Rayleigh-Bénard Convection of Non-Newtonian Power-Law Fluids with Temperature-Dependent Viscosity

<table>Evolution of <svg height="17.25" id="M151" style="vertical-align:-4.15506pt" version="1.1" viewbox="0 0 45.362499 17.25" width="45.362499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.013)"><path d="M162 -163h-61v866h61v-866z" id="x7C"></path></g><g transform="matrix(.017,-0,0,-.017,4.533,12.013)"><path d="M642 419q-50 -36 -71 -94q-28 -78 -68 -170q-39 -89 -114 -135q-34 -21 -83 -31l-45 -219l-31 -31q-8 7 -15 29.5t-2 45.5l35 172q-153 10 -153 169q0 28 6 84q7 68 7 88q0 48 -27 48q-18 0 -51 -28l-7 26q62 75 118 75q49 0 49 -90q0 -47 -5 -81q-10 -72 -10 -108
q0 -117 82 -133l87 455l62 19l7 -2l-99 -478q91 12 131 120q34 93 51 154q20 73 36 88q42 40 103 56z" id="x1D713"></path></g>
<g transform="matrix(.012,-0,0,-.012,15.65,17.05)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.012,-0,0,-.012,25.395,17.05)"><path d="M433 39l-85 -51q-27 0 -47 20q-16 16 -24 44q-5 -3 -24 -16.5t-27 -18.5t-23 -13.5t-25 -12t-19 -3.5q-52 0 -86 36.5t-34 85.5q0 71 81 99q128 43 155 65v17q0 54 -23 83.5t-62 29.5q-28 0 -45 -19t-29 -66q-7 -23 -29 -23q-15 0 -29 13t-14 30t31 40q25 19 68.5 40.5
t81.5 29.5q49 0 82 -27q45 -39 45 -123v-185q0 -60 35 -60q18 0 36 11zM275 84v156q-12 -6 -49 -23t-41 -19q-31 -14 -46 -31t-15 -43q0 -35 22.5 -55.5t48.5 -20.5q19 0 42 10.5t38 25.5z" id="x61"></path></g><g transform="matrix(.012,-0,0,-.012,30.619,17.05)"><path d="M474 0h-214v26q32 4 39 12t-5 28l-74 117q-39 -58 -64 -106q-16 -25 -10 -35t44 -16v-26h-186v26q46 6 64 17t48 49t85 120l-87 136q-23 36 -40 47t-58 16v26h199v-26q-31 -4 -36.5 -12.5t9.5 -29.5l59 -93q37 56 58 94q13 23 7 30t-40 11v26h183v-26q-47 -5 -65 -13.5
t-45 -39.5q-24 -29 -78 -110l97 -148q28 -43 46.5 -55.5t63.5 -18.5v-26z" id="x78"></path></g>
<g transform="matrix(.017,-0,0,-.017,40.825,12.013)"><path d="M162 -163h-61v866h61v-866z" id="x7C"></path></g>
</svg> with <svg height="11.375" id="M152" style="vertical-align:-0.16379pt" version="1.1" viewbox="0 0 18.450001 11.375" width="18.450001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M631 18l-4 -26q-24 0 -35 1q-57 3 -92.5 23t-69.5 68q-21 30 -101 160q-13 22 -30.5 31t-54.5 9h-32v-159q0 -63 14 -77.5t74 -19.5v-28h-262v28q62 5 76 19.5t14 77.5v402q0 62 -13 76t-73 19v28h248q109 0 159 -34q67 -42 67 -131q0 -115 -127 -172q27 -52 89 -144
q49 -71 81 -104q31 -36 72 -47zM212 316h48q73 0 108 29q55 42 55 124q0 76 -42 112t-104 36q-41 0 -54 -10q-11 -7 -11 -44v-247z" id="x52"></path></g><g transform="matrix(.017,-0,0,-.017,10.908,11.112)"><path d="M433 39l-85 -51q-27 0 -47 20q-16 16 -24 44q-5 -3 -24 -16.5t-27 -18.5t-23 -13.5t-25 -12t-19 -3.5q-52 0 -86 36.5t-34 85.5q0 71 81 99q128 43 155 65v17q0 54 -23 83.5t-62 29.5q-28 0 -45 -19t-29 -66q-7 -23 -29 -23q-15 0 -29 13t-14 30t31 40q25 19 68.5 40.5
t81.5 29.5q49 0 82 -27q45 -39 45 -123v-185q0 -60 35 -60q18 0 36 11zM275 84v156q-12 -6 -49 -23t-41 -19q-31 -14 -46 -31t-15 -43q0 -35 22.5 -55.5t48.5 -20.5q19 0 42 10.5t38 25.5z" id="x61"></path></g>
</svg> for <svg height="11.125" id="M153" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 41.112499 11.125" width="41.112499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" id="x1D45A"></path></g><g transform="matrix(.017,-0,0,-.017,18.183,10.862)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,32.887,10.862)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g>
</svg> (solid line) and <svg height="11.125" id="M154" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 49.275002 11.125" width="49.275002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" id="x1D45A"></path></g><g transform="matrix(.017,-0,0,-.017,18.183,10.862)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,32.887,10.862)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,41.046,10.862)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g>
</svg> (dashdot line) and various values of <svg height="7.9499998" id="M155" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" id="x1D45B"></path></g>
</svg>.</table>

International Scholarly Research Notices

fig6

Figure 6

Figure 6: Rayleigh-Bénard Convection of Non-Newtonian Power-Law Fluids with Temperature-Dependent Viscosity 