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id="toc-Relationship_to_continuum_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_to_continuum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relationship to continuum mechanics</span> </div> </a> <ul id="toc-Relationship_to_continuum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Assumptions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Assumptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Assumptions</span> </div> </a> <ul id="toc-Assumptions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Navier–Stokes_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Navier–Stokes_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Navier–Stokes equations</span> </div> </a> <ul id="toc-Navier–Stokes_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inviscid_and_viscous_fluids" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Inviscid_and_viscous_fluids"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Inviscid and viscous fluids</span> </div> </a> <ul id="toc-Inviscid_and_viscous_fluids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Newtonian_versus_non-Newtonian_fluids" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Newtonian_versus_non-Newtonian_fluids"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Newtonian versus non-Newtonian fluids</span> </div> </a> <button aria-controls="toc-Newtonian_versus_non-Newtonian_fluids-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Newtonian versus non-Newtonian fluids subsection</span> </button> <ul id="toc-Newtonian_versus_non-Newtonian_fluids-sublist" class="vector-toc-list"> <li id="toc-Equations_for_a_Newtonian_fluid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equations_for_a_Newtonian_fluid"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Equations for a Newtonian fluid</span> </div> </a> <ul id="toc-Equations_for_a_Newtonian_fluid-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </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" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Fluid mechanics</span></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 70 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-70" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">70 languages</span> </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/Stromingsleer" title="Stromingsleer – Afrikaans" lang="af" hreflang="af" data-title="Stromingsleer" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%85%D9%88%D8%A7%D8%A6%D8%B9" title="ميكانيكا الموائع – Arabic" lang="ar" hreflang="ar" data-title="ميكانيكا الموائع" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Mec%C3%A1nica_de_fluy%C3%ADos" title="Mecánica de fluyíos – Asturian" lang="ast" hreflang="ast" data-title="Mecánica de fluyíos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Hidroaeromexanika" title="Hidroaeromexanika – Azerbaijani" lang="az" hreflang="az" data-title="Hidroaeromexanika" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%AC%E0%A6%BE%E0%A6%B9%E0%A7%80_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="প্রবাহী বলবিজ্ঞান – Bangla" lang="bn" hreflang="bn" data-title="প্রবাহী বলবিজ্ঞান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Li%C3%BB-th%C3%A9_le%CC%8Dk-ha%CC%8Dk" title="Liû-thé le̍k-ha̍k – Minnan" lang="nan" hreflang="nan" data-title="Liû-thé le̍k-ha̍k" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%96%D0%B4%D1%80%D0%B0%D0%B0%D1%8D%D1%80%D0%B0%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Гідрааэрамеханіка – Belarusian" lang="be" hreflang="be" data-title="Гідрааэрамеханіка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D1%84%D0%BB%D1%83%D0%B8%D0%B4%D0%B8%D1%82%D0%B5" title="Механика на флуидите – Bulgarian" lang="bg" hreflang="bg" data-title="Механика на флуидите" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Mehanika_fluida" title="Mehanika fluida – Bosnian" lang="bs" hreflang="bs" data-title="Mehanika fluida" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mec%C3%A0nica_dels_fluids" title="Mecànica dels fluids – Catalan" lang="ca" hreflang="ca" data-title="Mecànica dels fluids" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Mechanika_tekutin" title="Mechanika tekutin – Czech" lang="cs" hreflang="cs" data-title="Mechanika tekutin" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fluidmekanik" title="Fluidmekanik – Danish" lang="da" hreflang="da" data-title="Fluidmekanik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Str%C3%B6mungsmechanik" title="Strömungsmechanik – German" lang="de" hreflang="de" data-title="Strömungsmechanik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Vooliste_mehaanika" title="Vooliste mehaanika – Estonian" lang="et" hreflang="et" data-title="Vooliste mehaanika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE_%CF%84%CF%89%CE%BD_%CF%81%CE%B5%CF%85%CF%83%CF%84%CF%8E%CE%BD" title="Μηχανική των ρευστών – Greek" lang="el" hreflang="el" data-title="Μηχανική των ρευστών" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mec%C3%A1nica_de_fluidos" title="Mecánica de fluidos – Spanish" lang="es" hreflang="es" data-title="Mecánica de fluidos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Me%C4%A5aniko_de_fluidoj" title="Meĥaniko de fluidoj – Esperanto" lang="eo" hreflang="eo" data-title="Meĥaniko de fluidoj" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Fluidoen_mekanika" title="Fluidoen mekanika – Basque" lang="eu" hreflang="eu" data-title="Fluidoen mekanika" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%D8%B3%DB%8C%D8%A7%D9%84%D8%A7%D8%AA" title="مکانیک سیالات – Persian" lang="fa" hreflang="fa" data-title="مکانیک سیالات" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9canique_des_fluides" title="Mécanique des fluides – French" lang="fr" hreflang="fr" data-title="Mécanique des fluides" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Meicnic_shreabh%C3%A1nach" title="Meicnic shreabhánach – Irish" lang="ga" hreflang="ga" data-title="Meicnic shreabhánach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Mec%C3%A1nica_de_flu%C3%ADdos" title="Mecánica de fluídos – Galician" lang="gl" hreflang="gl" data-title="Mecánica de fluídos" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Li%C3%B9-th%C3%AD_li%CC%8Dt-ho%CC%8Dk" title="Liù-thí li̍t-ho̍k – Hakka Chinese" lang="hak" hreflang="hak" data-title="Liù-thí li̍t-ho̍k" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A0%EC%B2%B4%EC%97%AD%ED%95%99" title="유체역학 – Korean" lang="ko" hreflang="ko" data-title="유체역학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%AB%D5%A4%D6%80%D5%B8%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1" title="Հիդրոմեխանիկա – Armenian" lang="hy" hreflang="hy" data-title="Հիդրոմեխանիկա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A4%B0%E0%A4%B2_%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="तरल यांत्रिकी – Hindi" lang="hi" hreflang="hi" data-title="तरल यांत्रिकी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Mehanika_fluida" title="Mehanika fluida – Croatian" lang="hr" hreflang="hr" data-title="Mehanika fluida" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Fluido_mekaniko" title="Fluido mekaniko – Ido" lang="io" hreflang="io" data-title="Fluido mekaniko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mekanika_fluida" title="Mekanika fluida – Indonesian" lang="id" hreflang="id" data-title="Mekanika fluida" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Meccanica_dei_fluidi" title="Meccanica dei fluidi – Italian" lang="it" hreflang="it" data-title="Meccanica dei fluidi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%AA_%D7%94%D7%96%D7%95%D7%A8%D7%9E%D7%99%D7%9D" title="מכניקת הזורמים – Hebrew" lang="he" hreflang="he" data-title="מכניקת הזורמים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Mekanika_fluida" title="Mekanika fluida – Javanese" lang="jv" hreflang="jv" data-title="Mekanika fluida" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B8%D0%B4%D1%80%D0%BE%D0%B0%D1%8D%D1%80%D0%BE%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Гидроаэромеханика – Kazakh" lang="kk" hreflang="kk" data-title="Гидроаэромеханика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B8%D0%B4%D1%80%D0%BE%D0%B0%D1%8D%D1%80%D0%BE%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Гидроаэромеханика – Kyrgyz" lang="ky" hreflang="ky" data-title="Гидроаэромеханика" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Hidroaeromeh%C4%81nika" title="Hidroaeromehānika – Latvian" lang="lv" hreflang="lv" data-title="Hidroaeromehānika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%81raml%C3%A1stan" title="Áramlástan – Hungarian" lang="hu" hreflang="hu" data-title="Áramlástan" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A6%E0%B5%8D%E0%B4%B0%E0%B4%B5_%E0%B4%AC%E0%B4%B2%E0%B4%A4%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="ദ്രവ ബലതന്ത്രം – Malayalam" lang="ml" hreflang="ml" data-title="ദ്രവ ബലതന്ത്രം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B5%E0%A4%BE%E0%A4%B9_%E0%A4%B5%E0%A4%BF%E0%A4%9C%E0%A5%8D%E0%A4%9E%E0%A4%BE%E0%A4%A8" title="प्रवाह विज्ञान – Marathi" lang="mr" hreflang="mr" data-title="प्रवाह विज्ञान" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Mekanik_bendalir" title="Mekanik bendalir – Malay" lang="ms" hreflang="ms" data-title="Mekanik bendalir" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A8%D0%B8%D0%BD%D0%B3%D1%8D%D0%BD%D0%B8%D0%B9_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA" title="Шингэний механик – Mongolian" lang="mn" hreflang="mn" data-title="Шингэний механик" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stromingsleer" title="Stromingsleer – Dutch" lang="nl" hreflang="nl" data-title="Stromingsleer" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%B5%81%E4%BD%93%E5%8A%9B%E5%AD%A6" title="流体力学 – Japanese" lang="ja" hreflang="ja" data-title="流体力学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fluidmekanikk" title="Fluidmekanikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Fluidmekanikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fluidmekanikk" title="Fluidmekanikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Fluidmekanikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Fluid_mechanics" title="Fluid mechanics – Oromo" lang="om" hreflang="om" data-title="Fluid mechanics" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D9%85%D8%A7%DB%8C%D8%B9%D8%A7%D8%AA%D9%88_%D9%85%DB%8C%D8%AE%D8%A7%D9%86%DB%8C%DA%A9" title="د مایعاتو میخانیک – Pashto" lang="ps" hreflang="ps" data-title="د مایعاتو میخانیک" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Mechanika_p%C5%82yn%C3%B3w" title="Mechanika płynów – Polish" lang="pl" hreflang="pl" data-title="Mechanika płynów" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Mec%C3%A2nica_dos_fluidos" title="Mecânica dos fluidos – Portuguese" lang="pt" hreflang="pt" data-title="Mecânica dos fluidos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Mecanica_fluidelor" title="Mecanica fluidelor – Romanian" lang="ro" hreflang="ro" data-title="Mecanica fluidelor" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B8%D0%B4%D1%80%D0%BE%D0%B0%D1%8D%D1%80%D0%BE%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Гидроаэромеханика – Russian" lang="ru" hreflang="ru" data-title="Гидроаэромеханика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Mekanika_e_fluideve" title="Mekanika e fluideve – Albanian" lang="sq" hreflang="sq" data-title="Mekanika e fluideve" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B6%BB%E0%B6%BD_%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%8A%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%80" title="තරල යාන්ත්‍ර විද්‍යාව – Sinhala" lang="si" hreflang="si" data-title="තරල යාන්ත්‍ර විද්‍යාව" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fluid_mechanics" title="Fluid mechanics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Fluid mechanics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Mechanika_tekut%C3%ADn" title="Mechanika tekutín – Slovak" lang="sk" hreflang="sk" data-title="Mechanika tekutín" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Mehanika_teko%C4%8Din" title="Mehanika tekočin – Slovenian" lang="sl" hreflang="sl" data-title="Mehanika tekočin" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Mehanika_fluida" title="Mehanika fluida – Serbian" lang="sr" hreflang="sr" data-title="Mehanika fluida" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Mehanika_fluida" title="Mehanika fluida – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Mehanika fluida" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Virtausmekaniikka" title="Virtausmekaniikka – Finnish" lang="fi" hreflang="fi" data-title="Virtausmekaniikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Str%C3%B6mningsmekanik" title="Strömningsmekanik – Swedish" lang="sv" hreflang="sv" data-title="Strömningsmekanik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%BE%E0%AE%AF%E0%AF%8D%E0%AE%AE_%E0%AE%B5%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="பாய்ம விசையியல் – Tamil" lang="ta" hreflang="ta" data-title="பாய்ம விசையியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Tamikanikt_n_iccig" title="Tamikanikt n iccig – Tachelhit" lang="shi" hreflang="shi" data-title="Tamikanikt n iccig" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%84%E0%B8%AB%E0%B8%A5" title="กลศาสตร์ของไหล – Thai" lang="th" hreflang="th" data-title="กลศาสตร์ของไหล" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Ak%C4%B1%C5%9Fkanlar_mekani%C4%9Fi" title="Akışkanlar mekaniği – Turkish" lang="tr" hreflang="tr" data-title="Akışkanlar mekaniği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%96%D0%B4%D1%80%D0%BE%D0%B0%D0%B5%D1%80%D0%BE%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Гідроаеромеханіка – Ukrainian" lang="uk" hreflang="uk" data-title="Гідроаеромеханіка" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B3%DB%8C%D8%A7%D9%84%DB%8C_%D9%85%DB%8C%DA%A9%D8%A7%D9%86%DB%8C%D8%A7%D8%AA" title="سیالی میکانیات – Urdu" lang="ur" hreflang="ur" data-title="سیالی میکانیات" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_ch%E1%BA%A5t_l%C6%B0u" title="Cơ học chất lưu – Vietnamese" lang="vi" hreflang="vi" data-title="Cơ học chất lưu" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pluwido_nga_mekanika" title="Pluwido nga mekanika – Waray" lang="war" hreflang="war" data-title="Pluwido nga mekanika" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%B5%81%E4%BD%93%E5%8A%9B%E5%AD%A6" title="流体力学 – Wu" lang="wuu" hreflang="wuu" data-title="流体力学" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B5%81%E9%AB%94%E5%8A%9B%E5%AD%B8" title="流體力學 – Cantonese" lang="yue" hreflang="yue" data-title="流體力學" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B5%81%E4%BD%93%E5%8A%9B%E5%AD%A6" title="流体力学 – Chinese" lang="zh" hreflang="zh" data-title="流体力学" data-language-autonym="中文" data-language-local-name="Chinese" 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D{\frac {d\varphi }{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1856f88def2056f28ed27c7d31180a6240820ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.874ex; height:5.509ex;" alt="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"></span><div class="sidebar-caption"><a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></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)">Laws</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Conservations</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Mass</a></li> <li><a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">Momentum</a></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Energy</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Inequalities</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Clausius–Duhem (entropy)</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)"><a href="/wiki/Solid_mechanics" title="Solid mechanics">Solid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Deformation_(physics)" title="Deformation (physics)">Deformation</a></li> <li><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a> <ul><li><a href="/wiki/Linear_elasticity" title="Linear