no simple but correct formulas for it. The simplest exact expressions are the <a href="/wiki/Green%E2%80%93Kubo_relations" title="Green–Kubo relations">Green–Kubo relations</a> for the linear shear viscosity or the transient time correlation function expressions derived by Evans and Morriss in 1988.<a href="#cite_note-FOOTNOTEEvansMorriss1988-34">[34]</a> Although these expressions are each exact, calculating the viscosity of a dense fluid using these relations currently requires the use of <a href="/wiki/Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a> computer simulations. Somewhat more progress can be made for a dilute gas, as elementary assumptions about how gas molecules move and interact lead to a basic understanding of the molecular origins of viscosity. More sophisticated treatments can be constructed by systematically coarse-graining the <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> of the gas molecules. An example of such a treatment is <a href="/wiki/Chapman%E2%80%93Enskog_theory" title="Chapman–Enskog theory">Chapman–Enskog theory</a>, which derives expressions for the viscosity of a dilute gas from the <a href="/wiki/Boltzmann_equation" title="Boltzmann equation">Boltzmann equation</a>.<a href="#cite_note-FOOTNOTEChapmanCowling1970-17">[17]</a> <div class="mw-heading mw-heading3"><h3 id="Pure_gases">Pure gases</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=12" title="Edit section: Pure gases">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">Kinetic theory of gases</a></div> <dl><dd><table class="toccolours collapsible collapsed" width="60%" style="text-align:left"> <tbody><tr> <th>Elementary calculation of viscosity for a dilute gas </th></tr> <tr> <td> Consider a dilute gas moving parallel to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" />-axis with velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dff407583551a7838f4c98386a715c505eda3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.294ex; height:2.843ex;" alt="{\displaystyle u(y)}" /> that depends only on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /> coordinate. To simplify the discussion, the gas is assumed to have uniform temperature and density. Under these assumptions, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> velocity of a molecule passing through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=0}" /> is equal to whatever velocity that molecule had when its mean free path <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /> began. Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /> is typically small compared with macroscopic scales, the average <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> velocity of such a molecule has the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(0)\pm \alpha \lambda {\frac {du}{dy}}(0),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>±</mo> <mi>α</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(0)\pm \alpha \lambda {\frac {du}{dy}}(0),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10a120df59636832be51a18e5ba311222c13a6eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.985ex; height:5.843ex;" alt="{\displaystyle u(0)\pm \alpha \lambda {\frac {du}{dy}}(0),}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> is a numerical constant on the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />. (Some authors estimate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =2/3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =2/3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ab1d44e6ccc10f965182d28785519c14977438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.073ex; height:2.843ex;" alt="{\displaystyle \alpha =2/3}" />;<a href="#cite_note-FOOTNOTEBirdStewartLightfoot2007-19">[19]</a><a href="#cite_note-FOOTNOTEBellacMortessagneBatrouni2004-35">[35]</a> on the other hand, a more careful calculation for rigid elastic spheres gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \simeq 0.998}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≃</mo> <mn>0.998</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \simeq 0.998}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c134c1b18f32d353735634edc8e95835abc042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.883ex; height:2.176ex;" alt="{\displaystyle \alpha \simeq 0.998}" />.) Next, because half the molecules on either side are moving towards <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=0}" />, and doing so on average with half the <a href="/wiki/Kinetic_theory_of_gases#Speed_of_molecules" title="Kinetic theory of gases">average molecular speed</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (8k_{\text{B}}T/\pi m)^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>8</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>π</mi> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (8k_{\text{B}}T/\pi m)^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47955fa8d841a314696f0e7867c3e3f489435e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.448ex; height:3.343ex;" alt="{\displaystyle (8k_{\text{B}}T/\pi m)^{1/2}}" />, the momentum flux from either side is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \left(u(0)\pm \alpha \lambda {\frac {du}{dy}}(0)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>ρ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>8</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> <mrow> <mi>π</mi> <mi>m</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>±</mo> <mi>α</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \left(u(0)\pm \alpha \lambda {\frac {du}{dy}}(0)\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64afb56fa717a5bd6d9d67e8ba914c262ecb8990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.918ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{4}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \left(u(0)\pm \alpha \lambda {\frac {du}{dy}}(0)\right).}" /></dd></dl> The net momentum flux at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=0}" /> is the difference of the two: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1}{2}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \alpha \lambda {\frac {du}{dy}}(0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>8</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> <mrow> <mi>π</mi> <mi>m</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <mi>α</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \alpha \lambda {\frac {du}{dy}}(0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8716c1706577c471eeb0242ce09bd4e9b139fa51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.776ex; height:6.176ex;" alt="{\displaystyle -{\frac {1}{2}}\rho \cdot {\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}\cdot \alpha \lambda {\frac {du}{dy}}(0).}" /></dd></dl> According to the definition of viscosity, this momentum flux should be equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mu {\frac {du}{dy}}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mu {\frac {du}{dy}}(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abdd799ba684f16a40a6db9dd3066456517b47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.563ex; height:5.843ex;" alt="{\displaystyle -\mu {\frac {du}{dy}}(0)}" />, which leads to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>α</mi> <mi>ρ</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> <mrow> <mi>π</mi> <mi>m</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/408f0ca64eb3aac3e5a585dec09e2ece507a4269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.757ex; height:6.176ex;" alt="{\displaystyle \mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}}.}" /></dd></dl> </td></tr></tbody></table></dd></dl> Viscosity in gases arises principally from the <a href="/wiki/Molecular_diffusion" title="Molecular diffusion">molecular diffusion</a> that transports momentum between layers of flow. An elementary calculation for a dilute gas at temperature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /> and density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" /> gives <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>α</mi> <mi>ρ</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> <mrow> <mi>π</mi> <mi>m</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0129230e0b100a5db640fd85a2239afd9ee41fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.757ex; height:6.176ex;" alt="{\displaystyle \mu =\alpha \rho \lambda {\sqrt {\frac {2k_{\text{B}}T}{\pi m}}},}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\text{B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9582c7c795d2def2c061f0dfa3a6f0fb3dd2de44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\text{B}}}" /> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /> the molecular mass, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> a numerical constant on the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />. The quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" />, the <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a>, measures the average distance a molecule travels between collisions. Even without a priori knowledge of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" />, this expression has nontrivial implications. In particular, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /> is typically inversely proportional to density and increases with temperature, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> itself should increase with temperature and be independent of density at fixed temperature. In fact, both of these predictions persist in more sophisticated treatments, and accurately describe experimental observations. By contrast, liquid viscosity typically decreases with temperature.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot2007-19">[19]</a><a href="#cite_note-FOOTNOTEBellacMortessagneBatrouni2004-35">[35]</a> For rigid elastic spheres of diameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /> can be computed, giving <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ={\frac {\alpha }{\pi ^{3/2}}}{\frac {\sqrt {k_{\text{B}}mT}}{\sigma ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>m</mi> <mi>T</mi> </msqrt> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ={\frac {\alpha }{\pi ^{3/2}}}{\frac {\sqrt {k_{\text{B}}mT}}{\sigma ^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89cf0b4edbdfd2a3c37c782ae88ac2d0c0385d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.459ex; height:6.509ex;" alt="{\displaystyle \mu ={\frac {\alpha }{\pi ^{3/2}}}{\frac {\sqrt {k_{\text{B}}mT}}{\sigma ^{2}}}.}" /></dd></dl> In this case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /> is independent of temperature, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \propto T^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∝</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \propto T^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf4eec5fd44de3461bf9ab6c72a545de66dcd351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.918ex; height:3.343ex;" alt="{\displaystyle \mu \propto T^{1/2}}" />. For more complicated molecular models, however, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /> depends on temperature in a non-trivial way, and simple kinetic arguments as used here are inadequate. More fundamentally, the notion of a mean free path becomes imprecise for particles that interact over a finite range, which limits the usefulness of the concept for describing real-world gases.