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

<table><span class="svg-td" style="width: 8.2375002px;"><svg height="9.8374996" id="M178" style="vertical-align:-0.1125pt" version="1.1" viewbox="0 0 8.2375002 9.8374996" width="8.2375002" 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>
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</g>
</svg></span> curve by the HAM with <span class="svg-td" style="width: 70.737503px;"><svg height="9.7875004" id="M179" style="vertical-align:-0.1575pt" version="1.1" viewbox="0 0 70.737503 9.7875004" width="70.737503" 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">𝑥</tspan>
<tspan style="font-size: 11.21px; " x="9.2690163" y="0">=</tspan>
<tspan style="font-size: 11.21px; " x="20.062321" y="0">0</tspan>
<tspan style="font-size: 11.21px; " x="25.666321" y="0">.</tspan>
<tspan style="font-size: 11.21px; " x="28.468321" y="0">0</tspan>
<tspan style="font-size: 11.21px; " x="34.072319" y="0">0</tspan>
<tspan style="font-size: 11.21px; " x="39.676319" y="0">0</tspan>
<tspan style="font-size: 11.21px; " x="45.280319" y="0">0</tspan>
<tspan style="font-size: 11.21px; " x="50.884319" y="0">1</tspan>
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</svg></span>, <span class="svg-td" style="width: 59.700001px;"><svg height="9.8999996" id="M180" style="vertical-align:-0.2475pt" version="1.1" viewbox="0 0 59.700001 9.8999996" width="59.700001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.9)">
<g transform="translate(72,-64.08)">
<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">4</tspan>
<tspan style="font-size: 11.21px; " x="36.448418" y="0">4</tspan>
<tspan style="font-size: 11.21px; " x="42.052418" y="0">9</tspan>
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</svg></span>, <span class="svg-td" style="width: 34.424999px;"><svg height="9.6000004" id="M181" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 34.424999 9.6000004" width="34.424999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.6)">
<g transform="translate(72,-64.32)">
<text transform="matrix(1,0,0,-1,-71.95,64.37)">
<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">2</tspan>
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</svg></span>, <span class="svg-td" style="width: 38.587502px;"><svg height="9.7875004" id="M182" 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>
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</svg></span>, and <span class="svg-td" style="width: 53.950001px;"><svg height="9.9624996" id="M183" 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>
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</svg></span>, by 10th-order approximation solution.</table>

Mathematical Problems in Engineering

fig8

Figure 8

Figure 8: Application of Homotopy Analysis Method to the Unsteady Squeezing Flow of a Second-Grade Fluid between Circular Plates 