Magnetohydrodynamic double-diffusive natural convection for nanofluid within a trapezoidal enclosure

Abstract

Double-diffusive natural convection flow in a trapezoidal cavity with various aspect ratios in the presence of water-based nanofluid and applied magnetic field in the direction perpendicular to the bottom and top parallel walls is investigated. The bottom and top parallel walls are considered to be insulated, whereas left and right walls are assumed to be uniformly heated and cold, respectively. The numerical computation is carried out to find the streamlines, isotherms, isoconcentrations, average Nusselt number, and average Sherwood number. This study is done for various values of Rayleigh number $(10^{5}\le Ra \le 10^{7})$, Hartmann number $(0\le Ha \le 120)$, various aspect ratios $(0.5\le A \le 2)$, the solid volume fraction $(0\le \varphi \le 0.1)$, and the inclination angle of cavity $(\phi )$. It is found that the strength of vortex decreases/increases as the magnetic field parameter/aspect ratio increases. It is also found that increase in the Rayleigh number causes natural convection due to the increase in the buoyancy forces. In nanofluid, mass transfer ratio is more effective than base fluid.

Appendix

$$\begin{aligned} P_{1}= & {} \frac{\mu _\mathrm{nf}}{\rho _\mathrm{nf}\alpha _\mathrm{f}},~~P_{2}=\frac{\sigma _\mathrm{nf}\rho _\mathrm{f}}{\sigma _\mathrm{f}\rho _\mathrm{nf}} Ha^{2},~~P_{3} = \frac{(\rho \beta )_\mathrm{nf}}{\rho _\mathrm{nf}\beta _\mathrm{f}} Ra Pr,~~P_{4}=\frac{\alpha _\mathrm{nf}}{\alpha _\mathrm{f}}, \\ B= & {} U G+ 0.5 V E A-H~P_{4},~ P_{6}=U G + 0.5 V E A-\frac{H}{Le},\\ G= & {} \frac{1}{1+2\eta A \cot \phi },~~E=-\frac{2 G \cot \phi (2 \xi -1)}{A},~~F = G^{2}[1+ \cot ^{2}\phi (2 \xi -1)^{2}], \\ H= & {} 4 G^{2} \cot ^{2}\phi (2\xi -1),~~ M=-U G+ V G(2\xi -1)\cot \phi + P_{1} H,\\ T_{1}= & {} -8P_{1} F G^{2}(1-2\xi )\cot ^{2}\phi +P_{1} F H \\&+F M- 4 A G^{3}\cot \phi [1+(1-2\xi )^{2}\cot ^{2}\phi ],\\ T_{2}= & {} -\frac{8}{A}P_{1} F G \cot \phi - 8P_{1} E G^{2}(1-2\xi )\cot ^{2}\phi \\&+P_{1} E H-\frac{8}{A}G^{3} P_{1} \cot \phi [1+(1-2\xi )^{2}\cot ^{2}\phi ]+M E-\frac{V E}{A},\\ T_{3}= & {} -\frac{4}{A}P_{1} E G \cot \phi -\frac{8}{A^{2}}P_{1} G^{2}(1-2\xi )\cot ^{2}\phi +\frac{P_{1} H}{A^{2}}+\frac{M}{A^{2}}-\frac{V E}{A},\\ T_{4}= & {} 24 G^{2}P_{1} \cot ^{2}\phi +32P_{1} E G^{3} A(1-2\xi )\cot ^{3}\phi \\&+\,24 G^{4} P_{1} \cot ^{2} \phi [1+(1-2\xi )^{2}\cot ^{2}\phi ] \\&-\,4 M G^{2} (1-2\xi )\cot ^{2}\phi +M H+4 V G^{3} \cot \phi [1+(1-2\xi )^{2}\cot ^{2}\phi ],\\ T_{5}= & {} \frac{2 P_{1} F}{A^{2}}+P_{1} E^{2}, \\ T_{6}= & {} 16G^{2}P_{1} E \cot ^{2}\phi +\frac{32}{A}P_{1} G^{3}(1-2\xi )\cot ^{3}\phi -\frac{4}{A}M G \cot \phi \\&+\,\frac{4}{A}V G^{2}(1-2\xi )\cot ^{2}\phi -\frac{V H}{A},\\ T_{7}= & {} -32P_{1} E G^{3}A \cot ^{3}\phi -96P_{1} G^{4}(1-2\xi )\cot ^{4}\phi \\&+\,8M G^{2}\cot ^{2}\phi -16V G^{3}(1-2\xi )\cot ^{3}\phi , \\ \frac{\partial \psi }{\partial \xi }= & {} \frac{1}{2h}(\psi _{i+1,j}-\psi _{i-1,j}) + O(h^{2}),\\ \frac{\partial \psi }{\partial \eta }= & {} \frac{1}{2h}(\psi _{i,j+1}-\psi _{i,j-1})+ O(h^{2}), \\ \frac{\partial ^{2} \psi }{\partial \xi ^{2}}= & {} \frac{1}{h^{2}}(\psi _{i+1,j}-2\psi _{i,j}+\psi _{i-1,j})+ O(h^{2}), \end{aligned}$$

