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Available in 17 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-17" 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">17 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%BD%D0%B0_%D0%94%D0%B5%D0%B2%D0%BE%D1%80%D0%B0" title="Число на Девора – Bulgarian" lang="bg" hreflang="bg" data-title="Число на Девора" 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/Deborin_broj" title="Deborin broj – Bosnian" lang="bs" hreflang="bs" data-title="Deborin broj" 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/Nombre_de_Deborah" title="Nombre de Deborah – Catalan" lang="ca" hreflang="ca" data-title="Nombre de Deborah" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Deborah-Zahl" title="Deborah-Zahl – German" lang="de" hreflang="de" data-title="Deborah-Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_Deborah" title="Número de Deborah – Spanish" lang="es" hreflang="es" data-title="Número de Deborah" 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/Debora_nombro" title="Debora nombro – Esperanto" lang="eo" hreflang="eo" data-title="Debora nombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AF%D8%A8%D9%88%D8%B1%D8%A7" title="عدد دبورا – Persian" lang="fa" hreflang="fa" data-title="عدد دبورا" 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/Nombre_de_Deborah" title="Nombre de Deborah – French" lang="fr" hreflang="fr" data-title="Nombre de Deborah" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8D%B0%EB%B2%84%EB%9F%AC_%EC%88%98" title="데버러 수 – Korean" lang="ko" hreflang="ko" data-title="데버러 수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_di_Deborah" title="Numero di Deborah – Italian" lang="it" hreflang="it" data-title="Numero di Deborah" 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%A1%D7%A4%D7%A8_%D7%93%D7%91%D7%95%D7%A8%D7%94" title="מספר דבורה – Hebrew" lang="he" hreflang="he" data-title="מספר דבורה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%87%E3%83%9C%E3%83%A9%E6%95%B0" title="デボラ数 – Japanese" lang="ja" hreflang="ja" data-title="デボラ数" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczba_Debory" title="Liczba Debory – Polish" lang="pl" hreflang="pl" data-title="Liczba Debory" 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/N%C3%BAmero_de_Deborah" title="Número de Deborah – Portuguese" lang="pt" hreflang="pt" data-title="Número de Deborah" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%94%D0%B5%D0%B1%D0%BE%D1%80%D1%8B" title="Число Деборы – Russian" lang="ru" hreflang="ru" data-title="Число Деборы" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Deborah_say%C4%B1s%C4%B1" title="Deborah sayısı – Turkish" lang="tr" hreflang="tr" data-title="Deborah sayısı" 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>The <b>Deborah number</b> (<b>De</b>) is a <a href="/wiki/Dimensionless_number" class="mw-redirect" title="Dimensionless number">dimensionless number</a>, often used in <a href="/wiki/Rheology" title="Rheology">rheology</a> to characterize the fluidity of materials under specific flow conditions. It quantifies the observation that given enough time even a solid-like material might flow, or a fluid-like material can act solid when it is deformed rapidly enough. Materials that have low relaxation times flow easily and as such show relatively rapid stress decay. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deborah_number&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Deborah number is the ratio of fundamentally different characteristic times. The Deborah number is defined as the ratio of the time it takes for a material to adjust to applied stresses or deformations, and the characteristic time scale of an experiment (or a computer simulation) probing the response of the material: </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 \mathrm {De} ={\frac {t_{\mathrm {c} }}{t_{\mathrm {p} }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {De} ={\frac {t_{\mathrm {c} }}{t_{\mathrm {p} }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4471eef4801bef45c056a475f1062f64c23d55c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.375ex; height:5.843ex;" alt="{\displaystyle \mathrm {De} ={\frac {t_{\mathrm {c} }}{t_{\mathrm {p} }}},}"></span></dd></dl> <p>where <span class="texhtml"><i>t</i><sub>c</sub></span> stands for the relaxation time and <span class="texhtml"><i>t</i><sub>p</sub></span> for the "time of observation", typically taken to be the time scale of the process.