<a href="#cite_note-FOOTNOTEChapmanCowling1970103-36">[36]</a> <div class="mw-heading mw-heading4"><h4 id="Chapman–Enskog_theory">Chapman–Enskog theory</h4>[<a href="/w/index.php?title=Viscosity&action=edit&section=13" title="Edit section: Chapman–Enskog theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Chapman%E2%80%93Enskog_theory" title="Chapman–Enskog theory">Chapman–Enskog theory</a></div> A technique developed by <a href="/wiki/Sydney_Chapman_(mathematician)" title="Sydney Chapman (mathematician)">Sydney Chapman</a> and <a href="/wiki/David_Enskog" title="David Enskog">David Enskog</a> in the early 1900s allows a more refined calculation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" />.<a href="#cite_note-FOOTNOTEChapmanCowling1970-17">[17]</a> It is based on the <a href="/wiki/Boltzmann_equation" title="Boltzmann equation">Boltzmann equation</a>, which provides a statistical description of a dilute gas in terms of intermolecular interactions.<a href="#cite_note-FOOTNOTECercignani1975-37">[37]</a> The technique allows accurate calculation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> for molecular models that are more realistic than rigid elastic spheres, such as those incorporating intermolecular attractions. Doing so is necessary to reproduce the correct temperature dependence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" />, which experiments show increases more rapidly than the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a2f8e9d493c53592f6eae070aeb0f5ca398a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle T^{1/2}}" /> trend predicted for rigid elastic spheres.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot2007-19">[19]</a> Indeed, the Chapman–Enskog analysis shows that the predicted temperature dependence can be tuned by varying the parameters in various molecular models. A simple example is the Sutherland model,<a href="#cite_note-38">[a]</a> which describes rigid elastic spheres with weak mutual attraction. In such a case, the attractive force can be treated <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbatively</a>, which leads to a simple expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" />: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ={\frac {5}{16\sigma ^{2}}}\left({\frac {k_{\text{B}}mT}{\pi }}\right)^{\!\!1/2}\ \left(1+{\frac {S}{T}}\right)^{\!\!-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mrow> <mn>16</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>m</mi> <mi>T</mi> </mrow> <mi>π</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ={\frac {5}{16\sigma ^{2}}}\left({\frac {k_{\text{B}}mT}{\pi }}\right)^{\!\!1/2}\ \left(1+{\frac {S}{T}}\right)^{\!\!-1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d21b449e12fc6a2557849fa76c8506620bb82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.193ex; height:6.676ex;" alt="{\displaystyle \mu ={\frac {5}{16\sigma ^{2}}}\left({\frac {k_{\text{B}}mT}{\pi }}\right)^{\!\!1/2}\ \left(1+{\frac {S}{T}}\right)^{\!\!-1},}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /> is independent of temperature, being determined only by the parameters of the intermolecular attraction. To connect with experiment, it is convenient to rewrite as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu _{0}\left({\frac {T}{T_{0}}}\right)^{\!\!3/2}\ {\frac {T_{0}+S}{T+S}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>S</mi> </mrow> <mrow> <mi>T</mi> <mo>+</mo> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{0}\left({\frac {T}{T_{0}}}\right)^{\!\!3/2}\ {\frac {T_{0}+S}{T+S}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aea6949185e0ed3d8ef8137c064f2e8ab77af3cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.364ex; height:6.676ex;" alt="{\displaystyle \mu =\mu _{0}\left({\frac {T}{T_{0}}}\right)^{\!\!3/2}\ {\frac {T_{0}+S}{T+S}},}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}" /> is the viscosity at temperature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}" />. This expression is usually named Sutherland's formula.<a href="#cite_note-FOOTNOTESutherland1893507–531-39">[38]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> is known from experiments at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=T_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=T_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed837f1758b02ae3912d00544954a950d76be7ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.147ex; height:2.509ex;" alt="{\displaystyle T=T_{0}}" /> and at least one other temperature, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /> can be calculated. Expressions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> obtained in this way are qualitatively accurate for a number of simple gases. Slightly more sophisticated models, such as the <a href="/wiki/Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones potential</a>, or the more flexible <a href="/wiki/Mie_potential" title="Mie potential">Mie potential</a>, may provide better agreement with experiments, but only at the cost of a more opaque dependence on temperature. A further advantage of these more complex interaction potentials is that they can be used to develop accurate models for a wide variety of properties using the same potential parameters. In situations where little experimental data is available, this makes it possible to obtain model parameters from fitting to properties such as pure-fluid <a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">vapour-liquid equilibria</a>, before using the parameters thus obtained to predict the viscosities of interest with reasonable accuracy. In some systems, the assumption of <a href="/wiki/Spherical_symmetry" class="mw-redirect" title="Spherical symmetry">spherical symmetry</a> must be abandoned, as is the case for vapors with highly <a href="/wiki/Polar_molecules" class="mw-redirect" title="Polar molecules">polar molecules</a> like <a href="/wiki/Properties_of_water" title="Properties of water">H2O</a>. In these cases, the Chapman–Enskog analysis is significantly more complicated.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200725–27-40">[39]</a><a href="#cite_note-FOOTNOTEChapmanCowling1970235–237-41">[40]</a> <div class="mw-heading mw-heading4"><h4 id="Bulk_viscosity">Bulk viscosity</h4>[<a href="/w/index.php?title=Viscosity&action=edit&section=14" title="Edit section: Bulk viscosity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Bulk_viscosity" class="mw-redirect" title="Bulk viscosity">Bulk viscosity</a></div> In the kinetic-molecular picture, a non-zero bulk viscosity arises in gases whenever there are non-negligible relaxational timescales governing the exchange of energy between the translational energy of molecules and their internal energy, e.g. <a href="/wiki/Rotational_energy" title="Rotational energy">rotational</a> and <a href="/wiki/Vibration" title="Vibration">vibrational</a>. As such, the bulk viscosity is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /> for a monatomic ideal gas, in which the internal energy of molecules is negligible, but is nonzero for a gas like <a href="/wiki/Carbon_dioxide" title="Carbon dioxide">carbon dioxide</a>, whose molecules possess both rotational and vibrational energy.<a href="#cite_note-FOOTNOTEChapmanCowling1970197,_214–216-42">[41]</a><a href="#cite_note-FOOTNOTECramer2012066102-2-43">[42]</a> <div class="mw-heading mw-heading3"><h3 id="Pure_liquids">Pure liquids</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=15" title="Edit section: Pure liquids">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Temperature_dependence_of_viscosity#Liquids" title="Temperature dependence of viscosity">Temperature dependence of viscosity § Liquids</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Viscosity_video_science_museum.ogv/220px--Viscosity_video_science_museum.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="293" data-durationhint="15" data-mwtitle="Viscosity_video_science_museum.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Viscosity_video_science_museum.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Viscosity_video_science_museum.ogv/Viscosity_video_science_museum.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="360" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/6/61/Viscosity_video_science_museum.ogv" type="video/ogg; codecs="theora"" data-width="480" data-height="640" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Viscosity_video_science_museum.ogv/Viscosity_video_science_museum.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="180" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Viscosity_video_science_museum.ogv/Viscosity_video_science_museum.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="270" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Viscosity_video_science_museum.ogv/Viscosity_video_science_museum.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="270" data-height="360" /></video><figcaption>Video showing three liquids with different viscosities</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><video id="mwe_player_1" poster="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv/220px--09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="30" data-mwtitle="09._Вискозност_на_течности.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/15/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/15/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/15/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/1/15/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv" type="video/ogg; codecs="theora, vorbis"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/15/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/15/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/15/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/15/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv/09._%D0%92%D0%B8%D1%81%D0%BA%D0%BE%D0%B7%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82%D0%B8.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video><figcaption>Experiment showing the behavior of a viscous fluid with blue dye for visibility</figcaption></figure> In contrast with gases, there is no simple yet accurate picture for the molecular origins of viscosity in liquids. At the simplest level of description, the relative motion of adjacent layers in a liquid is opposed primarily by attractive molecular forces acting across the layer boundary. In this picture, one (correctly) expects viscosity to decrease with increasing temperature. This is because increasing temperature increases the random thermal motion of the molecules, which makes it easier for them to overcome their attractive interactions.<a href="#cite_note-FOOTNOTEReidSherwood1958202-44">[43]</a> Building on this visualization, a simple theory can be constructed in analogy with the discrete structure of a solid: groups of molecules in a liquid are visualized as forming "cages" which surround and enclose single molecules.