A Linear Stability Analysis of Thermal Convection in a  Fluid Layer with Simultaneous Rotation and Magnetic Field Acting in Different Directions

Critical Rayleigh numbers and wave numbers at marginal stability for the onset of stationary convection when both bounding surfaces are free. Gravity vector <svg height="11.65" id="M426" style="vertical-align:-2.8119pt" version="1.1" viewbox="0 0 8.625 11.65" width="8.625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,8.087)"><path d="M254 68h52q81 0 129 -32.5t48 -87.5q0 -72 -69 -113t-192 -41q-194 0 -194 92q0 62 96 75q-47 17 -64.5 35t-17.5 47q0 44 28.5 71.5t90.5 45.5q-56 16 -90 56.5t-34 94.5q0 73 56 117q58 44 147 44q53 0 112 -22h129v-53h-80q40 -36 40 -98q0 -93 -92 -131
q-24 -10 -41 -12q-22 -2 -86 -2q-30 0 -49.5 -13.5t-19.5 -35.5q0 -21 21 -29t80 -8zM306 312q0 70 -15.5 100t-51.5 30t-50.5 -29t-14.5 -102q0 -70 15 -99.5t51 -29.5t51 29.5t15 100.5zM315 -52h-164q-21 -17 -28 -28t-7 -27q0 -67 134 -67q71 0 112.5 18t41.5 48
q0 56 -89 56z" id="x1D420"></path></g>
</svg> parallel to the magnetic field vector <svg height="11.6125" id="M427" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 13.35 11.6125" width="13.35" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path d="M759 25v-25h-339v25q54 6 71.5 20.5t17.5 52.5v228h-241v-228q0 -40 17 -54q16 -13 72 -19v-25h-336v25q54 8 69 20q16 13 16 53v479q0 41 -16.5 55t-68.5 19v25h336v-25q-54 -4 -71.5 -18.5t-17.5 -55.5v-204h241v204q0 41 -17.5 55.5t-71.5 18.5v25h339v-25
q-51 -4 -69.5 -19t-18.5 -55v-479q0 -40 16 -53q15 -12 72 -20z" id="x1D407"></path></g>
</svg>. <svg height="14.3125" id="M428" style="vertical-align:-2.2932pt" version="1.1" viewbox="0 0 13.175 14.3125" width="13.175" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path d="M745 361q0 -134 -83.5 -233t-214.5 -130l16 -11q97 -67 250 -132l-8 -23q-76 3 -131 16q-81 19 -242 125l-20 13q-129 8 -209 91t-80 208q0 160 116 271t289 111q136 0 226.5 -83t90.5 -223zM645 356q0 127 -57.5 201.5t-169.5 74.5q-126 0 -210.5 -104.5t-84.5 -248.5
q0 -97 46 -166.5t129 -87.5l84 15l29 -19q104 21 169 121.5t65 213.5z" id="x1D444"></path></g>
</svg> is the dimensionless magnetic parameter. <svg height="11.3125" id="M429" style="vertical-align:-0.1092pt" version="1.1" viewbox="0 0 11.075 11.3125" width="11.075" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
</svg> is the Rayleigh number obtained by the Tau Chebyshev method. <svg height="15.4375" id="M430" style="vertical-align:-3.41936pt" version="1.1" viewbox="0 0 26.025 15.4375" width="26.025" 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="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
<g transform="matrix(.012,-0,0,-.012,11.013,15.187)"><path d="M614 175l29 -10q-33 -109 -57 -154q-121 -26 -184 -26q-90 0 -160.5 29t-112.5 77t-63.5 105.5t-21.5 119.5q0 157 108 253t277 96q36 0 71.5 -5t69 -13.5t36.5 -8.5q15 -102 20 -150l-29 -8q-20 79 -66.5 114t-128.5 35q-119 0 -187.5 -86t-68.5 -207
q0 -140 73.5 -227.5t188.5 -87.5q73 0 119.5 37.5t86.5 116.5z" id="x43"></path></g><g transform="matrix(.012,-0,0,-.012,18.973,15.187)"><path d="M513 0h-211v26q44 5 55 17t11 60v166q0 123 -98 123q-59 0 -104 -51v-238q0 -48 10 -60.5t55 -16.5v-26h-213l1 26q46 4 57 16t11 61v477q0 48 -9.5 58.5t-58.5 14.5v24q63 9 147 35v-336q73 73 143 73q64 0 101 -42t37 -129v-175q0 -49 11 -61t55 -16v-26z" id="x68"></path></g>
</svg> is the Rayleigh number obtained by Chandrasekhar [<a href="/journals/mpe/2013/236901/#B6">1</a>].

Mathematical Problems in Engineering

tab7

Table 7

Table 7: A Linear Stability Analysis of Thermal Convection in a  Fluid Layer with Simultaneous Rotation and Magnetic Field Acting in Different Directions 

Table 7 | A Linear Stability Analysis of Thermal Convection in a  Fluid Layer with Simultaneous Rotation and Magnetic Field Acting in Different Directions 