Squeeze flow - Wikipedia

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.mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Confusing plainlinks metadata ambox ambox-style ambox-confusing" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be <a href="/wiki/Wikipedia:Vagueness" title="Wikipedia:Vagueness">confusing or unclear</a> to readers</b>. In particular, not written clearly for those unfamiliar with the subject..<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarify the article</a>. There might be a discussion about this on <a href="/wiki/Talk:Squeeze_flow" title="Talk:Squeeze flow">the talk page</a>.</span> <span class="date-container"><i>(<span class="date">April 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><b>Squeeze flow</b> (also called squeezing flow, squeezing film flow, or squeeze flow theory) is a type of flow in which a material is pressed out or deformed between two parallel plates or objects. First explored in 1874 by <a href="/wiki/Josef_Stefan" title="Josef Stefan">Josef Stefan</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> squeeze flow describes the outward movement of a droplet of material, its area of contact with the plate surfaces, and the effects of internal and external factors such as temperature, <a href="/wiki/Viscoelasticity" title="Viscoelasticity">viscoelasticity</a>, and heterogeneity of the material.<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> Several squeeze flow models exist to describe <a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a> and <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian fluids</a> undergoing squeeze flow under various geometries and conditions. Numerous applications across scientific and engineering disciplines including <a href="/wiki/Rheometry" title="Rheometry">rheometry</a>, welding engineering, and <a href="/wiki/Materials_science" title="Materials science">materials science</a> provide examples of squeeze flow in practical use. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_Assumptions">Basic Assumptions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=1" title="Edit section: Basic Assumptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Conservation of mass</a> (expressed as a <a href="/wiki/Continuity_equation" title="Continuity equation">continuity equation</a>), the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier-Stokes equations</a> for conservation of momentum, and the <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> provide the foundations for calculating and modeling squeeze flow. Boundary conditions for such calculations include assumptions of an <a href="/wiki/Incompressible_flow" title="Incompressible flow">incompressible fluid</a>, a two-dimensional system, neglecting of <a href="/wiki/Body_force" title="Body force">body forces</a>, and neglecting of inertial forces. </p><p>Relating applied force to material thickness: </p><p><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 F=-{\frac {4*L^{3}*\eta *W}{h^{3}}}{dh \over dt}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>∗</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>∗</mo> <mi>η</mi> <mo>∗</mo> <mi>W</mi> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>h</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=-{\frac {4*L^{3}*\eta *W}{h^{3}}}{dh \over dt}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08df71750475817aae82bbb58167d67847190946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.863ex; height:6.009ex;" alt="{\displaystyle F=-{\frac {4*L^{3}*\eta *W}{h^{3}}}{dh \over dt}}"></span> </p><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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is the applied squeezing force, <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 2L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/697324514412b5263f3f50bd3e267623d3fa1ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.745ex; height:2.176ex;" alt="{\displaystyle 2L}"></span> is the initial length of the droplet, <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is the fluid viscosity, <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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> is the width of the assumed rectangular plate, <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 2h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961075004bd1ffd53a4685965e9eb3a9b691f645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.501ex; height:2.176ex;" alt="{\displaystyle 2h}"></span> is the final height of the droplet, and <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 {dh \over dt}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>h</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dh \over dt}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f40579f58172a47aa1cfc831ab63ef6b2966f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.391ex; height:5.509ex;" alt="{\displaystyle {dh \over dt}}"></span> is the change in droplet height over time. To simplify most calculations, the applied force is assumed to be constant. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Asperity.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/96/Asperity.jpg" decoding="async" width="180" height="74" class="mw-file-element" data-file-width="180" data-file-height="74" /></a><figcaption>Visualization of surface-to-surface contact; asperity highlighted in red.