elasticity">linear</a></li></ul></li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a></li> <li><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a></li> <li><a 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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 class="mw-selflink selflink">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> <b>·</b> <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Dynamics</a></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a> <b>·</b> <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> <b>·</b> <a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a></li> <li><a href="/wiki/Viscosity" title="Viscosity">Viscosity</a> <ul><li>(<a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a> <b>·</b> <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> <b>·</b> <a href="/wiki/Mixing_(process_engineering)" title="Mixing (process engineering)">Mixing</a> <b>·</b> <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> <p><b>Fluid mechanics</b> is the branch of <a href="/wiki/Physics" title="Physics">physics</a> concerned with the <a href="/wiki/Mechanics" title="Mechanics">mechanics</a> of <a href="/wiki/Fluid" title="Fluid">fluids</a> (<a href="/wiki/Liquid" title="Liquid">liquids</a>, <a href="/wiki/Gas" title="Gas">gases</a>, and <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasmas</a>) and the <a href="/wiki/Force" title="Force">forces</a> on them.<sup id="cite_ref-White2011_1-0" class="reference"><a href="#cite_note-White2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 3">: 3 </span></sup> It has applications in a wide range of disciplines, including <a href="/wiki/Mechanical_engineering" title="Mechanical engineering">mechanical</a>, <a href="/wiki/Aerospace_engineering" title="Aerospace engineering">aerospace</a>, <a href="/wiki/Civil_engineering" title="Civil engineering">civil</a>, <a href="/wiki/Chemical_engineering" title="Chemical engineering">chemical</a>, and <a href="/wiki/Biomedical_engineering" title="Biomedical engineering">biomedical engineering</a>, as well as <a href="/wiki/Geophysics" title="Geophysics">geophysics</a>, <a href="/wiki/Oceanography" title="Oceanography">oceanography</a>, <a href="/wiki/Meteorology" title="Meteorology">meteorology</a>, <a href="/wiki/Astrophysics" title="Astrophysics">astrophysics</a>, and <a href="/wiki/Biology" title="Biology">biology</a>. </p><p>It can be divided into <a href="/wiki/Fluid_statics" class="mw-redirect" title="Fluid statics">fluid statics</a>, the study of fluids at rest; and <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, the study of the effect of forces on fluid motion.<sup id="cite_ref-White2011_1-1" class="reference"><a href="#cite_note-White2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 3">: 3 </span></sup> It is a branch of <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a <i>macroscopic</i> viewpoint rather than from <i>microscopic</i>. </p><p>Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved and are best addressed by <a href="/wiki/Numerical_methods" class="mw-redirect" title="Numerical methods">numerical methods</a>, typically using computers. A modern discipline, called <a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a> (CFD), is devoted to this approach.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Particle_image_velocimetry" title="Particle image velocimetry">Particle image velocimetry</a>, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/History_of_fluid_mechanics" title="History of fluid mechanics">History of fluid mechanics</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For a chronological guide, see <a href="/wiki/Timeline_of_fluid_and_continuum_mechanics" title="Timeline of fluid and continuum mechanics">Timeline of fluid and continuum mechanics</a>.</div> <p>The study of fluid mechanics goes back at least to the days of <a href="/wiki/Ancient_Greece" title="Ancient Greece">ancient Greece</a>, when <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> investigated fluid statics and <a href="/wiki/Buoyancy" title="Buoyancy">buoyancy</a> and formulated his famous law known now as the <a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a>, which was published in his work <i><a href="/wiki/On_Floating_Bodies" title="On Floating Bodies">On Floating Bodies</a></i>—generally considered to be the first major work on fluid mechanics. Iranian scholar <a href="/wiki/Abu_Rayhan_Biruni" class="mw-redirect" title="Abu Rayhan Biruni">Abu Rayhan Biruni</a> and later <a href="/wiki/Al-Khazini" title="Al-Khazini">Al-Khazini</a> applied <a href="/wiki/Experiment" title="Experiment">experimental</a> <a href="/wiki/Scientific_method" title="Scientific method">scientific methods</a> to fluid mechanics.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Rapid advancement in fluid mechanics began with <a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> (observations and experiments), <a href="/wiki/Evangelista_Torricelli" title="Evangelista Torricelli">Evangelista Torricelli</a> (invented the <a href="/wiki/Barometer" title="Barometer">barometer</a>), <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> (investigated <a href="/wiki/Viscosity" title="Viscosity">viscosity</a>) and <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> (researched <a href="/wiki/Hydrostatics" title="Hydrostatics">hydrostatics</a>, formulated <a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a>), and was continued by <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> with the introduction of mathematical fluid dynamics in <i>Hydrodynamica</i> (1739). </p><p>Inviscid flow was further analyzed by various mathematicians (<a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">Jean le Rond d'Alembert</a>, <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a>, <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>, <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a>) and viscous flow was explored by a multitude of <a href="/wiki/Engineers" class="mw-redirect" title="Engineers">engineers</a> including <a href="/wiki/Jean_L%C3%A9onard_Marie_Poiseuille" title="Jean Léonard Marie Poiseuille">Jean Léonard Marie Poiseuille</a> and <a href="/wiki/Gotthilf_Hagen" title="Gotthilf Hagen">Gotthilf Hagen</a>. Further mathematical justification was provided by <a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Claude-Louis Navier</a> and <a href="/wiki/George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">George Gabriel Stokes</a> in the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a>, and <a href="/wiki/Boundary_layers" class="mw-redirect" title="Boundary layers">boundary layers</a> were investigated (<a href="/wiki/Ludwig_Prandtl" title="Ludwig Prandtl">Ludwig Prandtl</a>, <a href="/wiki/Theodore_von_K%C3%A1rm%C3%A1n" title="Theodore von Kármán">Theodore von Kármán</a>), while various scientists such as <a href="/wiki/Osborne_Reynolds" title="Osborne Reynolds">Osborne Reynolds</a>, <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a>, and <a href="/wiki/Geoffrey_Ingram_Taylor" class="mw-redirect" title="Geoffrey Ingram Taylor">Geoffrey Ingram Taylor</a> advanced the understanding of fluid viscosity and <a href="/wiki/Turbulence" title="Turbulence">turbulence</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Main_branches">Main branches</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=2" title="Edit section: Main branches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Fluid_statics">Fluid statics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=3" title="Edit section: Fluid statics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Fluid_statics" class="mw-redirect" title="Fluid statics">Fluid statics</a></div> <p><a href="/wiki/Fluid_statics" class="mw-redirect" title="Fluid statics">Fluid statics</a> or <b>hydrostatics</b> is the branch of fluid mechanics that studies <a href="/wiki/Fluid" title="Fluid">fluids</a> at rest. It embraces the study of the conditions under which fluids are at rest in <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">stable</a> <a href="/wiki/Hydrostatic_equilibrium" title="Hydrostatic equilibrium">equilibrium</a>; and is contrasted with <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, the study of fluids in motion. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why <a href="/wiki/Atmospheric_pressure" title="Atmospheric pressure">atmospheric pressure</a> changes with <a href="/wiki/Altitude" title="Altitude">altitude</a>, why wood and <a href="/wiki/Oil" title="Oil">oil</a> float on water, and why the surface of water is always level whatever the shape of its container. Hydrostatics is fundamental to <a href="/wiki/Hydraulics" title="Hydraulics">hydraulics</a>, the <a href="/wiki/Engineering" title="Engineering">engineering</a> of equipment for storing, transporting and using <a href="/wiki/Fluids" class="mw-redirect" title="Fluids">fluids</a>. It is also relevant to some aspects of <a href="/wiki/Geophysics" title="Geophysics">geophysics</a> and <a href="/wiki/Astrophysics" title="Astrophysics">astrophysics</a> (for example, in understanding <a href="/wiki/Plate_tectonics" title="Plate tectonics">plate tectonics</a> and anomalies in the <a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">Earth's gravitational field</a>), to <a href="/wiki/Meteorology" title="Meteorology">meteorology</a>, to <a href="/wiki/Medicine" title="Medicine">medicine</a> (in the context of <a href="/wiki/Blood_pressure" title="Blood pressure">blood pressure</a>), and many other fields. </p> <div class="mw-heading mw-heading3"><h3 id="Fluid_dynamics">Fluid dynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=4" title="Edit section: Fluid dynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Fluid_dynamics" title="Fluid dynamics">Fluid dynamics</a></div> <p><i><a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Fluid dynamics</a></i> is a subdiscipline of fluid mechanics that deals with <i>fluid flow</i>—the science of liquids and gases in motion.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Fluid dynamics offers a systematic structure—which underlies these <a href="/wiki/Practical_disciplines" class="mw-redirect" title="Practical disciplines">practical disciplines</a>—that embraces empirical and semi-empirical laws derived from <a href="/wiki/Flow_measurement" title="Flow measurement">flow measurement</a> and used to solve practical problems. The solution to a <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a> problem typically involves calculating various properties of the fluid, such as <a href="/wiki/Velocity" title="Velocity">velocity</a>, <a href="/wiki/Pressure" title="Pressure">pressure</a>, <a href="/wiki/Density" title="Density">density</a>, and <a href="/wiki/Temperature" title="Temperature">temperature</a>, as functions of space and time. It has several subdisciplines itself, including <i><a href="/wiki/Aerodynamics" title="Aerodynamics">aerodynamics</a></i><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> (the study of air and other gases in motion) and <i>hydrodynamics</i><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating <a href="/wiki/Force" title="Force">forces</a> and <a href="/wiki/Moment_(physics)" title="Moment (physics)">movements</a> on <a href="/wiki/Aircraft" title="Aircraft">aircraft</a>, determining the <a href="/wiki/Mass_flow_rate" title="Mass flow rate">mass flow rate</a> of <a href="/wiki/Petroleum" title="Petroleum">petroleum</a> through pipelines, predicting evolving <a href="/wiki/Weather" title="Weather">weather</a> patterns, understanding <a href="/wiki/Nebula" title="Nebula">nebulae</a> in <a href="/wiki/Interstellar_space" class="mw-redirect" title="Interstellar space">interstellar space</a> and modeling <a href="/wiki/Explosions" class="mw-redirect" title="Explosions">explosions</a>. Some fluid-dynamical principles are used in <a href="/wiki/Traffic_engineering_(transportation)" title="Traffic engineering (transportation)">traffic engineering</a> and crowd dynamics. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_continuum_mechanics">Relationship to continuum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=5" title="Edit section: Relationship to continuum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fluid mechanics is a subdiscipline of <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>, as illustrated in the following table. </p> <table class="wikitable"> <tbody><tr> <td rowspan="4"><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a><br /><small>The study of the physics of continuous materials</small> </td> <td rowspan="2"><a href="/wiki/Solid_mechanics" title="Solid mechanics">Solid mechanics</a><br /><small>The study of the physics of continuous materials with a defined rest shape.</small> </td> <td colspan="2"><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a><br /><small>Describes materials that return to their rest shape after applied <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stresses</a> are removed.</small> </td></tr> <tr> <td><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a><br /><small>Describes materials that permanently deform after a sufficient applied stress.</small> </td> <td rowspan="2"><a href="/wiki/Rheology" title="Rheology">Rheology</a><br /><small>The study of materials with both solid and fluid characteristics.</small> </td></tr> <tr> <td rowspan="2"><b>Fluid mechanics</b><br /><small>The study of the physics of continuous materials which deform when subjected to a force.</small> </td> <td><a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">Non-Newtonian fluid</a><br /><small>Do not undergo strain rates proportional to the applied shear stress.</small> </td></tr> <tr> <td colspan="2"><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluids</a> undergo strain rates proportional to the applied shear stress. </td></tr></tbody></table> <p>In a mechanical view, a fluid is a substance that does not support <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a>; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress. </p> <div class="mw-heading mw-heading2"><h2 id="Assumptions">Assumptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=6" title="Edit section: Assumptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Reynolds.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Reynolds.svg/220px-Reynolds.svg.png" decoding="async" width="220" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Reynolds.svg/330px-Reynolds.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Reynolds.svg/440px-Reynolds.svg.png 2x" data-file-width="256" data-file-height="210" /></a><figcaption>Balance for some integrated fluid quantity in a <a href="/wiki/Control_volume" title="Control volume">control volume</a> enclosed by a <a href="/wiki/Control_surface_(fluid_dynamics)" class="mw-redirect" title="Control surface (fluid dynamics)">control surface</a>.</figcaption></figure> <p>The assumptions inherent to a fluid mechanical treatment of a physical system can be expressed in terms of mathematical equations. Fundamentally, every fluid mechanical system is assumed to obey: </p> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Conservation of mass</a></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Conservation of energy</a></li> <li><a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">Conservation of momentum</a></li> <li><a href="/wiki/Knudsen_number" title="Knudsen number">The continuum assumption</a></li></ul> <p>For example, the assumption that mass is conserved means that for any fixed <a href="/wiki/Control_volume" title="Control volume">control volume</a> (for example, a spherical volume)—enclosed by a <a href="/wiki/Control_surface_(fluid_dynamics)" class="mw-redirect" title="Control surface (fluid dynamics)">control surface</a>—the <a href="/wiki/Derivative" title="Derivative">rate of change</a> of the mass contained in that volume is equal to the rate at which mass is passing through the surface from <i>outside</i> to <i>inside</i>, minus the rate at which mass is passing from <i>inside</i> to <i>outside</i>. This can be expressed as an <a href="/wiki/Continuity_equation#Integral_form" title="Continuity equation">equation in integral form</a> over the control volume.