<sup id="cite_ref-Poole_1-0" class="reference"><a href="#cite_note-Poole-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The numerator, <a href="/wiki/Relaxation_time" class="mw-redirect" title="Relaxation time">relaxation time</a>, is the time needed for a reference amount of deformation to occur under a suddenly applied reference load (a more fluid-like material will therefore require less time to flow, giving a lower Deborah number relative to a solid subjected to the same loading rate). </p><p>The denominator, material time,<sup id="cite_ref-TAImaterialProperties_2-0" class="reference"><a href="#cite_note-TAImaterialProperties-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is the amount of time required to reach a given reference strain (a faster loading rate will therefore reach the reference strain sooner, giving a higher Deborah number). </p><p>Equivalently, the relaxation time is the time required for the stress induced, by a suddenly applied reference strain, to reduce by a certain reference amount. The relaxation time is actually based on the rate of relaxation that exists at the moment of the suddenly applied load. </p><p>This incorporates both the elasticity and viscosity of the material. At lower Deborah numbers, the material behaves in a more fluidlike manner, with an associated Newtonian viscous flow. At higher Deborah numbers, the material behavior enters the non-Newtonian regime, increasingly dominated by elasticity and demonstrating solidlike behavior.<sup id="cite_ref-Reiner1964_3-0" class="reference"><a href="#cite_note-Reiner1964-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> </p><p>For example, for a Hookean elastic solid, the relaxation time <span class="texhtml"><i>t</i><sub>c</sub></span> will be infinite and it will vanish for a Newtonian viscous fluid. For liquid water, <span class="texhtml"><i>t</i><sub>c</sub></span> is typically 10<sup>−12</sup> s, for lubricating oils passing through gear teeth at high pressure it is of the order of 10<sup>−6</sup> s and for polymers undergoing plastics processing, the relaxation time will be of the order of a few seconds. Therefore, depending on the situation, these liquids may exhibit elastic properties, departing from purely viscous behavior.<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> </p><p>While <span class="texhtml">De</span> is similar to the <a href="/wiki/Weissenberg_number" title="Weissenberg number">Weissenberg number</a> and is often confused with it in technical literature, they have different physical interpretations. The Weissenberg number indicates the degree of anisotropy or orientation generated by the deformation, and is appropriate to describe flows with a constant stretch history, such as simple shear. In contrast, the Deborah number should be used to describe flows with a non-constant stretch history, and physically represents the rate at which elastic energy is stored or released.<sup id="cite_ref-Poole_1-1" class="reference"><a href="#cite_note-Poole-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <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=Deborah_number&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> The Deborah number was originally proposed by <a href="/wiki/Markus_Reiner" title="Markus Reiner">Markus Reiner</a>, a professor at <a href="/wiki/Technion_%E2%80%93_Israel_Institute_of_Technology" title="Technion – Israel Institute of Technology">Technion</a> in <a href="/wiki/Israel" title="Israel">Israel</a>, who chose the name inspired by a verse in the <a href="/wiki/Bible" title="Bible">Bible</a>, stating "The mountains flowed before the Lord" in a song by the prophetess <a href="/wiki/Deborah" title="Deborah">Deborah</a> in the <a href="/wiki/Book_of_Judges" title="Book of Judges">Book of Judges</a>;<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> הָרִ֥ים נָזְל֖וּ מִפְּנֵ֣י יְהוָ֑ה <i>hā-rîm nāzəlū <a href="/wiki/Divine_countenance" title="Divine countenance">mippənê</a> <a href="/wiki/Yahweh" title="Yahweh">Yahweh</a></i>).<sup id="cite_ref-Reiner1964_3-1" class="reference"><a href="#cite_note-Reiner1964-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Millgram2018_7-0" class="reference"><a href="#cite_note-Millgram2018-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In his 1964 paper (a reproduction of his after-dinner speech to the Fourth International Congress on Rheology in 1962),<sup id="cite_ref-:0_8-0" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><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> <a href="/wiki/Markus_Reiner" title="Markus Reiner">Markus Reiner</a> further elucidated the name's origin:<sup id="cite_ref-:0_8-1" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></p><blockquote><p>“Deborah knew two things.  First, that the mountains flow, as everything flows. But, secondly, that they flowed before the Lord, and not before man, for the simple reason that man in his short lifetime cannot see them flowing, while the time of observation of God is infinite. We may therefore well define a nondimensional number the Deborah number D = time of relaxation/time of observation.”