Electrohydrodynamic Instability of Two Thin Viscous Leaky Dielectric Fluid Films in a Porous Medium

<table>Variation of growth rate <svg height="7.1875" id="M451" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 10.7625 7.1875" width="10.7625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,7.1875)">
<g transform="translate(72,-66.25)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜔</tspan>
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</g>
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</svg> with wavenumber <svg height="10.7375" id="M452" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 8.6000004 10.7375" width="8.6000004" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.7375)">
<g transform="translate(72,-63.41)">
<text transform="matrix(1,0,0,-1,-71.95,63.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑘</tspan>
</text>
</g>
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</svg> for various values of <svg height="10.925" id="M453" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 13.5625 10.925" width="13.5625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.925)">
<g transform="translate(72,-63.26)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜖</tspan>
</text>
<text transform="matrix(1,0,0,-1,-66.07,63.31)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
</g>
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</svg> when <svg height="10.9125" id="M454" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 46.612499 10.9125" width="46.612499" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜀</tspan>
<tspan style="font-size: 12.50px; " x="9.5147829" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.555172" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="27.806671" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="30.932423" y="0">4</tspan>
</text>
</g>
</g>
</svg>, <svg height="14.7875" id="M455" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 41.887501 14.7875" width="41.887501" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7875)">
<g transform="translate(72,-60.17)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑘</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.17,60.22)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-56.83,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.040389" y="0">5</tspan>
</text>
</g>
</g>
</svg>, <svg height="20.975" id="M456" style="vertical-align:-4.72992pt" version="1.1" viewbox="0 0 51.962502 20.975" width="51.962502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,20.975)">
<g transform="translate(72,-55.22)">
<text transform="matrix(1,0,0,-1,-71.95,59.98)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑆</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.95,66.03)">
<tspan style="font-size: 8.75px; " x="-0.87" y="10.76">1</tspan>
<tspan style="font-size: 8.75px; " x="0" y="0">(</tspan>
<tspan style="font-size: 8.75px; " x="2.9144161" y="0">0</tspan>
<tspan style="font-size: 8.75px; " x="7.2904158" y="0">)</tspan>
</text>
<text transform="matrix(1,0,0,-1,-48.77,59.98)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">0</tspan>
</text>
</g>
</g>
</svg>, <svg height="20.975" id="M457" style="vertical-align:-4.72992pt" version="1.1" viewbox="0 0 51.962502 20.975" width="51.962502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,20.975)">
<g transform="translate(72,-55.22)">
<text transform="matrix(1,0,0,-1,-71.95,59.98)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑆</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.95,66.03)">
<tspan style="font-size: 8.75px; " x="-0.87" y="10.76">2</tspan>
<tspan style="font-size: 8.75px; " x="0" y="0">(</tspan>
<tspan style="font-size: 8.75px; " x="2.9144161" y="0">0</tspan>
<tspan style="font-size: 8.75px; " x="7.2904158" y="0">)</tspan>
</text>
<text transform="matrix(1,0,0,-1,-48.77,59.98)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">0</tspan>
</text>
</g>
</g>
</svg>, <svg height="14.6" id="M458" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 40.762501 14.6" width="40.762501" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜖</tspan>
</text>
<text transform="matrix(1,0,0,-1,-66.07,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-57.73,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.040389" y="0">4</tspan>
</text>
</g>
</g>
</svg>, <svg height="13.7375" id="M459" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 47.775002 13.7375" width="47.775002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.7375)">
<g transform="translate(72,-61.01)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan>
<tspan style="font-size: 12.50px; " x="10.440005" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.480394" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="28.731894" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="31.857643" y="0">5</tspan>
</text>
</g>
</g>
</svg>, <svg height="14.6" id="M460" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 41.9375 14.6" width="41.9375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.34,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑟</tspan>
</text>
<text transform="matrix(1,0,0,-1,-56.79,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">1</tspan>
</text>
</g>
</g>
</svg>, and <svg height="14.7875" id="M461" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 69.425003 14.7875" width="69.425003" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7875)">
<g transform="translate(72,-60.17)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜖</tspan>
</text>
<text transform="matrix(1,0,0,-1,-66.07,60.22)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-57.73,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.040389" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="18.291889" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="23.493137" y="0">5</tspan>
<tspan style="font-size: 12.50px; " x="29.744637" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="34.958389" y="0">7</tspan>
</text>
</g>
</g>
</svg>, respectively.</table>

International Scholarly Research Notices

fig12

Figure 12

Figure 12: Electrohydrodynamic Instability of Two Thin Viscous Leaky Dielectric Fluid Films in a Porous Medium 