<sup id="cite_ref-Batchelor1967_11-0" class="reference"><a href="#cite_note-Batchelor1967-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 74">: 74 </span></sup> </p><p>The <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="continuum_assumption"></span><span id="Continuum_assumption"></span><span class="vanchor-text">continuum assumption</span></span></b> is an idealization of <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a> under which fluids can be treated as <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>, even though, on a microscopic scale, they are composed of <a href="/wiki/Molecules" class="mw-redirect" title="Molecules">molecules</a>. Under the continuum assumption, macroscopic (observed/measurable) properties such as density, pressure, temperature, and bulk velocity are taken to be well-defined at "infinitesimal" volume elements—small in comparison to the characteristic length scale of the system, but large in comparison to molecular length scale. Fluid properties can vary continuously from one volume element to another and are average values of the molecular properties. The continuum hypothesis can lead to inaccurate results in applications like supersonic speed flows, or molecular flows on nano scale.<sup id="cite_ref-Greenkorn2018_12-0" class="reference"><a href="#cite_note-Greenkorn2018-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Those problems for which the continuum hypothesis fails can be solved using <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>. To determine whether or not the continuum hypothesis applies, the <a href="/wiki/Knudsen_number" title="Knudsen number">Knudsen number</a>, defined as the ratio of the molecular <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a> to the characteristic length <a href="/wiki/Scale_(ratio)" title="Scale (ratio)">scale</a>, is evaluated. Problems with Knudsen numbers below 0.1 can be evaluated using the continuum hypothesis, but molecular approach (statistical mechanics) can be applied to find the fluid motion for larger Knudsen numbers. <span class="anchor" id="mathematical_fluid_mechanics"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Navier–Stokes_equations"><span id="Navier.E2.80.93Stokes_equations"></span>Navier–Stokes equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=7" title="Edit section: Navier–Stokes equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></div> <p>The <b>Navier–Stokes equations</b> (named after <a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Claude-Louis Navier</a> and <a href="/wiki/George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">George Gabriel Stokes</a>) are <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> that describe the force balance at a given point within a fluid. For an <a href="/wiki/Incompressible_fluid" class="mw-redirect" title="Incompressible fluid">incompressible fluid</a> with vector velocity field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"></span>, the Navier–Stokes equations are<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\nu \nabla ^{2}\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mi>ν</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\nu \nabla ^{2}\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/780aca0d22ef8974dc94f715b0f31b2a2f114cba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.473ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\nu \nabla ^{2}\mathbf {u} }"></span>.</dd></dl> <p>These differential equations are the analogues for deformable materials to Newton's equations of motion for particles – the Navier–Stokes equations describe changes in <a href="/wiki/Momentum" title="Momentum">momentum</a> (<a href="/wiki/Force" title="Force">force</a>) in response to <a href="/wiki/Pressure" title="Pressure">pressure</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> and viscosity, parameterized by the <a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>. Occasionally, <a href="/wiki/Body_force" title="Body force">body forces</a>, such as the gravitational force or Lorentz force are added to the equations. </p><p>Solutions of the Navier–Stokes equations for a given physical problem must be sought with the help of <a href="/wiki/Calculus" title="Calculus">calculus</a>. In practical terms, only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow in which the <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> is small. For more complex cases, especially those involving <a href="/wiki/Turbulence" title="Turbulence">turbulence</a>, such as global weather systems, aerodynamics, hydrodynamics and many more, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This branch of science is called <a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Inviscid_and_viscous_fluids">Inviscid and viscous fluids</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=8" title="Edit section: Inviscid and viscous fluids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <b>inviscid fluid</b> has no <a href="/wiki/Viscosity" title="Viscosity">viscosity</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7229c47b5bc20ef0a1371a4f3c09459ccb6909ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =0}"></span>. In practice, an inviscid flow is an <a href="/wiki/Ideal_fluid" class="mw-redirect" title="Ideal fluid">idealization</a>, one that facilitates mathematical treatment. In fact, purely inviscid flows are only known to be realized in the case of <a href="/wiki/Superfluidity" title="Superfluidity">superfluidity</a>. Otherwise, fluids are generally <b>viscous</b>, a property that is often most important within a <a href="/wiki/Boundary_layer" title="Boundary layer">boundary layer</a> near a solid surface,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> where the flow must match onto the <a href="/wiki/No-slip_condition" title="No-slip condition">no-slip condition</a> at the solid. In some cases, the mathematics of a fluid mechanical system can be treated by assuming that the fluid outside of boundary layers is inviscid, and then <a href="/wiki/Method_of_matched_asymptotic_expansions" title="Method of matched asymptotic expansions">matching</a> its solution onto that for a thin <a href="/wiki/Laminar_flow" title="Laminar flow">laminar</a> boundary layer. </p><p>For fluid flow over a porous boundary, the fluid velocity can be discontinuous between the free fluid and the fluid in the porous media (this is related to the Beavers and Joseph condition). Further, it is useful at low <a href="/wiki/Speed_of_sound" title="Speed of sound">subsonic</a> speeds to assume that gas is <a href="/wiki/Incompressible_fluid" class="mw-redirect" title="Incompressible fluid">incompressible</a>—that is, the density of the gas does not change even though the speed and <a href="/wiki/Static_pressure" title="Static pressure">static pressure</a> change. </p> <div class="mw-heading mw-heading2"><h2 id="Newtonian_versus_non-Newtonian_fluids">Newtonian versus non-Newtonian fluids</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=9" title="Edit section: Newtonian versus non-Newtonian fluids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>Newtonian fluid</b> (named after <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>) is defined to be a <a href="/wiki/Fluid" title="Fluid">fluid</a> whose <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a> is linearly proportional to the <a href="/wiki/Velocity" title="Velocity">velocity</a> <a href="/wiki/Gradient" title="Gradient">gradient</a> in the direction <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the plane of shear. This definition means regardless of the forces acting on a fluid, it <i>continues to flow</i>. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the <a href="/wiki/Drag_(physics)" title="Drag (physics)">drag</a> of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare <a href="/wiki/Friction" title="Friction">friction</a>). Important fluids, like water as well as most gasses, behave—to good approximation—as a Newtonian fluid under normal conditions on Earth.<sup id="cite_ref-Batchelor1967_11-1" class="reference"><a href="#cite_note-Batchelor1967-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 145">: 145 </span></sup> </p><p>By contrast, stirring a <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian fluid</a> can leave a "hole" behind. This will gradually fill up over time—this behavior is seen in materials such as pudding, <a href="/wiki/Non-newtonian_fluid#Oobleck" class="mw-redirect" title="Non-newtonian fluid">oobleck</a>, or <a href="/wiki/Sand" title="Sand">sand</a> (although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip <a href="/wiki/Paint" title="Paint">paints</a>). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property—for example, most fluids with long molecular chains can react in a non-Newtonian manner.<sup id="cite_ref-Batchelor1967_11-2" class="reference"><a href="#cite_note-Batchelor1967-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 145">: 145 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Equations_for_a_Newtonian_fluid">Equations for a Newtonian fluid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=10" title="Edit section: Equations for a Newtonian fluid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a></div> <p>The constant of proportionality between the viscous stress tensor and the velocity gradient is known as the <a href="/wiki/Viscosity" title="Viscosity">viscosity</a>. A simple equation to describe incompressible Newtonian fluid behavior is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =-\mu {\frac {\mathrm {d} u}{\mathrm {d} n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>u</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =-\mu {\frac {\mathrm {d} u}{\mathrm {d} n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08fa87d1e22b9c1c87b540b3258df91c828a4be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.034ex; height:5.509ex;" alt="{\displaystyle \tau =-\mu {\frac {\mathrm {d} u}{\mathrm {d} n}}}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> is the shear stress exerted by the fluid ("<a href="/wiki/Drag_(physics)" title="Drag (physics)">drag</a>"),</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> is the fluid viscosity—a constant of proportionality, and</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>u</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/171086ff60505f3030f24a15be159444e16de0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.523ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} n}}}"></span> is the velocity gradient perpendicular to the direction of shear.</dd></dl> <p>For a Newtonian fluid, the viscosity, by definition, depends only on <a href="/wiki/Temperature" title="Temperature">temperature</a>, not on the forces acting upon it. If the fluid is <a href="/wiki/Incompressible_fluid" class="mw-redirect" title="Incompressible fluid">incompressible</a> the equation governing the viscous stress (in <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinates</a>) is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)}"> <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> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da69264bc1a85e120443b36251fb5f348bc351d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.319ex; height:6.509ex;" alt="{\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span> is the shear stress on the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{th}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{th}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c454bb35f050556d361ee85e06ca923b16a3bf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.575ex; height:2.676ex;" alt="{\displaystyle i^{th}}"></span> face of a fluid element in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{th}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j^{th}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bfd24f79d376ca8156511feb951588f358a928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.758ex; height:3.009ex;" alt="{\displaystyle j^{th}}"></span> direction</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span> is the velocity in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{th}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{th}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c454bb35f050556d361ee85e06ca923b16a3bf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.575ex; height:2.676ex;" alt="{\displaystyle i^{th}}"></span> direction</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db47cb3d2f9496205a17a6856c91c1d3d363ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}"></span> is the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{th}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j^{th}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bfd24f79d376ca8156511feb951588f358a928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.758ex; height:3.009ex;" alt="{\displaystyle j^{th}}"></span> direction coordinate.</dd></dl> <p>If the fluid is not incompressible the general form for the viscous stress in a Newtonian fluid is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{ij}\nabla \cdot \mathbf {v} \right)+\kappa \delta _{ij}\nabla \cdot \mathbf {v} }"> <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> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>κ</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{ij}\nabla \cdot \mathbf {v} \right)+\kappa \delta _{ij}\nabla \cdot \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacbe693989661ada1e626b1252ef0e2e1d2a20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.409ex; height:6.509ex;" alt="{\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{ij}\nabla \cdot \mathbf {v} \right)+\kappa \delta _{ij}\nabla \cdot \mathbf {v} }"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> is the second viscosity coefficient (or bulk viscosity). If a fluid does not obey this relation, it is termed a <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian fluid</a>, of which there are several types. Non-Newtonian fluids can be either plastic, Bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic. </p><p>In some applications, another rough broad division among fluids is made: ideal and non-ideal fluids. An ideal fluid is non-viscous and offers no resistance whatsoever to a shearing force. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. One example of this is the flow far from solid surfaces. In many cases, the viscous effects are concentrated near the solid boundaries (such as in boundary layers) while in regions of the flow field far away from the boundaries the viscous effects can be neglected and the fluid there is treated as it were inviscid (ideal flow). When the viscosity is neglected, the term containing the viscous stress tensor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\tau } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\tau } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc46f8785eec5db11fc73af8c5f02473cd0fd71" 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 \mathbf {\tau } }"></span> in the Navier–Stokes equation vanishes. The equation reduced in this form is called the <a href="/wiki/Euler_equations_(fluid_dynamics)" title="Euler equations (fluid dynamics)">Euler equation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output 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method</a></li> <li><a href="/wiki/Stokesian_dynamics" title="Stokesian dynamics">Stokesian dynamics</a></li> <li><a href="/wiki/Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed-particle hydrodynamics</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-White2011-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-White2011_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-White2011_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><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 id="CITEREFWhite2011" class="citation book cs1">White, Frank M. (2011). <i>Fluid Mechanics</i> (7th ed.). McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352934-9" title="Special:BookSources/978-0-07-352934-9"><bdi>978-0-07-352934-9</bdi></a>.