## 3 Numerical Approach and Benchmarking of the Code
The governing equations are developed as follows ([PERSON] [47])  
Energy equation (\(\Phi=\mathrm{T}\)):  
\[\frac{\partial\omega_{1}}{\partial\alpha}+\frac{\partial}{\partial \kappa}\Big{[}\mathrm{U}_{1}\omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1 }}{\partial\kappa}\Big{]}+\frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{2} \omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1}}{\partial\kappa}\Big{]}+ \frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{3}-\Gamma_{\omega}\frac{ \partial\omega_{1}}{\partial\kappa}\Big{]}=\omega_{1}\frac{\partial\omega_{1 }}{\partial\kappa}+\omega_{2}\frac{\partial\Psi_{1}}{\partial\gamma}+\omega_{ 3}\frac{\partial\Omega_{1}}{\partial\alpha} \tag{13}\]4  
With \(\mathrm{b_{p}}=C_{p}^{0}\frac{\Delta x\Delta y\Delta z}{\Delta t}\)  
Vorticity equation: x-component (\(\Phi=\omega_{1}\)):  
\[\frac{\partial\omega_{1}}{\partial\alpha}+\frac{\partial}{\partial \kappa}\Big{[}\mathrm{U}_{1}\omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1 }}{\partial\kappa}\Big{]}+\frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{2} \omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1}}{\partial\kappa}\Big{]}+ \frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{3}-\Gamma_{\omega}\frac{ \partial\omega_{1}}{\partial\kappa}\Big{]}=\omega_{1}\frac{\partial\omega_{1 }}{\partial\kappa}+\omega_{2}\frac{\partial\Psi_{1}}{\partial\gamma}+\omega_{ 3}\frac{\partial\Omega_{1}}{\partial\alpha} \tag{13}\]5  
With:  
\[\frac{\partial\omega_{1}}{\partial\alpha}+\frac{\partial}{\partial \kappa}\Big{[}\mathrm{U}_{1}\omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1 }}{\partial\kappa}\Big{]}+\frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{2} \omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1}}{\partial\kappa}\Big{]}+ \frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{3}-\Gamma_{\omega}\frac{ \partial\omega_{1}}{\partial\kappa}\Big{]}=\omega_{1}\frac{\partial\omega_{1 }}{\partial\kappa}+\omega_{2}\frac{\partial\Psi_{1}}{\partial\gamma}+\omega_{ 3}\frac{\partial\Omega_{1}}{\partial\alpha} \tag{13}\]6  
\[\frac{\partial\omega_{1}}{\partial\alpha}+\frac{\partial}{\partial \kappa}\Big{[}\mathrm{U}_{1}\omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1 }}{\partial\kappa}\Big{]}+\frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{2} \omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1}}{\partial\kappa}\Big{]}+ \frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{3}-\Gamma_{\omega}\frac{ \partial\omega_{1}}{\partial\kappa}\Big{]}=\omega_{1}\frac{\partial\omega_{1 }}{\partial\kappa}+\omega_{2}\frac{\partial\Psi_{1}}{\partial\gamma}+\omega_{ 3}\frac{\partial\Omega_{1}}{\partial\alpha} \tag{13}\]7  
\[\frac{\partial\omega_{1}}{\partial\alpha}+\frac{\partial}{\partial \kappa}\Big{[}\mathrm{U}_{1}\omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1 }}{\partial\kappa}\Big{]}+\frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{2} \omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1}}{\partial\kappa}\Big{]}+ \frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{3}-\Gamma_{\omega}\frac{ \partial\omega_{1}}{\partial\kappa}\Big{]}=\omega_{1}\frac{\partial\omega_{1 }}{\partial\kappa}+\omega_{2}\frac{\partial\Psi_{1}}{\partial\gamma}+\omega_{ 3}\frac{\partial\Omega_{1}}{\partial\alpha} \tag{13}\]8  
Vorticity equation: y-component (\(\Phi=\omega_{2}\)):  
\[\frac{\partial\omega_{1}}{\partial\alpha}+\frac{\partial}{\partial \kappa}\Big{[}\mathrm{U}_{1}\omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1 }}{\partial\kappa}\Big{]}+\frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{2} \omega_{1}-\Gamma_{\omega}\frac{\partial\omega_{1}}{\partial\kappa}\Big{]}+ \frac{\partial}{\partial\gamma}\Big{[}\mathrm{U}_{3}-\Gamma_{\omega}\frac{ \partial\omega_{1}}{\partial\kappa}\Big{]}=\omega_{1}\frac{\partial\omega_{1 }}{\partial\kappa}+\omega_{2}\frac{\partial\Psi_{1}}{\partial\gamma}+\omega_{ 3}\frac{\partial\Omega_{1}}{\partial\alpha} \tag{13}\]9  
With:  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]0  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]1  
Vorticity equation: z-component (\(\Phi=\omega_{3}\)):  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]2  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]3  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]4  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]5  
Microrotation equation: x-component (\(\Phi=\mathrm{h_{1}}\)):  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]6  
With \(\mathrm{b_{p}}=\mathrm{h_{1p}^{0}}\frac{\Delta x\Delta y\Delta z}{\Delta t}+ \mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{t}}{\rho_{at}}\Big{)}\Big{(} \omega_{1p}-\mathrm{h_{1p}}\Big{)}\)  
Microrotation equation: y-component (\(\Phi=\mathrm{h_{2}}\)):  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]7  
With \(\mathrm{b_{p}}=\mathrm{h_{2p}^{0}}\frac{\Delta x\Delta y\Delta z}{\Delta t}+ \mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{t}}{\rho_{at}}\Big{)}\Big{(} \omega_{2p}-\mathrm{h_{2p}}\Big{)}\)  
Vector potential equations:  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]8 \[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{R}\mathrm{Pr}\Big{(}\frac{\rho_{1}\rho_{\rm{d}}}{\left(\rho\right) \Psi_{1}}\Big{)}\Big{(}\frac{\partial\tau}{\partial\kappa}-\mathrm{N}\frac{ \partial\Gamma}{\delta x}\Big{)}+\mathrm{Pr}\,\mathrm{Ha}^{2}\big{(}\frac{\rho _{1}}{\rho_{\rm{d}}}\Big{)}\Big{(}\frac{\sigma_{1}}{\sigma_{1}}\Big{)}\Big{(} \frac{\partial\upsilon_{1}}{\partial\kappa}\Big{)}\]9\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{Pr}\,\mathrm{Ha}^{2}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}}\Big{)}\Big{(} \frac{\partial\tau}{\partial\kappa}\Big{)}\]0  
The equations governing the problem are discretized by applying the control volume method.