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Newtonian_fluids">Newtonian fluids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=2" title="Edit section: Newtonian fluids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several equations accurately model <a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a> droplet sizes under different initial conditions. </p><p> Consideration of a single <a href="/wiki/Asperity_(materials_science)" title="Asperity (materials science)">asperity</a>, or surface protrusion, allows for measurement of a very specific cross-section of a droplet. To measure macroscopic squeeze flow effects, models exist for two the most common surfaces: circular and rectangular plate squeeze flows.</p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Axisymmetric_Squeeze_Flow.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Axisymmetric_Squeeze_Flow.png/417px-Axisymmetric_Squeeze_Flow.png" decoding="async" width="417" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Axisymmetric_Squeeze_Flow.png/626px-Axisymmetric_Squeeze_Flow.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Axisymmetric_Squeeze_Flow.png/834px-Axisymmetric_Squeeze_Flow.png 2x" data-file-width="1826" data-file-height="453" /></a><figcaption>Single asperity squeeze flow diagram at initial and follow-on conditions; plates (assumed to be semi-infinite, in gray), droplet (green).</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Single_asperity">Single asperity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=3" title="Edit section: Single asperity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For single asperity squeeze flow: </p><p><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 {h_{0}}{h}}=\left(1+{\frac {5*F*t*h_{0}^{2}}{4*\eta *W*L_{0}^{3}}}\right)^{1/5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>∗</mo> <mi>F</mi> <mo>∗</mo> <mi>t</mi> <mo>∗</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>4</mn> <mo>∗</mo> <mi>η</mi> <mo>∗</mo> <mi>W</mi> <mo>∗</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {h_{0}}{h}}=\left(1+{\frac {5*F*t*h_{0}^{2}}{4*\eta *W*L_{0}^{3}}}\right)^{1/5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c712c4d8dfe6349da68e78f2fc2326afd73f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.535ex; height:8.009ex;" alt="{\displaystyle {\frac {h_{0}}{h}}=\left(1+{\frac {5*F*t*h_{0}^{2}}{4*\eta *W*L_{0}^{3}}}\right)^{1/5}}"></span> </p><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 2h_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2h_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5ce4c1a19f8e8085ec10d780f962c477dff763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.556ex; height:2.509ex;" alt="{\displaystyle 2h_{0}}"></span> is the initial height of the droplet, <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 2h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961075004bd1ffd53a4685965e9eb3a9b691f645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.501ex; height:2.176ex;" alt="{\displaystyle 2h}"></span> is the final height of the droplet, <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is the applied squeezing force, <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> is the squeezing time, <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is the fluid viscosity, <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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> is the width of the assumed rectangular plate, and <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 2L_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2L_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5c5b7f1ebf7af6faa72bdb5814adf67cffa3cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.8ex; height:2.509ex;" alt="{\displaystyle 2L_{0}}"></span> is the initial length of the droplet.<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Based on conservation of mass calculations, the droplet width is inversely proportional to droplet height; as the width increases, the height decreases in response to squeezing forces.<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Circular_Plate_Squeeze_Flow.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Circular_Plate_Squeeze_Flow.png/417px-Circular_Plate_Squeeze_Flow.png" decoding="async" width="417" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Circular_Plate_Squeeze_Flow.png/626px-Circular_Plate_Squeeze_Flow.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Circular_Plate_Squeeze_Flow.png/834px-Circular_Plate_Squeeze_Flow.png 2x" data-file-width="1912" data-file-height="460" /></a><figcaption>Circular plate squeeze flow diagram at initial and follow-on conditions; circular plates (gray), droplet (hash marked ellipse between both plates). </figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Circular_plate">Circular plate</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=4" title="Edit section: Circular plate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For circular plate squeeze flow: </p><p><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 {h_{0}}{h}}=\left(1+{\frac {16*F*t*h_{0}^{2}}{3*\pi *\eta *R^{4}}}\right)^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>16</mn> <mo>∗</mo> <mi>F</mi> <mo>∗</mo> <mi>t</mi> <mo>∗</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>3</mn> <mo>∗</mo> <mi>π</mi> <mo>∗</mo> <mi>η</mi> <mo>∗</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {h_{0}}{h}}=\left(1+{\frac {16*F*t*h_{0}^{2}}{3*\pi *\eta *R^{4}}}\right)^{1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d479852284402d3239cf9df8ba44a53740bf8fe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.429ex; height:8.009ex;" alt="{\displaystyle {\frac {h_{0}}{h}}=\left(1+{\frac {16*F*t*h_{0}^{2}}{3*\pi *\eta *R^{4}}}\right)^{1/2}}"></span> </p><p><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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the radius of the circular plate.