</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=7th&rft.pub=McGraw-Hill&rft.date=2011&rft.isbn=978-0-07-352934-9&rft.aulast=White&rft.aufirst=Frank+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuYeohLiu2012" class="citation book cs1">Tu, Jiyuan; Yeoh, Guan Heng; Liu, Chaoqun (Nov 21, 2012). <i>Computational Fluid Dynamics: A Practical Approach</i>. Butterworth-Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0080982434" title="Special:BookSources/978-0080982434"><bdi>978-0080982434</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Fluid+Dynamics%3A+A+Practical+Approach&rft.pub=Butterworth-Heinemann&rft.date=2012-11-21&rft.isbn=978-0080982434&rft.aulast=Tu&rft.aufirst=Jiyuan&rft.au=Yeoh%2C+Guan+Heng&rft.au=Liu%2C+Chaoqun&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642,</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Batchelor, C. K., & Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge University Press.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Bertin, J. J., & Smith, M. L. (1998). Aerodynamics for engineers (Vol. 5). Upper Saddle River, NJ: Prentice Hall.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Anderson Jr, J. D. (2010). Fundamentals of aerodynamics. Tata McGraw-Hill Education.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Houghton, E. L., & Carpenter, P. W. (2003). Aerodynamics for engineering students. Elsevier.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Milne-Thomson, L. M. (1973). Theoretical aerodynamics. Courier Corporation.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Milne-Thomson, L. M. (1996). Theoretical hydrodynamics. Courier Corporation.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Birkhoff, G. (2015). Hydrodynamics. Princeton University Press.</span> </li> <li id="cite_note-Batchelor1967-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Batchelor1967_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Batchelor1967_11-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Batchelor1967_11-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBatchelor1967" class="citation book cs1"><a href="/wiki/George_Batchelor" title="George Batchelor">Batchelor, George K.</a> (1967). <i>An Introduction to Fluid Dynamics</i>. Cambridge University Press. p. 74. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-66396-2" title="Special:BookSources/0-521-66396-2"><bdi>0-521-66396-2</bdi></a>.</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+Fluid+Dynamics&rft.pages=74&rft.pub=Cambridge+University+Press&rft.date=1967&rft.isbn=0-521-66396-2&rft.aulast=Batchelor&rft.aufirst=George+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Greenkorn2018-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Greenkorn2018_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreenkorn2018" class="citation book cs1">Greenkorn, Robert (3 October 2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pjFRDwAAQBAJ&q=%22Breakdown+of+continuum+assumption%22&pg=PA18"><i>Momentum, Heat, and Mass Transfer Fundamentals</i></a>. CRC Press. p. 18. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4822-9297-8" title="Special:BookSources/978-1-4822-9297-8"><bdi>978-1-4822-9297-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Momentum%2C+Heat%2C+and+Mass+Transfer+Fundamentals&rft.pages=18&rft.pub=CRC+Press&rft.date=2018-10-03&rft.isbn=978-1-4822-9297-8&rft.aulast=Greenkorn&rft.aufirst=Robert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpjFRDwAAQBAJ%26q%3D%2522Breakdown%2Bof%2Bcontinuum%2Bassumption%2522%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Constantin, P., & Foias, C. (1988). Navier-stokes equations. University of Chicago Press.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Temam, R. (2001). Navier-Stokes equations: theory and numerical analysis (Vol. 343). <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Foias, C., Manley, O., Rosa, R., & Temam, R. (2001). Navier-Stokes equations and turbulence (Vol. 83). Cambridge University Press.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Girault, V., & Raviart, P. A. (2012). Finite element methods for Navier-Stokes equations: theory and algorithms (Vol. 5). Springer Science & Business Media.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Anderson, J. D., & Wendt, J. (1995). Computational fluid dynamics (Vol. 206). New York: McGraw-Hill.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Chung, T. J. (2010). Computational fluid dynamics. Cambridge University Press.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Blazek, J. (2015). Computational fluid dynamics: principles and applications. Butterworth-Heinemann.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Wesseling, P. (2009). Principles of computational fluid dynamics (Vol. 29). Springer Science & Business Media.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Anderson, D., Tannehill, J. C., & Pletcher, R. H. (2016). Computational fluid mechanics and heat transfer. Taylor & Francis.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKunduCohenDowling2015" class="citation book cs1">Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R. (27 March 2015). "10". <i>Fluid Mechanics</i> (6th ed.). Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0124059351" title="Special:BookSources/978-0124059351"><bdi>978-0124059351</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=10&rft.btitle=Fluid+Mechanics&rft.edition=6th&rft.pub=Academic+Press&rft.date=2015-03-27&rft.isbn=978-0124059351&rft.aulast=Kundu&rft.aufirst=Pijush+K.&rft.au=Cohen%2C+Ira+M.&rft.au=Dowling%2C+David+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFalkovich2011" class="citation cs2">Falkovich, Gregory (2011), <i>Fluid Mechanics (A short course for physicists)</i>, Cambridge University Press, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511794353">10.1017/CBO9780511794353</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-00575-4" title="Special:BookSources/978-1-107-00575-4"><bdi>978-1-107-00575-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics+%28A+short+course+for+physicists%29&rft.pub=Cambridge+University+Press&rft.date=2011&rft_id=info%3Adoi%2F10.1017%2FCBO9780511794353&rft.isbn=978-1-107-00575-4&rft.aulast=Falkovich&rft.aufirst=Gregory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKunduCohen2008" class="citation cs2">Kundu, Pijush K.; Cohen, Ira M. (2008), <i>Fluid Mechanics</i> (4th revised ed.), Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-373735-9" title="Special:BookSources/978-0-12-373735-9"><bdi>978-0-12-373735-9</bdi></a></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=4th+revised&rft.pub=Academic+Press&rft.date=2008&rft.isbn=978-0-12-373735-9&rft.aulast=Kundu&rft.aufirst=Pijush+K.&rft.au=Cohen%2C+Ira+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCurrie1974" class="citation cs2">Currie, I. G. (1974), <i>Fundamental Mechanics of Fluids</i>, <a href="/wiki/McGraw-Hill,_Inc." class="mw-redirect" title="McGraw-Hill, Inc.">McGraw-Hill, Inc.</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-015000-1" title="Special:BookSources/0-07-015000-1"><bdi>0-07-015000-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Mechanics+of+Fluids&rft.pub=McGraw-Hill%2C+Inc.&rft.date=1974&rft.isbn=0-07-015000-1&rft.aulast=Currie&rft.aufirst=I.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMasseyWard-Smith2005" class="citation cs2">Massey, B.; Ward-Smith, J. (2005), <i>Mechanics of Fluids</i> (8th ed.), Taylor & Francis, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-36206-1" title="Special:BookSources/978-0-415-36206-1"><bdi>978-0-415-36206-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Fluids&rft.edition=8th&rft.pub=Taylor+%26+Francis&rft.date=2005&rft.isbn=978-0-415-36206-1&rft.aulast=Massey&rft.aufirst=B.