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200729–31-45">[44]</a> These cages can be occupied or unoccupied, and stronger molecular attraction corresponds to stronger cages. Due to random thermal motion, a molecule "hops" between cages at a rate which varies inversely with the strength of molecular attractions. In <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">equilibrium</a> these "hops" are not biased in any direction. On the other hand, in order for two adjacent layers to move relative to each other, the "hops" must be biased in the direction of the relative motion. The force required to sustain this directed motion can be estimated for a given shear rate, leading to <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \approx {\frac {N_{\text{A}}h}{V}}\operatorname {exp} \left(3.8{\frac {T_{\text{b}}}{T}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mi>h</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>3.8</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>b</mtext> </mrow> </msub> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \approx {\frac {N_{\text{A}}h}{V}}\operatorname {exp} \left(3.8{\frac {T_{\text{b}}}{T}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9ad28fb47b15a285aaedc90912f0ff5c82d660" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.325ex; height:6.176ex;" alt="{\displaystyle \mu \approx {\frac {N_{\text{A}}h}{V}}\operatorname {exp} \left(3.8{\frac {T_{\text{b}}}{T}}\right),}" /></td> <td></td> <td class="nowrap">1</td></tr></tbody></table> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\text{A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/037f8d054f445451d8fee1b440cddba989e66ea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.331ex; height:2.509ex;" alt="{\displaystyle N_{\text{A}}}" /> is the <a href="/wiki/Avogadro_constant" title="Avogadro constant">Avogadro constant</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" /> is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" /> is the volume of a <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a> of liquid, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\text{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>b</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb954fc0fb95dad925ac02b36d1f2192976383a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle T_{\text{b}}}" /> is the <a href="/wiki/Normal_boiling_point" class="mw-redirect" title="Normal boiling point">normal boiling point</a>. This result has the same form as the well-known empirical relation <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =Ae^{B/T},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =Ae^{B/T},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a68dde54867bc9ea90dcdfa1db0cd8004fa884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.432ex; height:3.343ex;" alt="{\displaystyle \mu =Ae^{B/T},}" /></td> <td></td> <td class="nowrap">2</td></tr></tbody></table> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /> are constants fit from data.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200729–31-45">[44]</a><a href="#cite_note-FOOTNOTEReidSherwood1958203–204-46">[45]</a> On the other hand, several authors express caution with respect to this model. Errors as large as 30% can be encountered using equation (<a href="#math_1">1</a>), compared with fitting equation (<a href="#math_2">2</a>) to experimental data.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200729–31-45">[44]</a> More fundamentally, the physical assumptions underlying equation (<a href="#math_1">1</a>) have been criticized.<a href="#cite_note-FOOTNOTEHildebrand1977-47">[46]</a> It has also been argued that the exponential dependence in equation (<a href="#math_1">1</a>) does not necessarily describe experimental observations more accurately than simpler, non-exponential expressions.<a href="#cite_note-FOOTNOTEHildebrand197737-48">[47]</a><a href="#cite_note-FOOTNOTEEgelstaff1992264-49">[48]</a> In light of these shortcomings, the development of a less ad hoc model is a matter of practical interest. Foregoing simplicity in favor of precision, it is possible to write rigorous expressions for viscosity starting from the fundamental equations of motion for molecules. A classic example of this approach is Irving–Kirkwood theory.<a href="#cite_note-FOOTNOTEIrvingKirkwood1949817–829-50">[49]</a> On the other hand, such expressions are given as averages over multiparticle <a href="/wiki/Correlation_function_(statistical_mechanics)" title="Correlation function (statistical mechanics)">correlation functions</a> and are therefore difficult to apply in practice. In general, empirically derived expressions (based on existing viscosity measurements) appear to be the only consistently reliable means of calculating viscosity in liquids.<a href="#cite_note-FOOTNOTEReidSherwood1958206–209-51">[50]</a> Local atomic structure changes observed in undercooled liquids on cooling below the equilibrium melting temperature either in terms of radial distribution function g(r)<a href="#cite_note-52">[51]</a> or structure factor S(Q)<a href="#cite_note-53">[52]</a> are found to be directly responsible for the liquid fragility: deviation of the temperature dependence of viscosity of the undercooled liquid from the Arrhenius equation (2) through modification of the activation energy for viscous flow. At the same time equilibrium liquids follow the Arrhenius equation. <div class="mw-heading mw-heading3"><h3 id="Mixtures_and_blends">Mixtures and blends</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=16" title="Edit section: Mixtures and blends">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Viscosity_models_for_mixtures" title="Viscosity models for mixtures">Viscosity models for mixtures</a></div> <div class="mw-heading mw-heading4"><h4 id="Gaseous_mixtures">Gaseous mixtures</h4>[<a href="/w/index.php?title=Viscosity&action=edit&section=17" title="Edit section: Gaseous mixtures">edit</a>]</div> The same molecular-kinetic picture of a single component gas can also be applied to a gaseous mixture. For instance, in the <a href="/wiki/Chapman%E2%80%93Enskog_theory" title="Chapman–Enskog theory">Chapman–Enskog</a> approach the viscosity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{mix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>mix</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{mix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d69e7b4b7a37088007f427ae7c8d98009133acf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.328ex; height:2.176ex;" alt="{\displaystyle \mu _{\text{mix}}}" /> of a binary mixture of gases can be written in terms of the individual component viscosities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{1,2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1,2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dadd1becbc242ccd6b3dec28dba3a5e08bb58b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.735ex; height:2.343ex;" alt="{\displaystyle \mu _{1,2}}" />, their respective volume fractions, and the intermolecular interactions.<a href="#cite_note-FOOTNOTEChapmanCowling1970-17">[17]</a> As for the single-component gas, the dependence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{mix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>mix</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{mix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d69e7b4b7a37088007f427ae7c8d98009133acf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.328ex; height:2.176ex;" alt="{\displaystyle \mu _{\text{mix}}}" /> on the parameters of the intermolecular interactions enters through various collisional integrals which may not be expressible in <a href="/wiki/Closed-form_expression#Symbolic_integration" title="Closed-form expression">closed form</a>. To obtain usable expressions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{mix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>mix</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{mix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d69e7b4b7a37088007f427ae7c8d98009133acf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.328ex; height:2.176ex;" alt="{\displaystyle \mu _{\text{mix}}}" /> which reasonably match experimental data, the collisional integrals may be computed numerically or from correlations.<a href="#cite_note-:0-54">[53]</a> In some cases, the collision integrals are regarded as fitting parameters, and are fitted directly to experimental data.<a href="#cite_note-55">[54]</a> This is a common approach in the development of <a href="/w/index.php?title=Reference_equations&action=edit&redlink=1" class="new" title="Reference equations (page does not exist)">reference equations</a> for gas-phase viscosities. An example of such a procedure is the Sutherland approach for the single-component gas, discussed above. For gas mixtures consisting of simple molecules, <a href="/wiki/Revised_Enskog_theory" class="mw-redirect" title="Revised Enskog theory">Revised Enskog Theory</a> has been shown to accurately represent both the density- and temperature dependence of the viscosity over a wide range of conditions.<a href="#cite_note-56">[55]</a><a href="#cite_note-:0-54">[53]</a> <div class="mw-heading mw-heading4"><h4 id="Blends_of_liquids">Blends of liquids</h4>[<a href="/w/index.php?title=Viscosity&action=edit&section=18" title="Edit section: Blends of liquids">edit</a>]</div> As for pure liquids, the viscosity of a blend of liquids is difficult to predict from molecular principles. One method is to extend the molecular "cage" theory presented above for a pure liquid. This can be done with varying levels of sophistication. One expression resulting from such an analysis is the Lederer–Roegiers equation for a binary mixture: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \mu _{\text{blend}}={\frac {x_{1}}{x_{1}+\alpha x_{2}}}\ln \mu _{1}+{\frac {\alpha x_{2}}{x_{1}+\alpha x_{2}}}\ln \mu _{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>blend</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>α</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>α</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \mu _{\text{blend}}={\frac {x_{1}}{x_{1}+\alpha x_{2}}}\ln \mu _{1}+{\frac {\alpha x_{2}}{x_{1}+\alpha x_{2}}}\ln \mu _{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09e726b23a71eecf16a4b894dbd16aa7cad6b1cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:44.678ex; height:5.009ex;" alt="{\displaystyle \ln \mu _{\text{blend}}={\frac {x_{1}}{x_{1}+\alpha x_{2}}}\ln \mu _{1}+{\frac {\alpha x_{2}}{x_{1}+\alpha x_{2}}}\ln \mu _{2},}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /> is an empirical parameter, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1,2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1,2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a722bc42bd90aa8f4f6915385ccfe52a9f5f8c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.663ex; height:2.343ex;" alt="{\displaystyle x_{1,2}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{1,2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1,2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dadd1becbc242ccd6b3dec28dba3a5e08bb58b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.735ex; height:2.343ex;" alt="{\displaystyle \mu _{1,2}}" /> are the respective <a href="/wiki/Mole_fractions" class="mw-redirect" title="Mole fractions">mole fractions</a> and viscosities of the component liquids.<a href="#cite_note-FOOTNOTEZhmud201422-57">[56]</a> Since blending is an important process in the lubricating and oil industries, a variety of empirical and proprietary equations exist for predicting the viscosity of a blend.