<sup id="cite_ref-:0_3-2" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Rectangular_plate">Rectangular plate</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=5" title="Edit section: Rectangular plate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Rectangular_Plate_Squeeze_Flow.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Rectangular_Plate_Squeeze_Flow.png/417px-Rectangular_Plate_Squeeze_Flow.png" decoding="async" width="417" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Rectangular_Plate_Squeeze_Flow.png/626px-Rectangular_Plate_Squeeze_Flow.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Rectangular_Plate_Squeeze_Flow.png/834px-Rectangular_Plate_Squeeze_Flow.png 2x" data-file-width="1856" data-file-height="477" /></a><figcaption>Rectangular plate squeeze flow diagram at initial and follow-on conditions; rectangular plates (gray), droplet (green).</figcaption></figure> <p>For rectangular plate squeeze flow: </p><p><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 {h_{0}}{h}}=\left(1+{\frac {F*t*h_{0}^{2}}{2*\mu *W*L^{3}}}\right)^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo>∗</mo> <mi>t</mi> <mo>∗</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <mo>∗</mo> <mi>μ</mi> <mo>∗</mo> <mi>W</mi> <mo>∗</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {h_{0}}{h}}=\left(1+{\frac {F*t*h_{0}^{2}}{2*\mu *W*L^{3}}}\right)^{1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d683396ddfa667fde76e34fd96403561b7a3b4be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.767ex; height:8.009ex;" alt="{\displaystyle {\frac {h_{0}}{h}}=\left(1+{\frac {F*t*h_{0}^{2}}{2*\mu *W*L^{3}}}\right)^{1/2}}"></span> </p><p>These calculations assume a melt layer that has a length much larger than the sample width and thickness.<sup id="cite_ref-:0_3-3" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Non-Newtonian_fluids">Non-Newtonian fluids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=6" title="Edit section: Non-Newtonian fluids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Simplifying calculations for <a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluids</a> allows for basic analysis of squeeze flow, but many polymers can exhibit properties of <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian fluids</a>, such as <a href="/wiki/Viscoelasticity" title="Viscoelasticity">viscoelastic</a> characteristics, under <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformation</a>. The power law fluid model is sufficient to describe behaviors above the <a href="/wiki/Melting_point" title="Melting point">melting temperature</a> for <a href="/wiki/Semicrystalline_polymer" class="mw-redirect" title="Semicrystalline polymer">semicrystalline thermoplastics</a> or the <a href="/wiki/Glass_transition" title="Glass transition">glass transition temperature</a> for amorphous thermoplastics, and the <a href="/wiki/Bingham_plastic" title="Bingham plastic">Bingham fluid</a> model provides calculations based on variations in yield stress calculations.<sup id="cite_ref-:0_3-4" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_4-0" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Power_law_fluid">Power law fluid</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=7" title="Edit section: Power law fluid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For squeeze flow in a <a href="/wiki/Power-law_fluid" title="Power-law fluid">power law fluid</a>: </p><p><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 {h_{0}}{h}}=\left(1+t*({\frac {2n+3}{4n+2}})({\frac {(4*h_{0}*L_{0})^{n+1}*F*(n+2)}{(2*L_{0})^{2n+3}*W*m}})^{1/n}\right)^{n/2n+3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>t</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> <mrow> <mn>4</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo>∗</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∗</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>∗</mo> <mi>F</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>∗</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> <mo>∗</mo> <mi>W</mi> <mo>∗</mo> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {h_{0}}{h}}=\left(1+t*({\frac {2n+3}{4n+2}})({\frac {(4*h_{0}*L_{0})^{n+1}*F*(n+2)}{(2*L_{0})^{2n+3}*W*m}})^{1/n}\right)^{n/2n+3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bebc4ea8cf27bb520e7da6dae327872b42fadaa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:66.756ex; height:8.009ex;" alt="{\displaystyle {\frac {h_{0}}{h}}=\left(1+t*({\frac {2n+3}{4n+2}})({\frac {(4*h_{0}*L_{0})^{n+1}*F*(n+2)}{(2*L_{0})^{2n+3}*W*m}})^{1/n}\right)^{n/2n+3}}"></span> </p><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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> (or <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>) is the <i>flow consistency index</i> and <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is the dimensionless <i>flow behavior index</i>.