$$\begin{aligned} \frac{\partial ^{2} \psi }{\partial \eta ^{2}}= & {} \frac{1}{h^{2}}(\psi _{i,j+1}-2\psi _{i,j}+\psi _{i,j-1})+ O(h^{2}), \\ \frac{\partial ^{2} \psi }{\partial \xi \partial \eta }= & {} \frac{1}{4h^{2}}(\psi _{i-1,j-1}-\psi _{i+1,j-1}+\psi _{i+1,j+1}-\psi _{i-1,j+1})+ O(h^{2}),\\ \frac{\partial ^{3} \psi }{\partial \xi ^{2}\partial \eta }= & {} \frac{1}{2h^{3}}(2\psi _{i,j-1}-2\psi _{i,j+1}-\psi _{i-1,j-1}\\&-\psi _{i+1,j-1}+\psi _{i+1,j+1}+\psi _{i-1,j+1})+ O(h^{2}), \\ \frac{\partial ^{3} \psi }{\partial \xi ^{3}}= & {} \frac{1}{h^{2}}(\psi _{\xi i+1,j}-2\psi _{\xi i,j}+\psi _{\xi i-1,j})+ O(h^{2}),\\ \frac{\partial ^{3} \psi }{\partial \eta ^{3}}= & {} \frac{1}{h^{2}}(\psi _{\eta i,j+1}-2\psi _{\eta i,j}+\psi _{\eta i,j-1})+ O(h^{2}),\\ \frac{\partial ^{3} \psi }{\partial \xi \partial \eta ^{2}}= & {} \frac{1}{2h^{3}}[2\psi _{i-1,j}-2\psi _{i+1,j}-\psi _{i-1,j-1}\\&+\psi _{i+1,j-1}+\psi _{i+1,j+1}-\psi _{i-1,j+1}]+ O(h^{2}), \\ \frac{\partial ^{4} \psi }{\partial \xi ^{4}}= & {} \frac{6}{h^{4}}[h(\psi _{\xi i+1,j}-\psi _{\xi i-1,j})-2(\psi _{i+1,j}-2\psi _{i,j}+\psi _{i-1,j})]+ O(h^{2}),\\ \frac{\partial ^{4} \psi }{\partial \eta ^{4}}= & {} \frac{6}{h^{4}}[h(\psi _{\eta i,j+1}-\psi _{\eta i,j-1})-2(\psi _{i,j+1}-2\psi _{i,j}+\psi _{i,j-1})]+ O(h^{2}), \\ \frac{\partial ^{4} \psi }{\partial \xi ^{3}\partial \eta }= & {} \frac{1}{2 h^{3}}[2\psi _{\xi i,j-1}-2\psi _{\xi i,j+1}-\psi _{\xi i-1,j-1}\\&-\psi _{\xi i+1,j-1}+\psi _{\xi i+1,j+1}+\psi _{\xi i-1,j+1}]+ O(h^{2}), \\ \frac{\partial ^{4} \psi }{\partial \xi \partial \eta ^{3}}= & {} \frac{1}{2 h^{3}}[2\psi _{\eta i-1,j}-2\psi _{\eta i+1,j}-\psi _{\eta i-1,j-1}\\&+\psi _{\eta i+1,j-1}+\psi _{\eta i+1,j+1}-\psi _{\eta i-1,j+1}]+ O(h^{2}), \\ \frac{\partial ^{4} \psi }{\partial \xi ^{2}\partial \eta ^{2}}= & {} \frac{1}{h^{4}}[4\psi _{i,j}-2(\psi _{i-1,j}+\psi _{i+1,j}+\psi _{i,j-1}+\psi _{i,j+1})+\psi _{i-1,j-1}\\&+\psi _{i+1,j-1}+\psi _{i+1,j+1}-\psi _{i-1,j+1}]+ O(h^{2}). \end{aligned}$$

Here, h is the step length on a uniform rectangular mesh in the transformed domain.