<a href="#cite_note-FOOTNOTEZhmud201422-57">[56]</a> <div class="mw-heading mw-heading3"><h3 id="Solutions_and_suspensions">Solutions and suspensions</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=19" title="Edit section: Solutions and suspensions">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Aqueous_solutions">Aqueous solutions</h4>[<a href="/w/index.php?title=Viscosity&action=edit&section=20" title="Edit section: Aqueous solutions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Debye%E2%80%93H%C3%BCckel_theory" title="Debye–Hückel theory">Debye–Hückel theory</a> and <a href="/wiki/List_of_viscosities#Aqueous_solutions" title="List of viscosities">List of viscosities § Aqueous solutions</a></div> Depending on the <a href="/wiki/Solute" class="mw-redirect" title="Solute">solute</a> and range of concentration, an aqueous <a href="/wiki/Electrolyte" title="Electrolyte">electrolyte</a> solution can have either a larger or smaller viscosity compared with pure water at the same temperature and pressure. For instance, a 20% saline (<a href="/wiki/Sodium_chloride" title="Sodium chloride">sodium chloride</a>) solution has viscosity over 1.5 times that of pure water, whereas a 20% <a href="/wiki/Potassium_iodide" title="Potassium iodide">potassium iodide</a> solution has viscosity about 0.91 times that of pure water. An idealized model of dilute electrolytic solutions leads to the following prediction for the viscosity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8369676bc9600b382e8a6af5a9369f2e619cd5c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle \mu _{s}}" /> of a solution:<a href="#cite_note-FOOTNOTEViswanathGhoshPrasadDutt2007-58">[57]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mu _{s}}{\mu _{0}}}=1+A{\sqrt {c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mu _{s}}{\mu _{0}}}=1+A{\sqrt {c}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7224f5f7876df0d7b95bf1c2368e913b6cc282fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.726ex; height:5.343ex;" alt="{\displaystyle {\frac {\mu _{s}}{\mu _{0}}}=1+A{\sqrt {c}},}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}" /> is the viscosity of the solvent, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /> is the concentration, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> is a positive constant which depends on both solvent and solute properties. However, this expression is only valid for very dilute solutions, having <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /> less than 0.1 mol/L.<a href="#cite_note-FOOTNOTEAbdulagatovZeinalovaAzizov200675–88-59">[58]</a> For higher concentrations, additional terms are necessary which account for higher-order molecular correlations: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mu _{s}}{\mu _{0}}}=1+A{\sqrt {c}}+Bc+Cc^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> <mo>+</mo> <mi>B</mi> <mi>c</mi> <mo>+</mo> <mi>C</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mu _{s}}{\mu _{0}}}=1+A{\sqrt {c}}+Bc+Cc^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d577edc276e4285acc95d049264feaa7539a7973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.005ex; height:5.343ex;" alt="{\displaystyle {\frac {\mu _{s}}{\mu _{0}}}=1+A{\sqrt {c}}+Bc+Cc^{2},}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /> are fit from data. In particular, a negative value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /> is able to account for the decrease in viscosity observed in some solutions. Estimated values of these constants are shown below for sodium chloride and potassium iodide at temperature 25 °C (mol = <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a>, L = <a href="/wiki/Liter" class="mw-redirect" title="Liter">liter</a>).<a href="#cite_note-FOOTNOTEViswanathGhoshPrasadDutt2007-58">[57]</a> <table class="wikitable"> <tbody><tr> <th scope="col">Solute </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> (mol−1/2 L1/2) </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /> (mol−1 L) </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /> (mol−2 L2) </th></tr> <tr> <th scope="row"><a href="/wiki/Sodium_chloride" title="Sodium chloride">Sodium chloride</a> (NaCl) </th> <td>0.0062 </td> <td>0.0793 </td> <td>0.0080 </td></tr> <tr> <th scope="row"><a href="/wiki/Potassium_iodide" title="Potassium iodide">Potassium iodide</a> (KI) </th> <td>0.0047 </td> <td>−0.0755 </td> <td>0.0000 </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Suspensions">Suspensions</h4>[<a href="/w/index.php?title=Viscosity&action=edit&section=21" title="Edit section: Suspensions">edit</a>]</div> In a suspension of solid particles (e.g. <a href="/wiki/Micron" class="mw-redirect" title="Micron">micron</a>-size spheres suspended in oil), an effective viscosity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{eff}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{eff}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7111a2b595cddff7203465b91f2e1a5b60549f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.371ex; height:2.176ex;" alt="{\displaystyle \mu _{\text{eff}}}" /> can be defined in terms of stress and strain components which are averaged over a volume large compared with the distance between the suspended particles, but small with respect to macroscopic dimensions.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200731–33-60">[59]</a> Such suspensions generally exhibit non-Newtonian behavior. However, for dilute systems in steady flows, the behavior is Newtonian and expressions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{eff}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{eff}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7111a2b595cddff7203465b91f2e1a5b60549f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.371ex; height:2.176ex;" alt="{\displaystyle \mu _{\text{eff}}}" /> can be derived directly from the particle dynamics. In a very dilute system, with volume fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \lesssim 0.02}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>≲</mo> <mn>0.02</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \lesssim 0.02}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f406c6c20b93c9a68575447bc99e3fb5612570a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.618ex; height:2.843ex;" alt="{\displaystyle \phi \lesssim 0.02}" />, interactions between the suspended particles can be ignored. In such a case one can explicitly calculate the flow field around each particle independently, and combine the results to obtain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{eff}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{eff}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7111a2b595cddff7203465b91f2e1a5b60549f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.371ex; height:2.176ex;" alt="{\displaystyle \mu _{\text{eff}}}" />. For spheres, this results in the Einstein's effective viscosity formula: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+{\frac {5}{2}}\phi \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+{\frac {5}{2}}\phi \right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8141492fe77c207ecc4d5ecc5402428a73e254" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.154ex; height:6.176ex;" alt="{\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+{\frac {5}{2}}\phi \right),}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}" /> is the viscosity of the suspending liquid. The linear dependence on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" /> is a consequence of neglecting interparticle interactions. For dilute systems in general, one expects <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{eff}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{eff}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7111a2b595cddff7203465b91f2e1a5b60549f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.371ex; height:2.176ex;" alt="{\displaystyle \mu _{\text{eff}}}" /> to take the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+B\phi \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+B\phi \right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e3553335495441a9d1d7811c1871b71c48d5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.308ex; height:2.843ex;" alt="{\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+B\phi \right),}" /></dd></dl> where the coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /> may depend on the particle shape (e.g. spheres, rods, disks).<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200732-61">[60]</a> Experimental determination of the precise value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" /> is difficult, however: even the prediction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=5/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=5/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd09fbc65fc2887b9c47c0adff0719f5b660247e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.35ex; height:2.843ex;" alt="{\displaystyle B=5/2}" /> for spheres has not been conclusively validated, with various experiments finding values in the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.5\lesssim B\lesssim 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.5</mn> <mo>≲</mo> <mi>B</mi> <mo>≲</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.5\lesssim B\lesssim 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a4fe5715e37f1023e48165b10cd4eb3def44f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.095ex; height:2.843ex;" alt="{\displaystyle 1.5\lesssim B\lesssim 5}" />. This deficiency has been attributed to difficulty in controlling experimental conditions.<a href="#cite_note-FOOTNOTEMuellerLlewellinMader20091201–1228-62">[61]</a> In denser suspensions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{eff}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{eff}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7111a2b595cddff7203465b91f2e1a5b60549f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.371ex; height:2.176ex;" alt="{\displaystyle \mu _{\text{eff}}}" /> acquires a nonlinear dependence on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" />, which indicates the importance of interparticle interactions. Various analytical and semi-empirical schemes exist for capturing this regime. At the most basic level, a term quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" /> is added to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{eff}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{eff}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7111a2b595cddff7203465b91f2e1a5b60549f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.371ex; height:2.176ex;" alt="{\displaystyle \mu _{\text{eff}}}" />: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+B\phi +B_{1}\phi ^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eff</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>B</mi> <mi>ϕ</mi> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+B\phi +B_{1}\phi ^{2}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e05805430bb3102b9dcd1f6db0a7e226f35a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.727ex; height:3.343ex;" alt="{\displaystyle \mu _{\text{eff}}=\mu _{0}\left(1+B\phi +B_{1}\phi ^{2}\right),}" /></dd></dl> and the coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa091eb428443c9c5c5fcf32a69d3665c89e00c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{1}}" /> is fit from experimental data or approximated from the microscopic theory. However, some authors advise caution in applying such simple formulas since non-Newtonian behavior appears in dense suspensions (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \gtrsim 0.25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>≳</mo> <mn>0.25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \gtrsim 0.25}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d5061da46ecfc95f12debcf00f2203c52b97f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.618ex; height:2.843ex;" alt="{\displaystyle \phi \gtrsim 0.25}" /> for spheres),<a href="#cite_note-FOOTNOTEMuellerLlewellinMader20091201–1228-62">[61]</a> or in suspensions of elongated or flexible particles.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200731–33-60">[59]</a> There is a distinction between a suspension of solid particles, described above, and an <a href="/wiki/Emulsion" title="Emulsion">emulsion</a>. The latter is a suspension of tiny droplets, which themselves may exhibit internal circulation. The presence of internal circulation can decrease the observed effective viscosity, and different theoretical or semi-empirical models must be used.