&rft.au=Ward-Smith%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNazarenko2014" class="citation cs2">Nazarenko, Sergey (2014), <i>Fluid Dynamics via Examples and Solutions</i>, CRC Press (Taylor & Francis group), <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-43-988882-7" title="Special:BookSources/978-1-43-988882-7"><bdi>978-1-43-988882-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Dynamics+via+Examples+and+Solutions&rft.pub=CRC+Press+%28Taylor+%26+Francis+group%29&rft.date=2014&rft.isbn=978-1-43-988882-7&rft.aulast=Nazarenko&rft.aufirst=Sergey&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFluid+mechanics" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fluid_mechanics&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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data-file-height="512" /></span></span></span><span class="sister-link"><a href="https://en.wikiversity.org/wiki/Fluid_Mechanics_for_Mechanical_Engineers" class="extiw" title="v:Fluid Mechanics for Mechanical Engineers">Resources</a> from Wikiversity</span></li></ul></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.freebookcentre.net/Physics/Fluid-Mechanics-Books.html">Free Fluid Mechanics books</a></li> <li><a rel="nofollow" class="external text" href="http://arjournals.annualreviews.org/loi/fluid">Annual Review of Fluid Mechanics</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090119222743/http://arjournals.annualreviews.org/loi/fluid">Archived</a> 2009-01-19 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.cfd-online.com/Wiki/Main_Page">CFDWiki</a> – the Computational Fluid Dynamics reference wiki.</li> <li><a rel="nofollow" class="external text" href="http://www.interactiveflows.com/downloads/">Educational Particle Image Velocimetry</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170803103757/http://www.interactiveflows.com/downloads/">Archived</a> 2017-08-03 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – resources and demonstrations</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 .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output 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title="Entrance length (fluid dynamics)">Entrance length</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Dimensionless_numbers_in_fluid_mechanics" title="Dimensionless numbers in fluid mechanics">Dimensionless numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Archimedes_number" title="Archimedes number">Archimedes</a></li> <li><a href="/wiki/Atwood_number" title="Atwood number">Atwood</a></li> <li><a href="/wiki/Bagnold_number" title="Bagnold number">Bagnold</a></li> <li><a href="/wiki/Bejan_number#Fluid_mechanics_and_heat_transfer" title="Bejan number">Bejan</a></li> <li><a href="/wiki/Biot_number" title="Biot number">Biot</a></li> <li><a href="/wiki/E%C3%B6tv%C3%B6s_number" title="Eötvös number">Bond</a></li> <li><a href="/wiki/Brinkman_number" title="Brinkman number">Brinkman</a></li> <li><a href="/wiki/Capillary_number" title="Capillary number">Capillary</a></li> <li><a href="/wiki/Cauchy_number" title="Cauchy number">Cauchy</a></li> <li><a href="/wiki/Chandrasekhar_number" title="Chandrasekhar number">Chandrasekhar</a></li> <li><a href="/wiki/Damk%C3%B6hler_numbers" title="Damköhler numbers">Damköhler</a></li> <li><a href="/wiki/Darcy_number" title="Darcy number">Darcy</a></li> <li><a href="/wiki/Dean_number" title="Dean number">Dean</a></li> <li><a href="/wiki/Deborah_number" title="Deborah number">Deborah</a></li> <li><a href="/wiki/Dukhin_number" title="Dukhin number">Dukhin</a></li> <li><a href="/wiki/Eckert_number" title="Eckert number">Eckert</a></li> <li><a href="/wiki/Ekman_number" title="Ekman number">Ekman</a></li> <li><a href="/wiki/E%C3%B6tv%C3%B6s_number" title="Eötvös number">Eötvös</a></li> <li><a href="/wiki/Euler_number_(physics)" title="Euler number (physics)">Euler</a></li> <li><a href="/wiki/Froude_number" title="Froude number">Froude</a></li> <li><a href="/wiki/Galilei_number" title="Galilei number">Galilei</a></li> <li><a href="/wiki/Graetz_number" title="Graetz number">Graetz</a></li> <li><a href="/wiki/Grashof_number" title="Grashof number">Grashof</a></li> <li><a href="/wiki/G%C3%B6rtler_vortices" title="Görtler vortices">Görtler</a></li> <li><a href="/wiki/Hagen_number" title="Hagen number">Hagen</a></li> <li><a href="/wiki/Iribarren_number" title="Iribarren number">Iribarren</a></li> <li><a href="/wiki/Kapitza_number" title="Kapitza number">Kapitza</a></li> <li><a href="/wiki/Keulegan%E2%80%93Carpenter_number" title="Keulegan–Carpenter number">Keulegan–Carpenter</a></li> <li><a href="/wiki/Knudsen_number" title="Knudsen number">Knudsen</a></li> <li><a href="/wiki/Laplace_number" title="Laplace number">Laplace</a></li> <li><a href="/wiki/Lewis_number" title="Lewis number">Lewis</a></li> <li><a href="/wiki/Mach_number" title="Mach number">Mach</a></li> <li><a href="/wiki/Marangoni_number" title="Marangoni number"> Marangoni</a></li> <li><a href="/wiki/Morton_number" title="Morton number">Morton</a></li> <li><a href="/wiki/Nusselt_number" title="Nusselt number">Nusselt</a></li> <li><a href="/wiki/Ohnesorge_number" title="Ohnesorge number">Ohnesorge</a></li> <li><a href="/wiki/P%C3%A9clet_number" title="Péclet number">Péclet</a></li> <li><a href="/wiki/Prandtl_number" title="Prandtl number">Prandtl</a> <ul><li><a href="/wiki/Magnetic_Prandtl_number" title="Magnetic Prandtl number">magnetic</a></li> <li><a href="/wiki/Turbulent_Prandtl_number" title="Turbulent Prandtl number">turbulent</a></li></ul></li> <li><a href="/wiki/Rayleigh_number" title="Rayleigh number">Rayleigh</a></li> <li><a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds</a> <ul><li><a href="/wiki/Magnetic_Reynolds_number" title="Magnetic Reynolds number">magnetic</a></li></ul></li> <li><a href="/wiki/Richardson_number" title="Richardson number">Richardson</a></li> <li><a href="/wiki/Roshko_number" title="Roshko number">Roshko</a></li> <li><a href="/wiki/Rossby_number" title="Rossby number">Rossby</a></li> <li><a href="/wiki/Rouse_number" title="Rouse number">Rouse</a></li> <li><a href="/wiki/Schmidt_number" title="Schmidt number">Schmidt</a></li> <li><a href="/wiki/Scruton_number" title="Scruton number">Scruton</a></li> <li><a href="/wiki/Sherwood_number" title="Sherwood number">Sherwood</a></li> <li><a href="/wiki/Shields_parameter" title="Shields parameter">Shields</a></li> <li><a href="/wiki/Stanton_number" title="Stanton number">Stanton</a></li> <li><a href="/wiki/Stokes_number" title="Stokes number">Stokes</a></li> <li><a href="/wiki/Strouhal_number" title="Strouhal number">Strouhal</a></li> <li><a href="/wiki/Stuart_number" title="Stuart number">Stuart</a></li> <li><a href="/wiki/Laplace_number" title="Laplace number">Suratman</a></li> <li><a href="/wiki/Taylor_number" title="Taylor number">Taylor</a></li> <li><a href="/wiki/Ursell_number" title="Ursell number">Ursell</a></li> <li><a href="/wiki/Weber_number" title="Weber number">Weber</a></li> <li><a href="/wiki/Weissenberg_number" title="Weissenberg number">Weissenberg</a></li> <li><a href="/wiki/Womersley_number" title="Womersley number">Womersley</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_branches_of_physics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li 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title="Applied physics">Applied</a> <ul><li><a href="/wiki/Engineering_physics" title="Engineering physics">Engineering</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Approaches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Experimental_physics" title="Experimental physics">Experimental</a></li> <li><a href="/wiki/Theoretical_physics" title="Theoretical physics">Theoretical</a> <ul><li><a href="/wiki/Computational_physics" title="Computational physics">Computational</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Classical_physics" title="Classical physics">Classical</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical 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