<sup id="cite_ref-:0_3-5" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><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 m=m_{0}*exp\left({\frac {-E_{a}}{R*T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∗</mo> <mi>e</mi> <mi>x</mi> <mi>p</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi>R</mi> <mo>∗</mo> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=m_{0}*exp\left({\frac {-E_{a}}{R*T}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f684952990113d7eac5315e560463e1d029718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.25ex; height:6.176ex;" alt="{\displaystyle m=m_{0}*exp\left({\frac {-E_{a}}{R*T}}\right)}"></span> </p><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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the <i>flow consistency index, <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 m_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6ff51ee949104fe6fae553cfbdfba29d5fac1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle m_{0}}"></span></i>is the <i>initial flow consistency index</i>, <i><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 E_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a5c4699f75d498b504e59bfd8b5496aa20351a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.817ex; height:2.509ex;" alt="{\displaystyle E_{a}}"></span></i>is the <a href="/wiki/Activation_energy" title="Activation energy">activation energy</a>, <i><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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span></i>is the universal <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a>, and <i><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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></i>is the <a href="https://simple.wikipedia.org/wiki/Absolute_temperature" class="extiw" title="simple:Absolute temperature">absolute temperature</a>.<sup id="cite_ref-:0_3-6" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>During experimentation to determine the accuracy of the power law fluid model, observations showed that modeling slow squeeze flow generated inaccurate power law constants (<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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> and <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>) using a standard <a href="/wiki/Viscometer" title="Viscometer">viscometer</a>, and fast squeeze flow demonstrated that polymers may exhibit better lubrication than current constitutive models will predict.<sup id="cite_ref-:2_5-0" class="reference"><a href="#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The current empirical model for power law fluids is relatively accurate for modeling inelastic flows, but certain kinematic flow assumptions and incomplete understanding of polymeric lubrication properties tend to provide inaccurate modeling of power law fluids.<sup id="cite_ref-:2_5-1" class="reference"><a href="#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Bingham_fluid">Bingham fluid</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=8" title="Edit section: Bingham fluid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Bingham_plastic" title="Bingham plastic">Bingham fluids</a> exhibit uncommon characteristics during squeeze flow. While undergoing compression, Bingham fluids should fail to move and act as a solid until achieving a yield stress; however, as the parallel plates move closer together, the fluid shows some radial movement. One study proposes a “biviscosity” model where the Bingham fluid retains some unyielded regions that maintain solid-like properties, while other regions yield and allow for some compression and outward movement.<sup id="cite_ref-:1_4-1" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><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 ={\begin{cases}\eta _{2}*{du \over dy}+\tau _{1},&{\text{if }}\tau \geq \tau _{1}\\\eta _{1}*{du \over dy},&{\text{if }}\tau <\tau _{1}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>τ</mi> <mo>≥</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>τ</mi> <mo><</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\begin{cases}\eta _{2}*{du \over dy}+\tau _{1},&{\text{if }}\tau \geq \tau _{1}\\\eta _{1}*{du \over dy},&{\text{if }}\tau <\tau _{1}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470039eac9845205637c1f61ad9877ad632c8843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.156ex; height:8.176ex;" alt="{\displaystyle \tau ={\begin{cases}\eta _{2}*{du \over dy}+\tau _{1},&{\text{if }}\tau \geq \tau _{1}\\\eta _{1}*{du \over dy},&{\text{if }}\tau <\tau _{1}\end{cases}}}"></span> </p><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 \eta _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa14f274aac6c807aa93264beecd659e410bba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.21ex; height:2.176ex;" alt="{\displaystyle \eta _{2}}"></span> is the <i>known viscosity</i> of the Bingham fluid, <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 \eta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f05ea928db07c86bb11863e431b8406febd09d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.21ex; height:2.176ex;" alt="{\displaystyle \eta _{1}}"></span> is the <i>"paradoxical" viscosity</i> of the solid-like state, and <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 _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3e70085bbfec0d2ed0f898618bbb48db4ab7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.07ex; height:2.009ex;" alt="{\displaystyle \tau _{1}}"></span> is the <i>biviscosity region stress</i>.