</p></blockquote> <div class="mw-heading mw-heading2"><h2 id="Time-temperature_superposition">Time-temperature superposition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deborah_number&action=edit&section=3" title="Edit section: Time-temperature superposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Deborah number is particularly useful in conceptualizing the <a href="/wiki/Time%E2%80%93temperature_superposition" title="Time–temperature superposition">time–temperature superposition</a> principle. Time-temperature superposition has to do with altering experimental time scales using reference temperatures to extrapolate temperature-dependent mechanical properties of <a href="/wiki/Polymers" class="mw-redirect" title="Polymers">polymers</a>. A material at low temperature with a long experimental or <a href="/wiki/Relaxation_time" class="mw-redirect" title="Relaxation time">relaxation time</a> behaves like the same material at high temperature and short experimental or relaxation time if the Deborah number remains the same. This can be particularly useful when working with materials which relax on a long time scale under a certain temperature. The practical application of this idea arises in the <a href="/wiki/Williams%E2%80%93Landel%E2%80%93Ferry_equation" title="Williams–Landel–Ferry equation">Williams–Landel–Ferry equation</a>. Time-temperature superposition avoids the inefficiency of measuring a polymer's behavior over long periods of time at a specified temperature by utilizing the Deborah number.<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> </p> <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=Deborah_number&action=edit&section=4" 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"><ol class="references"> <li id="cite_note-Poole-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Poole_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Poole_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="CITEREFPoole2012" class="citation journal cs1">Poole, R J (2012). <a rel="nofollow" class="external text" href="http://pcwww.liv.ac.uk/~robpoole/PAPERS/POOLE_45.pdf">"The Deborah and Weissenberg numbers"</a> <span class="cs1-format">(PDF)</span>. <i>Rheology Bulletin</i>. <b>53</b> (2): 32–39.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rheology+Bulletin&rft.atitle=The+Deborah+and+Weissenberg+numbers&rft.volume=53&rft.issue=2&rft.pages=32-39&rft.date=2012&rft.aulast=Poole&rft.aufirst=R+J&rft_id=http%3A%2F%2Fpcwww.liv.ac.uk%2F~robpoole%2FPAPERS%2FPOOLE_45.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeborah+number" class="Z3988"></span></span> </li> <li id="cite_note-TAImaterialProperties-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-TAImaterialProperties_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranck" class="citation web cs1">Franck, A. <a rel="nofollow" class="external text" href="http://tainstruments.com/pdf/literature/AAN004_Viscoelasticity_and_DMA.pdf">"Viscoelasticity and dynamic mechanical testing"</a> <span class="cs1-format">(PDF)</span>. <i>TA Instruments</i>. TA Instruments Germany<span class="reference-accessdate">. Retrieved <span class="nowrap">26 March</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=TA+Instruments&rft.atitle=Viscoelasticity+and+dynamic+mechanical+testing&rft.aulast=Franck&rft.aufirst=A.&rft_id=http%3A%2F%2Ftainstruments.com%2Fpdf%2Fliterature%2FAAN004_Viscoelasticity_and_DMA.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeborah+number" class="Z3988"></span></span> </li> <li id="cite_note-Reiner1964-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Reiner1964_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Reiner1964_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReiner1964" class="citation cs2">Reiner, M. (1964), "The Deborah Number", <i>Physics Today</i>, <b>17</b> (1): 62, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PhT....17a..62R">1964PhT....17a..62R</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.3051374">10.1063/1.3051374</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=The+Deborah+Number&rft.volume=17&rft.issue=1&rft.pages=62&rft.date=1964&rft_id=info%3Adoi%2F10.1063%2F1.3051374&rft_id=info%3Abibcode%2F1964PhT....17a..62R&rft.aulast=Reiner&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeborah+number" class="Z3988"></span></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"><a rel="nofollow" class="external text" href="http://rrc.engr.wisc.edu/deborah.html">The Deborah Number</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110413144406/http://rrc.engr.wisc.edu/deborah.html">Archived</a> 2011-04-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarnesHuttonWalters1989" class="citation book cs1">Barnes, H.A.; Hutton, J.F.; Walters, K. (1989). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontorh00barn"><i>An introduction to rheology</i></a></span> (5. impr. ed.). Amsterdam: Elsevier. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontorh00barn/page/n12">5</a>–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-87140-4" title="Special:BookSources/978-0-444-87140-4"><bdi>978-0-444-87140-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+rheology&rft.place=Amsterdam&rft.pages=5-6&rft.edition=5.+impr.&rft.pub=Elsevier&rft.date=1989&rft.isbn=978-0-444-87140-4&rft.aulast=Barnes&rft.aufirst=H.A.&rft.au=Hutton%2C+J.F.&rft.au=Walters%2C+K.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontorh00barn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeborah+number" class="Z3988"></span></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"><a href="https://en.wikisource.org/wiki/Bible_(King_James)/Judges#5:5" class="extiw" title="s:Bible (King James)/Judges">Judges 5:5</a></span> </li> <li id="cite_note-Millgram2018-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Millgram2018_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMillgram2018" class="citation book cs1">Millgram, Hillel I. (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xUdODwAAQBAJ&pg=PA123"><i>Judges and Saviors, Deborah and Samson: Reflections of a World in Chaos</i></a>. Hamilton Books. pp. 123–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7618-6990-0" title="Special:BookSources/978-0-7618-6990-0"><bdi>978-0-7618-6990-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Judges+and+Saviors%2C+Deborah+and+Samson%3A+Reflections+of+a+World+in+Chaos&rft.pages=123-&rft.pub=Hamilton+Books&rft.date=2018&rft.isbn=978-0-7618-6990-0&rft.aulast=Millgram&rft.aufirst=Hillel+I.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxUdODwAAQBAJ%26pg%3DPA123&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeborah+number" class="Z3988"></span></span> </li> <li id="cite_note-:0-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReiner1964" class="citation journal cs1">Reiner, M. (1964-01-01). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1063/1.3051374">"The Deborah Number"</a>. <i>Physics Today</i>. <b>17</b> (1): 62. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PhT....17a..62R">1964PhT....17a..62R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.3051374">10.1063/1.3051374</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9228">0031-9228</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=The+Deborah+Number&rft.volume=17&rft.issue=1&rft.pages=62&rft.date=1964-01-01&rft.issn=0031-9228&rft_id=info%3Adoi%2F10.1063%2F1.3051374&rft_id=info%3Abibcode%2F1964PhT....17a..62R&rft.aulast=Reiner&rft.aufirst=M.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1063%2F1.3051374&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeborah+number" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhillips2012" class="citation journal cs1">Phillips, Tim (2012-12-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Farh-2012-0006">"The British Society of Rheology Midwinter Meeting: Complex Fluids and Complex Flows"</a>. <i>Applied Rheology</i>. <b>22</b> (2): 104–105. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Farh-2012-0006">10.1515/arh-2012-0006</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1617-8106">1617-8106</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Rheology&rft.atitle=The+British+Society+of+Rheology+Midwinter+Meeting%3A+Complex+Fluids+and+Complex+Flows&rft.volume=22&rft.issue=2&rft.pages=104-105&rft.date=2012-12-01&rft_id=info%3Adoi%2F10.1515%2Farh-2012-0006&rft.issn=1617-8106&rft.aulast=Phillips&rft.aufirst=Tim&rft_id=https%3A%2F%2Fdoi.org%2F10.1515%252Farh-2012-0006&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeborah+number" class="Z3988"></span></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">Rudin, Alfred, and Phillip Choi. The Elements of Polymer Science and Engineering. 3rd. Oxford: Academic Press, 2013. Print. Page 221.</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=Deborah_number&action=edit&section=5" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>J.S. Vrentas, C.M. Jarzebski, J.L. Duda (1975) <a rel="nofollow" class="external text" href="http://onlinelibrary.wiley.com/doi/10.1002/aic.690210510/abstract">"A Deborah number for diffusion in polymer-solvent systems"</a>, <a href="/wiki/AIChE" class="mw-redirect" title="AIChE">AIChE</a> Journal 21(5):894–901, weblink to Wiley Online Library.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul 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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 class="mw-selflink selflink">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><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Deborah_number&oldid=1258706137">https://en.wikipedia.org/w/index.php?title=Deborah_number&oldid=1258706137</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a 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