<a href="#cite_note-FOOTNOTEBirdStewartLightfoot200733-63">[62]</a> <div class="mw-heading mw-heading3"><h3 id="Amorphous_materials">Amorphous materials</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=22" title="Edit section: Amorphous materials">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Glassviscosityexamples.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Glassviscosityexamples.png/500px-Glassviscosityexamples.png" decoding="async" width="400" height="274" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Glassviscosityexamples.png/960px-Glassviscosityexamples.png 1.5x" data-file-width="2878" data-file-height="1972" /></a><figcaption>Common <a href="/wiki/Glass" title="Glass">glass</a> viscosity curves<a href="#cite_note-FOOTNOTEFluegel2007-64">[63]</a></figcaption></figure> In the high and low temperature limits, viscous flow in <a href="/wiki/Amorphous_solid" title="Amorphous solid">amorphous materials</a> (e.g. in <a href="/wiki/Glass" title="Glass">glasses</a> and melts)<a href="#cite_note-FOOTNOTEDoremus20027619–7629-65">[64]</a><a href="#cite_note-FOOTNOTEOjovanTravisHand2007415107-66">[65]</a><a href="#cite_note-FOOTNOTEOjovanLee20043803–3810-67">[66]</a> has the <a href="/wiki/Arrhenius_equation" title="Arrhenius equation">Arrhenius form</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =Ae^{Q/(RT)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =Ae^{Q/(RT)},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b19a42b7eae1095445dc759dcace47b00368b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.011ex; height:3.343ex;" alt="{\displaystyle \mu =Ae^{Q/(RT)},}" /></dd></dl> where Q is a relevant <a href="/wiki/Activation_energy" title="Activation energy">activation energy</a>, given in terms of molecular parameters; T is temperature; R is the molar <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a>; and A is approximately a constant. The activation energy Q takes a different value depending on whether the high or low temperature limit is being considered: it changes from a high value QH at low temperatures (in the glassy state) to a low value QL at high temperatures (in the liquid state). <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:B2O3_viscosoty.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/B2O3_viscosoty.jpg/250px-B2O3_viscosoty.jpg" decoding="async" width="250" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/B2O3_viscosoty.jpg/500px-B2O3_viscosoty.jpg 1.5x" data-file-width="939" data-file-height="627" /></a><figcaption><a href="/wiki/Common_logarithm" title="Common logarithm">Common logarithm</a> of viscosity against temperature for <a href="/wiki/Boron_trioxide" title="Boron trioxide">B2O3</a>, showing two regimes</figcaption></figure> For intermediate temperatures, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" /> varies nontrivially with temperature and the simple Arrhenius form fails. On the other hand, the two-exponential equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =AT\exp \left({\frac {B}{RT}}\right)\left[1+C\exp \left({\frac {D}{RT}}\right)\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>A</mi> <mi>T</mi> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>C</mi> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =AT\exp \left({\frac {B}{RT}}\right)\left[1+C\exp \left({\frac {D}{RT}}\right)\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38aa9224e9ac73624655cd20405e140af63a62eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.719ex; height:6.176ex;" alt="{\displaystyle \mu =AT\exp \left({\frac {B}{RT}}\right)\left[1+C\exp \left({\frac {D}{RT}}\right)\right],}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" /> are all constants, provides a good fit to experimental data over the entire range of temperatures, while at the same time reducing to the correct Arrhenius form in the low and high temperature limits. This expression, also known as Duouglas-Doremus-<a href="/wiki/Michael_Ojovan" title="Michael Ojovan">Ojovan</a> model,<a href="#cite_note-68">[67]</a> can be motivated from various theoretical models of amorphous materials at the atomic level.<a href="#cite_note-FOOTNOTEOjovanTravisHand2007415107-66">[65]</a> A two-exponential equation for the viscosity can be derived within the Dyre shoving model of supercooled liquids, where the Arrhenius energy barrier is identified with the high-frequency <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> times a characteristic shoving volume.<a href="#cite_note-FOOTNOTEDyreOlsenChristensen19962171-69">[68]</a><a href="#cite_note-FOOTNOTEHecksherDyre2015-70">[69]</a> Upon specifying the temperature dependence of the shear modulus via thermal expansion and via the repulsive part of the intermolecular potential, another two-exponential equation is retrieved:<a href="#cite_note-FOOTNOTEKrausserSamwerZaccone201513762-71">[70]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\exp {\left\{{\frac {V_{c}C_{G}}{k_{B}T}}\exp {\left[(2+\lambda )\alpha _{T}T_{g}\left(1-{\frac {T}{T_{g}}}\right)\right]}\right\}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\exp {\left\{{\frac {V_{c}C_{G}}{k_{B}T}}\exp {\left[(2+\lambda )\alpha _{T}T_{g}\left(1-{\frac {T}{T_{g}}}\right)\right]}\right\}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7c6670713177337446c22a7976e9664d2008526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.478ex; height:6.176ex;" alt="{\displaystyle \mu =\exp {\left\{{\frac {V_{c}C_{G}}{k_{B}T}}\exp {\left[(2+\lambda )\alpha _{T}T_{g}\left(1-{\frac {T}{T_{g}}}\right)\right]}\right\}}}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{G}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{G}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47f01459fd6bc92fead984540025151d0e431bf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.186ex; height:2.509ex;" alt="{\displaystyle C_{G}}" /> denotes the high-frequency <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> of the material evaluated at a temperature equal to the <a href="/wiki/Glass_transition" title="Glass transition">glass transition</a> temperature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{g}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e95177ce2147ed7b88f95acad4c46bfd195ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.379ex; height:2.843ex;" alt="{\displaystyle T_{g}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/338b595db7a169754c5e088d9881010c2225f597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.299ex; height:2.509ex;" alt="{\displaystyle V_{c}}" /> is the so-called shoving volume, i.e. it is the characteristic volume of the group of atoms involved in the shoving event by which an atom/molecule escapes from the cage of nearest-neighbours, typically on the order of the volume occupied by few atoms. Furthermore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bae4c2d7a1f5097997a285df47dd01fb03ac1dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.877ex; height:2.009ex;" alt="{\displaystyle \alpha _{T}}" /> is the <a href="/wiki/Thermal_expansion" title="Thermal expansion">thermal expansion</a> coefficient of the material, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /> is a parameter which measures the steepness of the power-law rise of the ascending flank of the first peak of the <a href="/wiki/Radial_distribution_function" title="Radial distribution function">radial distribution function</a>, and is quantitatively related to the repulsive part of the <a href="/wiki/Interatomic_potential" title="Interatomic potential">interatomic potential</a>.<a href="#cite_note-FOOTNOTEKrausserSamwerZaccone201513762-71">[70]</a> Finally, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f38f7b73e53fd7b5d9ca64bec3a1438cc0eade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.691ex; height:2.509ex;" alt="{\displaystyle k_{B}}" /> denotes the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>. <div class="mw-heading mw-heading3"><h3 id="Eddy_viscosity">Eddy viscosity</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=23" title="Edit section: Eddy viscosity">edit</a>]</div> In the study of <a href="/wiki/Turbulence" title="Turbulence">turbulence</a> in <a href="/wiki/Fluid" title="Fluid">fluids</a>, a common practical strategy is to ignore the small-scale <a href="/wiki/Vortex" title="Vortex">vortices</a> (or <a href="/wiki/Eddy_(fluid_dynamics)" title="Eddy (fluid dynamics)">eddies</a>) in the motion and to calculate a large-scale motion with an effective viscosity, called the "eddy viscosity", which characterizes the transport and dissipation of <a href="/wiki/Energy" title="Energy">energy</a> in the smaller-scale flow (see <a href="/wiki/Large_eddy_simulation" title="Large eddy simulation">large eddy simulation</a>).<a href="#cite_note-FOOTNOTEBirdStewartLightfoot2007163-72">[71]</a><a href="#cite_note-FOOTNOTELesieur20122–-73">[72]</a> In contrast to the viscosity of the fluid itself, which must be positive by the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>, the eddy viscosity can be negative.<a href="#cite_note-FOOTNOTESivashinskyYakhot19851040-74">[73]</a><a href="#cite_note-FOOTNOTEXieLevchenko2019045434-75">[74]</a> <div class="mw-heading mw-heading2"><h2 id="Prediction">Prediction</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=24" title="Edit section: Prediction">edit</a>]</div> Because viscosity depends continuously on temperature and pressure, it cannot be fully characterized by a finite number of experimental measurements. Predictive formulas become necessary if experimental values are not available at the temperatures and pressures of interest. This capability is important for thermophysical simulations, in which the temperature and pressure of a fluid can vary continuously with space and time. A similar situation is encountered for mixtures of pure fluids, where the viscosity depends continuously on the concentration ratios of the constituent fluids For the simplest fluids, such as dilute monatomic gases and their mixtures, <a href="/wiki/Ab_initio" title="Ab initio">ab initio</a> <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> computations can accurately predict viscosity in terms of fundamental atomic constants, i.e., without reference to existing viscosity measurements.<a href="#cite_note-FOOTNOTESharipovBenites2020-76">[75]</a> For the special case of dilute helium, <a href="/wiki/Measurement_uncertainty" title="Measurement uncertainty">uncertainties</a> in the ab initio calculated viscosity are two order of magnitudes smaller than uncertainties in experimental values.<a href="#cite_note-FOOTNOTERowlandAl_GhafriMay2020-77">[76]</a> For slightly more complex fluids and mixtures at moderate densities (i.e. <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">sub-critical densities</a>) <a href="/wiki/Revised_Enskog_theory" class="mw-redirect" title="Revised Enskog theory">Revised Enskog Theory</a> can be used to predict viscosities with some accuracy.<a href="#cite_note-:0-54">[53]</a> Revised Enskog Theory is predictive in the sense that predictions for viscosity can be obtained using parameters fitted to other, pure-fluid <a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">thermodynamic properties</a> or <a href="/wiki/Transport_phenomena" title="Transport phenomena">transport properties</a>, thus requiring no a priori experimental viscosity measurements. For most fluids, high-accuracy, first-principles computations are not feasible. Rather, theoretical or empirical expressions must be fit to existing viscosity measurements. If such an expression is fit to high-fidelity data over a large range of temperatures and pressures, then it is called a "reference correlation" for that fluid. Reference correlations have been published for many pure fluids; a few examples are <a href="/wiki/Water" title="Water">water</a>, <a href="/wiki/Carbon_dioxide" title="Carbon dioxide">carbon dioxide</a>, <a href="/wiki/Ammonia" title="Ammonia">ammonia</a>, <a href="/wiki/Benzene" title="Benzene">benzene</a>, and <a href="/wiki/Xenon" title="Xenon">xenon</a>.