<sup id="cite_ref-:1_4-2" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> To determine this new stress: </p><p><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 _{0}=\tau _{1}(1-\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{0}=\tau _{1}(1-\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77ec98b36daf0b363ecb5e17dca3d83ad471808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.995ex; height:2.843ex;" alt="{\displaystyle \tau _{0}=\tau _{1}(1-\epsilon )}"></span> </p><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 \tau _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56dad457e274b970f5d98b9dc40bef7f895c7f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.07ex; height:2.009ex;" alt="{\displaystyle \tau _{0}}"></span> is the <i>yield stress</i> and <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 \epsilon ={\frac {\eta _{2}}{\eta _{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon ={\frac {\eta _{2}}{\eta _{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c648bb1289642b032481b50e288bd492810a9ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.088ex; height:5.343ex;" alt="{\displaystyle \epsilon ={\frac {\eta _{2}}{\eta _{1}}}}"></span> is the dimensionless <i>viscosity ratio</i>. If <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 \epsilon =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6cea33fb1374a6510eb3a7062b52245de1b33ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon =1}"></span>, the fluid exhibits Newtonian behavior; as <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 \epsilon \rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon \rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68af83da5cb4596259c9ec2e76260e0bacee12ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle \epsilon \rightarrow 0}"></span>, the Bingham model applies.<sup id="cite_ref-:1_4-3" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=9" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Squeeze flow application is prevalent in several science and engineering fields. Modeling and experimentation assist with understanding the complexities of squeeze flow during processes such as rheological testing, <a href="/wiki/Hot_plate_welding" title="Hot plate welding">hot plate welding</a>, and composite material joining. </p> <div class="mw-heading mw-heading3"><h3 id="Rheological_testing">Rheological testing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=10" title="Edit section: Rheological testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Squeeze flow <a href="/wiki/Rheometry" title="Rheometry">rheometry</a> allows for evaluation of polymers under wide ranges of temperatures, shear rates, and flow indexes. Parallel plate <a href="/w/index.php?title=Plastometer&action=edit&redlink=1" class="new" title="Plastometer (page does not exist)">plastometers</a> provide analysis for high viscosity materials such as rubber and glass, cure times for epoxy resins, and fiber-filled suspension flows.<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> While <a href="/wiki/Viscometer" title="Viscometer">viscometers</a> provide useful results for squeeze flow measurements, testing conditions such as applied rotation rates, material composition, and fluid flow behaviors under shear may require the use of <a href="/wiki/Rheometer" title="Rheometer">rheometers</a> or other novel setups to obtain accurate data.<sup id="cite_ref-:2_5-2" class="reference"><a href="#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Squeeze_Out_Phenomena.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Squeeze_Out_Phenomena.png/323px-Squeeze_Out_Phenomena.png" decoding="async" width="323" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Squeeze_Out_Phenomena.png/485px-Squeeze_Out_Phenomena.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Squeeze_Out_Phenomena.png/646px-Squeeze_Out_Phenomena.png 2x" data-file-width="1986" data-file-height="989" /></a><figcaption>Hot plate welding; (a) Heating Phase, (b) Change-over Phase, (c) Joining Phase. Polymer materials (gray), heating element (red), melt layers (orange), squeeze out regions (orange, hash mark borders). </figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Hot_plate_welding">Hot plate welding</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=11" title="Edit section: Hot plate welding"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>During conventional hot plate welding, a successful joining phase depends on proper maintenance of squeeze flow to ensure that pressure and temperature create an ideal weld. Excessive pressure causes squeeze out of valuable material and weakens the bond due to fiber realignment in the melt layer,<sup id="cite_ref-:3_7-0" class="reference"><a href="#cite_note-:3-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> while failure to allow cooling to room temperature creates weak, brittle welds that crack or break completely during use.<sup id="cite_ref-:0_3-7" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Composite_material_joining">Composite material joining</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Squeeze_flow&action=edit&section=12" title="Edit section: Composite material joining"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prevalent in the aerospace and automotive industries, composites serve as expensive, yet mechanically strong, materials in the construction of several types of aircraft and vehicles. While aircraft parts are typically composed of <a href="/wiki/Thermosetting_polymer" title="Thermosetting polymer">thermosetting polymers</a>, <a href="/wiki/Thermoplastic" title="Thermoplastic">thermoplastics</a> may become an analog to permit increased manufacturing of these stronger materials through their melting abilities and relatively inexpensive raw materials. Characterization and testing of thermoplastic composites experiencing squeeze flow allow for study of fiber orientations within the melt and final products to determine weld strength.<sup id="cite_ref-:3_7-1" class="reference"><a href="#cite_note-:3-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Fiber strand length and size show significant effects on material strength,<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> and squeeze flow causes fibers to orient along the load direction while being perpendicular to the joining direction to achieve the same final properties as thermosetting composites.