Unsteady Magnetohydrodynamic Heat Transfer in a Semi-Infinite Porous Medium with Thermal Radiation Flux: Analytical and Numerical Study

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td align="left"><span style="width: 7.875px;"><svg height="9.8625002" id="M176" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 7.875 9.8625002" width="7.875" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.8625)">
<g transform="translate(72,-64.11)">
<text transform="matrix(1,0,0,-1,-71.95,66.45)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑦</tspan>
</text>
</g>
</g>
</svg></span></td><td align="center"><svg height="13.775" id="M177" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 23.325001 13.775" width="23.325001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.775)">
<g transform="translate(72,-60.98)">
<text transform="matrix(1,0,0,-1,-71.95,61.03)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑀</tspan>
</text>
<text transform="matrix(1,0,0,-1,-58.26,66.03)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg> 5.0 Laplace/NSM</td><td align="center"><svg height="13.775" id="M178" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 23.325001 13.775" width="23.325001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.775)">
<g transform="translate(72,-60.98)">
<text transform="matrix(1,0,0,-1,-71.95,61.03)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑀</tspan>
</text>
<text transform="matrix(1,0,0,-1,-58.26,66.03)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg> 8.0 Laplace/NSM</td><td align="center"><svg height="13.775" id="M179" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 23.325001 13.775" width="23.325001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.775)">
<g transform="translate(72,-60.98)">
<text transform="matrix(1,0,0,-1,-71.95,61.03)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑀</tspan>
</text>
<text transform="matrix(1,0,0,-1,-58.26,66.03)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg> 10.0 Laplace/NSM</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td align="left">0.0</td><td align="center">1.0/1.0</td><td align="center">1.0/1.0</td><td align="center">1.0/1.0</td></tr><tr><td align="left">0.5</td><td align="center">0.38047/0.35236</td><td align="center">0.29907/0.28788</td><td align="center">0.26028/0.25409</td></tr><tr><td align="left">1.0</td><td align="center">0.13844/0.10987</td><td align="center">0.09607/0.08472</td><td align="center">0.07858/0.07234</td></tr><tr><td align="left">1.5</td><td align="center">0.04490/0.03069</td><td align="center">0.02992/0.02458</td><td align="center">0.02424/0.02158</td></tr><tr><td align="left">2.0</td><td align="center">0.01205/0.00813</td><td align="center">0.00800/0.00694</td><td align="center">0.00651/0.00627</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Dimensionless spatial velocity distribution (<svg height="7.1624999" id="M169" style="vertical-align:-0.11285pt" version="1.1" viewbox="0 0 7.5374999 7.1624999" width="7.5374999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,7.1625)">
<g transform="translate(72,-66.27)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan>
</text>
</g>
</g>
</svg>) for <svg height="14.6" id="M170" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 55.612499 14.6" width="55.612499" 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">𝐺</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.77,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑟</tspan>
</text>
<text transform="matrix(1,0,0,-1,-55.23,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405136" y="0">0</tspan>
</text>
</g>
</g>
</svg> <svg height="14.2375" id="M171" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 17.125 14.2375" width="17.125" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.2375)">
<g transform="translate(72,-60.61)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐾</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.42,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑟</tspan>
</text>
</g>
</g>
</svg> = 1.0, <svg height="16.6" id="M172" style="vertical-align:-4.74141pt" version="1.1" viewbox="0 0 56.924999 16.6" width="56.924999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,16.6)">
<g transform="translate(72,-58.72)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐾</tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.42,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑝</tspan>
</text>
<text transform="matrix(1,0,0,-1,-54.17,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405136" y="0">0</tspan>
</text>
</g>
</g>
</svg>, <svg height="14.6" id="M173" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 53.424999 14.6" width="53.424999" 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">𝑃</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.52,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑟</tspan>
</text>
<text transform="matrix(1,0,0,-1,-56.98,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405136" y="0">7</tspan>
</text>
</g>
</g>
</svg>, and <svg height="10.9125" id="M174" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 43.9375 10.9125" width="43.9375" 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">𝑡</tspan>
<tspan style="font-size: 12.50px; " x="7.37677" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="19.417158" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="25.668659" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="28.794409" y="0">2</tspan>
</text>
</g>
</g>
</svg> for the effect of Hartmann number (<svg height="13.775" id="M175" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 23.325001 13.775" width="23.325001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.775)">
<g transform="translate(72,-60.98)">
<text transform="matrix(1,0,0,-1,-71.95,61.03)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑀</tspan>
</text>
<text transform="matrix(1,0,0,-1,-58.26,66.03)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg>).

Advances in Numerical Analysis

tab1

Table 1

Table 1: Unsteady Magnetohydrodynamic Heat Transfer in a Semi-Infinite Porous Medium with Thermal Radiation Flux: Analytical and Numerical Study 