<a href="#cite_note-FOOTNOTEHuberPerkinsLaeseckeFriend2009-78">[77]</a><a href="#cite_note-FOOTNOTELaeseckeMuzny2017-79">[78]</a><a href="#cite_note-FOOTNOTEMonogenidouAssaelHuber2018-80">[79]</a><a href="#cite_note-FOOTNOTEAvgeriAssaelHuberPerkins2014-81">[80]</a><a href="#cite_note-FOOTNOTEVelliadouTasidouAntoniadisAssael2021-82">[81]</a> Many of these cover temperature and pressure ranges that encompass gas, liquid, and <a href="/wiki/Supercritical_fluid" title="Supercritical fluid">supercritical</a> phases. Thermophysical modeling software often relies on reference correlations for predicting viscosity at user-specified temperature and pressure. These correlations may be <a href="/wiki/Proprietary_software" title="Proprietary software">proprietary</a>. Examples are <a href="/wiki/REFPROP" title="REFPROP">REFPROP</a><a href="#cite_note-83">[82]</a> (proprietary) and <a href="/w/index.php?title=CoolProp&action=edit&redlink=1" class="new" title="CoolProp (page does not exist)">CoolProp</a><a href="#cite_note-FOOTNOTEBellWronskiQuoilinLemort2014-84">[83]</a> (open-source). Viscosity can also be computed using formulas that express it in terms of the statistics of individual particle trajectories. These formulas include the <a href="/wiki/Green%E2%80%93Kubo_relations" title="Green–Kubo relations">Green–Kubo relations</a> for the linear shear viscosity and the transient time correlation function expressions derived by Evans and Morriss in 1988.<a href="#cite_note-FOOTNOTEEvansMorriss2007-85">[84]</a><a href="#cite_note-FOOTNOTEEvansMorriss1988-34">[34]</a> The advantage of these expressions is that they are formally exact and valid for general systems. The disadvantage is that they require detailed knowledge of particle trajectories, available only in computationally expensive simulations such as <a href="/wiki/Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a>. An accurate model for interparticle interactions is also required, which may be difficult to obtain for complex molecules.<a href="#cite_note-FOOTNOTEMaginnMesserlyCarlsonRoe2019-86">[85]</a> <div class="mw-heading mw-heading2"><h2 id="Selected_substances">Selected substances</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=25" title="Edit section: Selected substances">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:University_of_Queensland_Pitch_drop_experiment-white_bg.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/University_of_Queensland_Pitch_drop_experiment-white_bg.jpg/250px-University_of_Queensland_Pitch_drop_experiment-white_bg.jpg" decoding="async" width="170" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/University_of_Queensland_Pitch_drop_experiment-white_bg.jpg/330px-University_of_Queensland_Pitch_drop_experiment-white_bg.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/University_of_Queensland_Pitch_drop_experiment-white_bg.jpg/500px-University_of_Queensland_Pitch_drop_experiment-white_bg.jpg 2x" data-file-width="1012" data-file-height="1524" /></a><figcaption>In the University of Queensland <a href="/wiki/Pitch_drop_experiment" title="Pitch drop experiment">pitch drop experiment</a>, pitch has been dripping slowly through a funnel since 1927, at a rate of one drop roughly every decade. In this way the viscosity of pitch has been determined to be approximately 230 billion (2.3×1011) times that of water.<a href="#cite_note-FOOTNOTEEdgeworthDaltonParnell1984198–200-87">[86]</a></figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_viscosities" title="List of viscosities">List of viscosities</a></div> Observed values of viscosity vary over several orders of magnitude, even for common substances (see the order of magnitude table below). For instance, a 70% sucrose (sugar) solution has a viscosity over 400 times that of water, and 26,000 times that of air.<a href="#cite_note-FOOTNOTERumble2018-88">[87]</a> More dramatically, <a href="/wiki/Pitch_(resin)" title="Pitch (resin)">pitch</a> has been estimated to have a viscosity 230 billion times that of water.<a href="#cite_note-FOOTNOTEEdgeworthDaltonParnell1984198–200-87">[86]</a> <div class="mw-heading mw-heading3"><h3 id="Water">Water</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=26" title="Edit section: Water">edit</a>]</div> The <a href="/wiki/Dynamic_viscosity" class="mw-redirect" title="Dynamic viscosity">dynamic viscosity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /> of <a href="/wiki/Water" title="Water">water</a> is about 0.89 mPa·s at room temperature (25 °C). As a function of temperature in <a href="/wiki/Kelvin" title="Kelvin">kelvins</a>, the viscosity can be estimated using the semi-empirical <a href="/wiki/Vogel-Fulcher-Tammann_equation" class="mw-redirect" title="Vogel-Fulcher-Tammann equation">Vogel-Fulcher-Tammann equation</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =A\exp \left({\frac {B}{T-C}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>A</mi> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <mrow> <mi>T</mi> <mo>−</mo> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =A\exp \left({\frac {B}{T-C}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4a7cd3f9828de6e773ab671a203e820fa57b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.683ex; height:6.176ex;" alt="{\displaystyle \mu =A\exp \left({\frac {B}{T-C}}\right)}" /></dd></dl> where A = 0.02939 mPa·s, B = 507.88 K, and C = 149.3 K.<a href="#cite_note-FOOTNOTEViswanathNatarajan1989714–715-89">[88]</a> Experimentally determined values of the viscosity are also given in the table below. The values at 20 °C are a useful reference: there, the dynamic viscosity is about 1 cP and the kinematic viscosity is about 1 cSt. <table class="wikitable sortable"> <caption>Viscosity of water at various temperatures<a href="#cite_note-FOOTNOTERumble2018-88">[87]</a> </caption> <tbody><tr style="background:#efefef;"> <th>Temperature (°C) </th> <th>Viscosity (mPa·s or cP) </th></tr> <tr> <td align="center">10 </td> <td>1.305 9 </td></tr> <tr> <td align="center">20 </td> <td>1.001 6 </td></tr> <tr> <td align="center">30 </td> <td>0.797 22 </td></tr> <tr> <td align="center">50 </td> <td>0.546 52 </td></tr> <tr> <td align="center">70 </td> <td>0.403 55 </td></tr> <tr> <td align="center">90 </td> <td>0.314 17 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Air">Air</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=27" title="Edit section: Air">edit</a>]</div> Under standard atmospheric conditions (25 °C and pressure of 1 bar), the dynamic viscosity of air is 18.5 μPa·s, roughly 50 times smaller than the viscosity of water at the same temperature. Except at very high pressure, the viscosity of air depends mostly on the temperature. Among the many possible approximate formulas for the temperature dependence (see <a href="/wiki/Temperature_dependence_of_viscosity" title="Temperature dependence of viscosity">Temperature dependence of viscosity</a>), one is:<a href="#cite_note-90">[89]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\text{air}}=2.791\times 10^{-7}\times T^{0.7355}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>air</mtext> </mrow> </msub> <mo>=</mo> <mn>2.791</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>7</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0.7355</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\text{air}}=2.791\times 10^{-7}\times T^{0.7355}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4c94cb6ec9bcf300c4419070c90e131594b34e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.565ex; height:3.176ex;" alt="{\displaystyle \eta _{\text{air}}=2.791\times 10^{-7}\times T^{0.7355}}" /></dd></dl> which is accurate in the range −20 °C to 400 °C. For this formula to be valid, the temperature must be given in <a href="/wiki/Kelvins" class="mw-redirect" title="Kelvins">kelvins</a>; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\text{air}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>air</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\text{air}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a96d4b32901aa47df04ac6e06aabde98c5e19d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.312ex; height:2.176ex;" alt="{\displaystyle \eta _{\text{air}}}" /> then corresponds to the viscosity in Pa·s. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Runny_hunny.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Runny_hunny.jpg/250px-Runny_hunny.jpg" decoding="async" width="170" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Runny_hunny.jpg/330px-Runny_hunny.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Runny_hunny.jpg/500px-Runny_hunny.jpg 2x" data-file-width="1780" data-file-height="2700" /></a><figcaption><a href="/wiki/Honey" title="Honey">Honey</a> being drizzled</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Other_common_substances">Other common substances</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=28" title="Edit section: Other common substances">edit</a>]</div> <table class="wikitable"> <tbody><tr style="background:#efefef;"> <th>Substance </th> <th>Viscosity (mPa·s) </th> <th>Temperature (°C) </th> <th>Ref. </th></tr> <tr> <td><a href="/wiki/Benzene" title="Benzene">Benzene</a> </td> <td>0.604 </td> <td>25 </td> <td rowspan="3"><a href="#cite_note-FOOTNOTERumble2018-88">[87]</a> </td></tr> <tr> <td><a href="/wiki/Water" title="Water">Water</a> </td> <td>1.0016 </td> <td>20 </td></tr> <tr> <td><a href="/wiki/Mercury_(element)" title="Mercury (element)">Mercury</a> </td> <td>1.526 </td> <td>25 </td></tr> <tr> <td><a href="/wiki/Whole_milk" class="mw-redirect" title="Whole milk">Whole milk</a> </td> <td>2.12 </td> <td>20 </td> <td><a href="#cite_note-FOOTNOTEFellows2009-91">[90]</a> </td></tr> <tr> <td><a href="/wiki/Dark_beer" class="mw-redirect" title="Dark beer">Dark beer</a> </td> <td>2.53 </td> <td>20 </td> <td> </td></tr> <tr> <td><a href="/wiki/Olive_oil" title="Olive oil">Olive oil</a> </td> <td>56.2 </td> <td>26 </td> <td><a href="#cite_note-FOOTNOTEFellows2009-91">[90]</a> </td></tr> <tr> <td><a href="/wiki/Honey" title="Honey">Honey</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" /> 2,000–10,000 </td> <td>20 </td> <td><a href="#cite_note-FOOTNOTEYanniotisSkaltsiKaraburnioti2006372–377-92">[91]</a> </td></tr> <tr> <td><a href="/wiki/Ketchup" title="Ketchup">Ketchup</a><a href="#cite_note-nnf-93">[b]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" /> 5,000–20,000 </td> <td>25 </td> <td><a href="#cite_note-FOOTNOTEKoochekiGhandiRazaviMortazavi2009596–602-94">[92]</a> </td></tr> <tr> <td><a href="/wiki/Peanut_butter" title="Peanut butter">Peanut butter</a><a href="#cite_note-nnf-93">[b]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" /> 104–106 </td> <td> </td> <td><a href="#cite_note-FOOTNOTECiterneCarreauMoan200186–96-95">[93]</a> </td></tr> <tr> <td><a href="/wiki/Pitch_(resin)" title="Pitch (resin)">Pitch</a> </td> <td>2.3×1011 </td> <td>10–30 (variable) </td> <td><a href="#cite_note-FOOTNOTEEdgeworthDaltonParnell1984198–200-87">[86]</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Order_of_magnitude_estimates">Order of magnitude estimates</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=29" title="Edit section: Order of magnitude estimates">edit</a>]</div> The following table illustrates the range of viscosity values observed in common substances. Unless otherwise noted, a temperature of 25 °C and a pressure of 1 atmosphere are assumed. The values listed are representative estimates only, as they do not