<sup id="cite_ref-:3_7-2" class="reference"><a href="#cite_note-:3-7"><span class="cite-bracket">[</span>7<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=Squeeze_flow&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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="CITEREFUllahKhanFizaUllah2022" class="citation journal cs1">Ullah, Hakeem; Khan, Muhammad Arif; Fiza, Mehreen; Ullah, Kashif; Ayaz, Muhammad; Al-Mekhlafi, Seham M. (2022-03-31). <a rel="nofollow" class="external text" href="https://doi.org/10.1155%2F2022%2F1668206">"Analytical and Numerical Analysis of the Squeezed Unsteady MHD Nanofluid Flow in the Presence of Thermal Radiation"</a>. <i>Journal of Nanomaterials</i>. <b>2022</b> (1): e1668206. <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/10.1155%2F2022%2F1668206">10.1155/2022/1668206</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/1687-4110">1687-4110</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Nanomaterials&rft.atitle=Analytical+and+Numerical+Analysis+of+the+Squeezed+Unsteady+MHD+Nanofluid+Flow+in+the+Presence+of+Thermal+Radiation&rft.volume=2022&rft.issue=1&rft.pages=e1668206&rft.date=2022-03-31&rft_id=info%3Adoi%2F10.1155%2F2022%2F1668206&rft.issn=1687-4110&rft.aulast=Ullah&rft.aufirst=Hakeem&rft.au=Khan%2C+Muhammad+Arif&rft.au=Fiza%2C+Mehreen&rft.au=Ullah%2C+Kashif&rft.au=Ayaz%2C+Muhammad&rft.au=Al-Mekhlafi%2C+Seham+M.&rft_id=https%3A%2F%2Fdoi.org%2F10.1155%252F2022%252F1668206&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASqueeze+flow" 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">Engmann, J., Servais, C., & Burbidge, A. S. (2005). Squeeze flow theory and applications to rheometry: A review. <i>Journal of Non-Newtonian Fluid Mechanics,</i> <i>132</i>(1-3), 1-27. doi:10.1016/j.jnnfm.2005.08.007</span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:0_3-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-:0_3-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-:0_3-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-:0_3-7"><sup><i><b>h</b></i></sup></a></span> <span class="reference-text">Grewell, D., Benatar, A., & Park, J. (2013). <i>Plastics and composites welding handbook</i>. New York: Hanser.</span> </li> <li id="cite_note-:1-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:1_4-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">Wilson, S. (1993). Squeezing flow of a Bingham material. <i>Journal of Non-Newtonian Fluid Mechanics,</i> <i>47</i>, 211-219. doi:10.1016/0377-0257(93)80051-c</span> </li> <li id="cite_note-:2-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:2_5-2"><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="CITEREFGrimm1978" class="citation journal cs1">Grimm, Roger J. (1978). "Squeezing flows of polymeric liquids". <i>AIChE Journal</i>. <b>24</b> (3): 427–439. <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/1978AIChE..24..427G">1978AIChE..24..427G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Faic.690240307">10.1002/aic.690240307</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/1547-5905">1547-5905</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=AIChE+Journal&rft.atitle=Squeezing+flows+of+polymeric+liquids&rft.volume=24&rft.issue=3&rft.pages=427-439&rft.date=1978&rft.issn=1547-5905&rft_id=info%3Adoi%2F10.1002%2Faic.690240307&rft_id=info%3Abibcode%2F1978AIChE..24..427G&rft.aulast=Grimm&rft.aufirst=Roger+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASqueeze+flow" 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">Macosko, C. W., & Larson, R. G. (1994). <i>Rheology: Principles, measurements, and applications</i>. New York: VCH.</span> </li> <li id="cite_note-:3-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-:3_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:3_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:3_7-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Fiebig, I., & Schoeppner, V. (2018). Factors influencing the fiber orientation in welding of fiber-reinforced thermoplastics. <i>Welding in the World,</i> <i>62</i>(5), 997-1012. doi:10.1007/s40194-018-0628-0</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">Picher-Martel, G., Levy, A., & Hubert, P. (2015). Compression molding of Carbon/Polyether ether ketone composites: Squeeze flow behavior of unidirectional and randomly oriented strands. <i>Polymer Composites,</i> <i>38</i>(9), 1828-1837. doi:10.1002/pc.23753</span> </li> </ol></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=Squeeze_flow&oldid=1249383196">https://en.wikipedia.org/w/index.php?title=Squeeze_flow&oldid=1249383196</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Welding" title="Category:Welding">Welding</a></li><li><a href="/wiki/Category:Plastics" title="Category:Plastics">Plastics</a></li><li><a href="/wiki/Category:Materials_science" title="Category:Materials science">Materials science</a></li><li><a href="/wiki/Category:Molding_processes" title="Category:Molding processes">Molding processes</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_April_2019" title="Category:Wikipedia articles needing clarification from April 2019">Wikipedia articles needing clarification from April 2019</a></li><li><a href="/wiki/Category:All_Wikipedia_articles_needing_clarification" title="Category:All Wikipedia articles needing clarification">All Wikipedia articles needing clarification</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 4 October 2024, at 16:50<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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