Application of Homotopy Analysis Method to the Unsteady Squeezing Flow of a Second-Grade Fluid between Circular Plates

<table>Obtained solution by the HAM with <span class="svg-td" style="width: 42.212502px;"><svg height="9.8374996" id="M168" style="vertical-align:-0.1125pt" version="1.1" viewbox="0 0 42.212502 9.8374996" width="42.212502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.8375)">
<g transform="translate(72,-64.13)">
<text transform="matrix(1,0,0,-1,-71.95,64.3)">
<tspan style="font-size: 11.21px; " x="0" y="0">ℏ</tspan>
<tspan style="font-size: 11.21px; " x="9.5940475" y="0">=</tspan>
<tspan style="font-size: 11.21px; " x="20.387352" y="0">−</tspan>
<tspan style="font-size: 11.21px; " x="28.064833" y="0">1</tspan>
</text>
</g>
</g>
</svg></span>, <span class="svg-td" style="width: 59.700001px;"><svg height="9.75" id="M169" style="vertical-align:-0.12375pt" version="1.1" viewbox="0 0 59.700001 9.75" width="59.700001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.75)">
<g transform="translate(72,-64.2)">
<text transform="matrix(1,0,0,-1,-71.95,64.37)">
<tspan style="font-size: 11.21px; " x="0" y="0">𝐾</tspan>
<tspan style="font-size: 11.21px; " x="11.645112" y="0">=</tspan>
<tspan style="font-size: 11.21px; " x="22.438416" y="0">2</tspan>
<tspan style="font-size: 11.21px; " x="28.042416" y="0">.</tspan>
<tspan style="font-size: 11.21px; " x="30.844416" y="0">1</tspan>
<tspan style="font-size: 11.21px; " x="36.448418" y="0">2</tspan>
<tspan style="font-size: 11.21px; " x="42.052418" y="0">1</tspan>
</text>
</g>
</g>
</svg></span>, <span class="svg-td" style="width: 44.9375px;"><svg height="9.9624996" id="M170" style="vertical-align:-0.1575pt" version="1.1" viewbox="0 0 44.9375 9.9624996" width="44.9375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.9625)">
<g transform="translate(72,-64.03)">
<text transform="matrix(1,0,0,-1,-71.95,64.24)">
<tspan style="font-size: 11.21px; " x="0" y="0">𝐿</tspan>
<tspan style="font-size: 11.21px; " x="11.03988" y="0">=</tspan>
<tspan style="font-size: 11.21px; " x="21.833183" y="0">1</tspan>
<tspan style="font-size: 11.21px; " x="27.437183" y="0">.</tspan>
<tspan style="font-size: 11.21px; " x="30.239183" y="0">5</tspan>
</text>
</g>
</g>
</svg></span>, <span class="svg-td" style="width: 38.587502px;"><svg height="9.7875004" id="M171" style="vertical-align:-0.1575pt" version="1.1" viewbox="0 0 38.587502 9.7875004" width="38.587502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.7875)">
<g transform="translate(72,-64.17)">
<text transform="matrix(1,0,0,-1,-71.95,64.37)">
<tspan style="font-size: 11.21px; " x="0" y="0">𝑀</tspan>
<tspan style="font-size: 11.21px; " x="14.368656" y="0">=</tspan>
<tspan style="font-size: 11.21px; " x="25.161961" y="0">0</tspan>
</text>
</g>
</g>
</svg></span>, and <span class="svg-td" style="width: 53.950001px;"><svg height="9.9624996" id="M172" style="vertical-align:-0.1575pt" version="1.1" viewbox="0 0 53.950001 9.9624996" width="53.950001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.9625)">
<g transform="translate(72,-64.03)">
<text transform="matrix(1,0,0,-1,-71.95,64.24)">
<tspan style="font-size: 11.21px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 11.21px; " x="12.642624" y="0">=</tspan>
<tspan style="font-size: 11.21px; " x="23.435928" y="0">0</tspan>
<tspan style="font-size: 11.21px; " x="29.039928" y="0">.</tspan>
<tspan style="font-size: 11.21px; " x="31.841928" y="0">0</tspan>
<tspan style="font-size: 11.21px; " x="37.445927" y="0">5</tspan>
</text>
</g>
</g>
</svg></span>, by 20th-order approximation solution.</table>

Mathematical Problems in Engineering

fig6

Figure 6

Figure 6: Application of Homotopy Analysis Method to the Unsteady Squeezing Flow of a Second-Grade Fluid between Circular Plates 