account for measurement uncertainties, variability in material definitions, or non-Newtonian behavior. <table class="wikitable"> <tbody><tr> <th>Factor (Pa·s) </th> <th>Description </th> <th>Examples </th> <th>Values (Pa·s) </th> <th>Ref. </th></tr> <tr> <td rowspan="2">10−6 </td> <td rowspan="2">Lower range of gaseous viscosity </td> <td><a href="/wiki/Butane" title="Butane">Butane</a> </td> <td>7.49 × 10−6 </td> <td><a href="#cite_note-FOOTNOTEKestinKhalifaWakeham1977-96">[94]</a> </td></tr> <tr> <td><a href="/wiki/Hydrogen" title="Hydrogen">Hydrogen</a> </td> <td>8.8 × 10−6 </td> <td><a href="#cite_note-FOOTNOTEAssaelKalyvaMonogenidouHuber2018-97">[95]</a> </td></tr> <tr> <td rowspan="2">10−5 </td> <td rowspan="2">Upper range of gaseous viscosity </td> <td><a href="/wiki/Krypton" title="Krypton">Krypton</a> </td> <td>2.538 × 10−5 </td> <td rowspan="2"><a href="#cite_note-FOOTNOTEKestinRoWakeham1972-98">[96]</a> </td></tr> <tr> <td><a href="/wiki/Neon" title="Neon">Neon</a> </td> <td>3.175 × 10−5 </td></tr> <tr> <td rowspan="3">10−4 </td> <td rowspan="3">Lower range of liquid viscosity </td> <td><a href="/wiki/Pentane" title="Pentane">Pentane</a> </td> <td>2.24 × 10−4 </td> <td><a href="#cite_note-FOOTNOTERumble2018-88">[87]</a> </td></tr> <tr> <td><a href="/wiki/Gasoline" title="Gasoline">Gasoline</a> </td> <td>6 × 10−4 </td> <td> </td></tr> <tr> <td><a href="/wiki/Water" title="Water">Water</a> </td> <td>8.90 × 10−4 </td> <td rowspan="3"><a href="#cite_note-FOOTNOTERumble2018-88">[87]</a> </td></tr> <tr> <td rowspan="5">10−3 </td> <td rowspan="5">Typical range for small-molecule Newtonian liquids </td> <td><a href="/wiki/Ethanol" title="Ethanol">Ethanol</a> </td> <td>1.074 × 10−3 </td></tr> <tr> <td><a href="/wiki/Mercury_(element)" title="Mercury (element)">Mercury</a> </td> <td>1.526 × 10−3 </td></tr> <tr> <td><a href="/wiki/Whole_milk" class="mw-redirect" title="Whole milk">Whole milk</a> (20 °C) </td> <td>2.12 × 10−3 </td> <td><a href="#cite_note-FOOTNOTEFellows2009-91">[90]</a> </td></tr> <tr> <td><a href="/wiki/Blood" title="Blood">Blood</a> </td> <td>3 × 10−3 to 6 × 10−3 </td> <td><a href="#cite_note-FOOTNOTERosensonMcCormickUretz1996-99">[97]</a> </td></tr> <tr> <td>Liquid <a href="/wiki/Steel" title="Steel">steel</a> (1550 °C) </td> <td>6 × 10−3 </td> <td><a href="#cite_note-FOOTNOTEZhaoWangYangLi2021-100">[98]</a> </td></tr> <tr> <td rowspan="10">10−2 – 100 </td> <td rowspan="10">Oils and long-chain hydrocarbons </td> <td><a href="/wiki/Linseed_oil" title="Linseed oil">Linseed oil</a> </td> <td>0.028 </td> <td> </td></tr> <tr> <td><a href="/wiki/Oleic_acid" title="Oleic acid">Oleic acid</a> </td> <td>0.036 </td> <td><a href="#cite_note-FOOTNOTESagdeevGabitovIsyanovKhairutdinov2019-101">[99]</a> </td></tr> <tr> <td><a href="/wiki/Olive_oil" title="Olive oil">Olive oil</a> </td> <td>0.084 </td> <td><a href="#cite_note-FOOTNOTEFellows2009-91">[90]</a> </td></tr> <tr> <td>SAE 10 <a href="/wiki/Motor_oil" title="Motor oil">Motor oil</a> </td> <td>0.085 to 0.14 </td> <td> </td></tr> <tr> <td><a href="/wiki/Castor_oil" title="Castor oil">Castor oil</a> </td> <td>0.1 </td> <td> </td></tr> <tr> <td>SAE 20 <a href="/wiki/Motor_oil" title="Motor oil">Motor oil</a> </td> <td>0.14 to 0.42 </td> <td> </td></tr> <tr> <td>SAE 30 <a href="/wiki/Motor_oil" title="Motor oil">Motor oil</a> </td> <td>0.42 to 0.65 </td> <td> </td></tr> <tr> <td>SAE 40 <a href="/wiki/Motor_oil" title="Motor oil">Motor oil</a> </td> <td>0.65 to 0.90 </td> <td> </td></tr> <tr> <td><a href="/wiki/Glycerine" class="mw-redirect" title="Glycerine">Glycerine</a> </td> <td>1.5 </td> <td> </td></tr> <tr> <td><a href="/wiki/Pancake_syrup" class="mw-redirect" title="Pancake syrup">Pancake syrup</a> </td> <td>2.5 </td> <td> </td></tr> <tr> <td rowspan="5">101 – 103 </td> <td rowspan="5">Pastes, gels, and other semisolids (generally non-Newtonian) </td> <td><a href="/wiki/Ketchup" title="Ketchup">Ketchup</a> </td> <td rowspan="2">≈ 101 </td> <td><a href="#cite_note-FOOTNOTEKoochekiGhandiRazaviMortazavi2009596–602-94">[92]</a> </td></tr> <tr> <td><a href="/wiki/Mustard_(condiment)" title="Mustard (condiment)">Mustard</a> </td> <td> </td></tr> <tr> <td><a href="/wiki/Sour_cream" title="Sour cream">Sour cream</a> </td> <td rowspan="2">≈ 102 </td> <td> </td></tr> <tr> <td><a href="/wiki/Peanut_butter" title="Peanut butter">Peanut butter</a> </td> <td><a href="#cite_note-FOOTNOTECiterneCarreauMoan200186–96-95">[93]</a> </td></tr> <tr> <td><a href="/wiki/Lard" title="Lard">Lard</a> </td> <td>≈ 103 </td> <td> </td></tr> <tr> <td>≈108 </td> <td>Viscoelastic polymers </td> <td><a href="/wiki/Pitch_(resin)" title="Pitch (resin)">Pitch</a> </td> <td>2.3×108 </td> <td><a href="#cite_note-FOOTNOTEEdgeworthDaltonParnell1984198–200-87">[86]</a> </td></tr> <tr> <td>≈1021 </td> <td>Certain solids under a viscoelastic description </td> <td><a href="/wiki/Mantle_(geology)" title="Mantle (geology)">Mantle (geology)</a> </td> <td>≈ 1019 to 1024 </td> <td><a href="#cite_note-FOOTNOTEWalzerHendelBaumgardner-102">[100]</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=30" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Dashpot" title="Dashpot">Dashpot</a></li> <li><a href="/wiki/Deborah_number" title="Deborah number">Deborah number</a></li> <li><a href="/wiki/Dilatant" title="Dilatant">Dilatant</a></li> <li><a href="/wiki/Herschel%E2%80%93Bulkley_fluid" title="Herschel–Bulkley fluid">Herschel–Bulkley fluid</a></li> <li><a href="/wiki/High_viscosity_mixer" title="High viscosity mixer">High viscosity mixer</a></li> <li><a href="/wiki/Hyperviscosity_syndrome" title="Hyperviscosity syndrome">Hyperviscosity syndrome</a></li> <li><a href="/wiki/Intrinsic_viscosity" title="Intrinsic viscosity">Intrinsic viscosity</a></li> <li><a href="/wiki/Inviscid_flow" title="Inviscid flow">Inviscid flow</a></li> <li><a href="/wiki/Joback_method" title="Joback method">Joback method</a> (estimation of liquid viscosity from molecular structure)</li> <li><a href="/wiki/Kaye_effect" title="Kaye effect">Kaye effect</a></li> <li><a href="/wiki/Microviscosity" title="Microviscosity">Microviscosity</a></li> <li><a href="/wiki/Morton_number" title="Morton number">Morton number</a></li> <li><a href="/wiki/Oil_pressure" title="Oil pressure">Oil pressure</a></li> <li><a href="/wiki/Quasi-solid" title="Quasi-solid">Quasi-solid</a></li> <li><a href="/wiki/Rheology" title="Rheology">Rheology</a></li> <li><a href="/wiki/Stokes_flow" title="Stokes flow">Stokes flow</a></li> <li><a href="/wiki/Superfluid_helium-4" title="Superfluid helium-4">Superfluid helium-4</a></li> <li><a href="/wiki/Viscoplasticity" title="Viscoplasticity">Viscoplasticity</a></li> <li><a href="/wiki/Viscosity_models_for_mixtures" title="Viscosity models for mixtures">Viscosity models for mixtures</a></li> <li><a href="/wiki/Zahn_cup" title="Zahn cup">Zahn cup</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=31" title="Edit section: References">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Footnotes">Footnotes</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=32" title="Edit section: Footnotes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-38"><a href="#cite_ref-38">^</a> The discussion which follows draws from <a href="#CITEREFChapmanCowling1970">Chapman & Cowling 1970</a>, pp. 232–237 </li> <li id="cite_note-nnf-93">^ <a href="#cite_ref-nnf_93-0">a</a> <a href="#cite_ref-nnf_93-1">b</a> These materials are highly <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian</a>. </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3>[<a href="/w/index.php?title=Viscosity&action=edit&section=33" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Britanica-1">^ <a href="#cite_ref-Britanica_1-0">a</a> <a href="#cite_ref-Britanica_1-1">b</a> <a href="#cite_ref-Britanica_1-2">c</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/viscosity">"Viscosity"</a>. 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Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-57598-8" title="Special:BookSources/978-3-319-57598-8"><bdi>978-3-319-57598-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Encyclopaedia+of+Historical+Metrology%2C+Weights%2C+and+Measures%3A+Volume+1&rft.btitle=Encyclopaedia+of+Historical+Metrology%2C+Weights%2C+and+Measures&rft.pub=Birkh%C3%A4user&rft.date=2018&rft.isbn=978-3-319-57598-8&rft.aulast=Gyllenbok&rft.aufirst=Jan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHannan2007" class="citation book cs1">Hannan, Henry (2007). Technician's Formulation Handbook for Industrial and Household Cleaning Products. 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Journal of Non-Crystalline Solids. 7th IDMRCS: Relaxation in Complex Systems. 407: 14–22. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015JNCS..407...14H">2015JNCS..407...14H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnoncrysol.2014.08.056">10.1016/j.jnoncrysol.2014.08.056</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-3093">0022-3093</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220215003710/https://www.sciencedirect.com/science/article/abs/pii/S0022309314004529">Archived</a> from the original on 2022-02-15. 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McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-112230-6" title="Special:BookSources/978-0-07-112230-6"><bdi>978-0-07-112230-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200315193956/https://books.google.com/books?id=GajCQgAACAAJ">Archived</a> from the original on 2020-03-15. Retrieved 2019-09-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Heat+Transfer&rft.pub=McGraw-Hill&rft.date=2002&rft.isbn=978-0-07-112230-6&rft.aulast=Holman&rft.aufirst=Jack+Philip&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGajCQgAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHuberPerkinsLaeseckeFriend2009" class="citation journal cs1">Huber, M. L.; Perkins, R. 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Retrieved 2008-06-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Society+of+Materials+Science+%28Japan%29&rft.atitle=Long-term+Creep+of+Rocks%3A+Results+with+Large+Specimens+Obtained+in+about+20+Years+and+Those+with+Small+Specimens+in+about+3+Years&rft.volume=27&rft.issue=293&rft.pages=%3Cspan+class%3D%22nowrap%22%3E157-%3C%2Fspan%3E161&rft.date=1978-02-15&rft.aulast=Kumagai&rft.aufirst=Naoichi&rft.au=Sasajima%2C+Sadao&rft.au=Ito%2C+Hidebumi&rft_id=https%3A%2F%2Ftranslate.google.com%2Ftranslate%3Fhl%3Den%26sl%3Dja%26u%3Dhttp%3A%2F%2Fci.nii.ac.jp%2Fnaid%2F110002299397%2F%26sa%3DX%26oi%3Dtranslate%26resnum%3D4%26ct%3Dresult%26prev%3D%2Fsearch%253Fq%253DIto%252BHidebumi%2526hl%253Den&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKwapiszewskaSzczepańskiKalwarczykMichalska2020" class="citation journal cs1">Kwapiszewska, Karina; Szczepański, Krzysztof; Kalwarczyk, Tomasz; Michalska, Bernadeta; Patalas-Krawczyk, Paulina; Szymański, Jędrzej; Andryszewski, Tomasz; Iwan, Michalina; Duszyński, Jerzy; Hołyst, Robert (2020). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7450658">"Nanoscale Viscosity of Cytoplasm Is Conserved in Human Cell Lines"</a>. The Journal of Physical Chemistry Letters. 11 (16): 6914–6920. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Facs.jpclett.0c01748">10.1021/acs.jpclett.0c01748</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7450658">7450658</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32787203">32787203</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Physical+Chemistry+Letters&rft.atitle=Nanoscale+Viscosity+of+Cytoplasm+Is+Conserved+in+Human+Cell+Lines&rft.volume=11&rft.issue=16&rft.pages=%3Cspan+class%3D%22nowrap%22%3E6914-%3C%2Fspan%3E6920&rft.date=2020&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7450658%23id-name%3DPMC&rft_id=info%3Apmid%2F32787203&rft_id=info%3Adoi%2F10.1021%2Facs.jpclett.0c01748&rft.aulast=Kwapiszewska&rft.aufirst=Karina&rft.au=Szczepa%C5%84ski%2C+Krzysztof&rft.au=Kalwarczyk%2C+Tomasz&rft.au=Michalska%2C+Bernadeta&rft.au=Patalas-Krawczyk%2C+Paulina&rft.au=Szyma%C5%84ski%2C+J%C4%99drzej&rft.au=Andryszewski%2C+Tomasz&rft.au=Iwan%2C+Michalina&rft.au=Duszy%C5%84ski%2C+Jerzy&rft.au=Ho%C5%82yst%2C+Robert&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7450658&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLaeseckeMuzny2017" class="citation journal cs1">Laesecke, Arno; Muzny, Chris D. 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Retrieved 2019-09-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics&rft.edition=2nd&rft.pub=Elsevier&rft.date=1987&rft.isbn=978-0-08-057073-0&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=Lifshitz%2C+E.M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeVKbCgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaginnMesserlyCarlsonRoe2019" class="citation journal cs1">Maginn, Edward J.; Messerly, Richard A.; Carlson, Daniel J.; Roe, Daniel R.; Elliott, J. Richard (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.33011%2Flivecoms.1.1.6324">"Best Practices for Computing Transport Properties 1. Self-Diffusivity and Viscosity from Equilibrium Molecular Dynamics [Article v1.0]"</a>. Living Journal of Computational Molecular Science. 1 (1). University of Colorado at Boulder. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.33011%2Flivecoms.1.1.6324">10.33011/livecoms.1.1.6324</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2575-6524">2575-6524</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:104357320">104357320</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Living+Journal+of+Computational+Molecular+Science&rft.atitle=Best+Practices+for+Computing+Transport+Properties+1.+Self-Diffusivity+and+Viscosity+from+Equilibrium+Molecular+Dynamics+%5BArticle+v1.0%5D&rft.volume=1&rft.issue=1&rft.date=2019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A104357320%23id-name%3DS2CID&rft.issn=2575-6524&rft_id=info%3Adoi%2F10.33011%2Flivecoms.1.1.6324&rft.aulast=Maginn&rft.aufirst=Edward+J.&rft.au=Messerly%2C+Richard+A.&rft.au=Carlson%2C+Daniel+J.&rft.au=Roe%2C+Daniel+R.&rft.au=Elliott%2C+J.+Richard&rft_id=https%3A%2F%2Fdoi.org%2F10.33011%252Flivecoms.1.1.6324&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMonogenidouAssaelHuber2018" class="citation journal cs1">Monogenidou, S. A.; Assael, M. J.; Huber, M. L. (2018). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6512859">"Reference Correlation for the Viscosity of Ammonia from the Triple Point to 725 K and up to 50 MPa"</a>. Journal of Physical and Chemical Reference Data. 47 (2). AIP Publishing: 023102. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018JPCRD..47b3102M">2018JPCRD..47b3102M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.5036724">10.1063/1.5036724</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0047-2689">0047-2689</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6512859">6512859</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/31092958">31092958</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physical+and+Chemical+Reference+Data&rft.atitle=Reference+Correlation+for+the+Viscosity+of+Ammonia+from+the+Triple+Point+to+725+K+and+up+to+50+MPa&rft.volume=47&rft.issue=2&rft.pages=023102&rft.date=2018&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6512859%23id-name%3DPMC&rft_id=info%3Abibcode%2F2018JPCRD..47b3102M&rft_id=info%3Apmid%2F31092958&rft_id=info%3Adoi%2F10.1063%2F1.5036724&rft.issn=0047-2689&rft.aulast=Monogenidou&rft.aufirst=S.+A.&rft.au=Assael%2C+M.+J.&rft.au=Huber%2C+M.+L.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6512859&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLesieur2012" class="citation book cs1">Lesieur, Marcel (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QILpCAAAQBAJ&pg=PR2">Turbulence in Fluids: Stochastic and Numerical Modelling</a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-009-0533-7" title="Special:BookSources/978-94-009-0533-7"><bdi>978-94-009-0533-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200314133321/https://books.google.com/books?id=QILpCAAAQBAJ&pg=PR2">Archived</a> from the original on 2020-03-14. Retrieved 2018-11-30.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Turbulence+in+Fluids%3A+Stochastic+and+Numerical+Modelling&rft.pub=Springer&rft.date=2012&rft.isbn=978-94-009-0533-7&rft.aulast=Lesieur&rft.aufirst=Marcel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQILpCAAAQBAJ%26pg%3DPR2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMewisWagner2012" class="citation book cs1">Mewis, Jan; Wagner, Norman J. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Et6kZGtdiFsC">Colloidal Suspension Rheology</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-51599-3" title="Special:BookSources/978-0-521-51599-3"><bdi>978-0-521-51599-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200314194224/https://books.google.com/books?id=Et6kZGtdiFsC">Archived</a> from the original on 2020-03-14. Retrieved 2018-12-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Colloidal+Suspension+Rheology&rft.pub=Cambridge+University+Press&rft.date=2012&rft.isbn=978-0-521-51599-3&rft.aulast=Mewis&rft.aufirst=Jan&rft.au=Wagner%2C+Norman+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEt6kZGtdiFsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMcNaughtWilkinson1997" class="citation book cs1">McNaught, A. D.; Wilkinson, A. (1997). "poise". IUPAC. Compendium of Chemical Terminology (the "Gold Book"). S. J. Chalk (2nd ed.). 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Journal of Non-Crystalline Solids. 107 (1): 14. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988JNCS..107...14S">1988JNCS..107...14S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-3093%2888%2990086-5">10.1016/0022-3093(88)90086-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Non-Crystalline+Solids&rft.atitle=Viscoelasticity+in+silica+gel&rft.volume=107&rft.issue=1&rft.pages=14&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2F0022-3093%2888%2990086-5&rft_id=info%3Abibcode%2F1988JNCS..107...14S&rft.aulast=Scherer&rft.aufirst=George+W.&rft.au=Pardenek%2C+Sandra+A.&rft.au=Swiatek%2C+Rose+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchroeder1999" class="citation book cs1">Schroeder, Daniel V. 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Retrieved 2018-11-30.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Thermal+Physics&rft.pub=Addison+Wesley&rft.date=1999&rft.isbn=978-0-201-38027-9&rft.aulast=Schroeder&rft.aufirst=Daniel+V.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1gosQgAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSharipovBenites2020" class="citation journal cs1">Sharipov, Felix; Benites, Victor J. (2020-07-01). "Transport coefficients of multi-component mixtures of noble gases based on ab initio potentials: Viscosity and thermal conductivity". Physics of Fluids. 32 (7). 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The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 36 (223): 507–531. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786449308620508">10.1080/14786449308620508</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1941-5982">1941-5982</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190720113013/https://web.stanford.edu/~cantwell/AA210A_Course_Material/Sutherland_Viscosity_Model.pdf">Archived</a> (PDF) from the original on 2019-07-20. 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American Association for the Advancement of Science (AAAS): eaba3747. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1912.06711">1912.06711</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020SciA....6.3747T">2020SciA....6.3747T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fsciadv.aba3747">10.1126/sciadv.aba3747</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2375-2548">2375-2548</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7182420">7182420</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32426470">32426470</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+Advances&rft.atitle=Minimal+quantum+viscosity+from+fundamental+physical+constants&rft.volume=6&rft.issue=17&rft.pages=eaba3747&rft.date=2020-04-22&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7182420%23id-name%3DPMC&rft_id=info%3Abibcode%2F2020SciA....6.3747T&rft_id=info%3Aarxiv%2F1912.06711&rft.issn=2375-2548&rft_id=info%3Adoi%2F10.1126%2Fsciadv.aba3747&rft_id=info%3Apmid%2F32426470&rft.aulast=Trachenko&rft.aufirst=K.&rft.au=Brazhkin%2C+V.+V.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7182420&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTrachenkoBrazhkin2021" class="citation journal cs1">Trachenko, Kostya; Brazhkin, Vadim V. 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Retrieved 2020-05-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=tec-science&rft.atitle=Viscosity+of+liquids+and+gases&rft.date=2020-03-25&rft.au=tec-science&rft_id=https%3A%2F%2Fwww.tec-science.com%2Fmechanics%2Fgases-and-liquids%2Fviscosity-of-liquids-and-gases%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AViscosity" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Viscosity&action=edit&section=35" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output 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the text of <a href="https://en.wikisource.org/wiki/The_New_Student%27s_Reference_Work" class="extiw" title="s:The New Student's Reference Work">The New Student's Reference Work</a> article "<a href="https://en.wikisource.org/wiki/The_New_Student%27s_Reference_Work/Viscosity_of_Liquids" class="extiw" title="s:The New Student's Reference Work/Viscosity of Liquids">Viscosity of Liquids</a>".</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735" /><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" 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href="https://www.engineersedge.com/fluid_flow/fluid_data.htm">Fluid Characteristics Chart</a> – a table of viscosities and vapor pressures for various fluids</li> <li><a rel="nofollow" class="external text" href="http://web.ics.purdue.edu/~alexeenk/GDT/index.html">Gas Dynamics Toolbox</a> – calculate coefficient of viscosity for mixtures of gases</li> <li><a rel="nofollow" class="external text" href="http://glassproperties.com/viscosity/ViscosityMeasurement.htm">Glass Viscosity Measurement</a> – viscosity measurement, viscosity units and fixpoints, glass viscosity calculation</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100113072736/http://www.diracdelta.co.uk/science/source/k/i/kinematic%20viscosity/source.html">Kinematic Viscosity</a> – conversion between kinematic and dynamic viscosity</li> <li><a rel="nofollow" class="external text" href="http://www.thermexcel.com/english/tables/eau_atm.htm">Physical Characteristics of Water</a> – a table of water viscosity as a function of temperature</li> <li><a rel="nofollow" class="external text" href="http://ddbonline.ddbst.de/VogelCalculation/VogelCalculationCGI.exe">Calculation of temperature-dependent dynamic viscosities for some common components</a></li> <li><a rel="nofollow" class="external text" href="https://www.astro.uu.se/~bf/course/numhd_course/2_5_2Artificial_viscosity.html">Artificial viscosity</a></li> <li><a rel="nofollow" class="external text" href="https://www.engineersedge.com/physics/viscosity_of_air_dynamic_and_kinematic_14483.htm">Viscosity of Air, Dynamic and Kinematic, Engineers Edge</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox 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