## 3 Numerical Treatments  
The numerical treatments used to solve the governing equations are the non-orthogonal grids FVM. The real physical domain \((X,Y)\) is converted to a rectangular domain \((\xi,\eta)\) using the following functions:  
\[\xi=\frac{X+Y\cot\Phi}{1+2Y\cot\Phi},\eta=\frac{Y}{A}. \tag{32}\]Using (32), the advection terms are given as:  
\[\frac{\partial(U\Omega)}{\partial X}+\frac{\partial(V\Omega)}{\partial Y}=\frac{1 }{J^{*}}\bigg{[}\frac{\partial}{\partial\xi}(\Omega U^{*})+\frac{\partial}{ \partial\eta}(\Omega V^{*})\bigg{]}, \tag{33}\]  
where  
\[U^{*}=\beta_{11}U+\beta_{21}V, \tag{34}\]  
\[V^{*}=\beta_{22}V+\beta_{12}U, \tag{35}\]  
\[\beta_{11}=\frac{\partial Y}{\partial\eta},\ \beta_{12}=-\frac{\partial Y}{ \partial\xi}, \tag{36}\]  
\[\beta_{21}=-\frac{\partial X}{\partial\eta},\ \beta_{22}=\frac{\partial X}{ \partial\xi}. \tag{37}\]  
Similar, the diffusive terms are expressed as:  
\[\begin{array}{ll}\frac{\partial}{\partial X}\Big{[}\Gamma^{*}\frac{ \partial\Omega}{\partial X}\Big{]}+\frac{\partial}{\partial Y}\Big{[}\Gamma^{* }\frac{\partial\Omega}{\partial Y}\Big{]}&=\frac{1}{I}\frac{\partial}{ \partial\xi}\Big{[}\Gamma^{*}\alpha_{11}\frac{\partial\Omega}{\partial\xi}+ \Gamma^{*}\alpha_{12}\frac{\partial\Omega}{\partial\eta}\Big{]}\\ &+\frac{1}{I^{*}}\frac{\partial}{\partial\eta}\Big{[}\Gamma^{*}\alpha_{22} \frac{\partial\Omega}{\partial\eta}+\Gamma^{*}\alpha_{12}\frac{\partial\Omega }{\partial\xi}\Big{]}\end{array} \tag{38}\]  
In Equation (38), \(\alpha_{11}\), \(\alpha_{22}\) are expressed as:  
\[\alpha_{11}=\frac{\alpha^{*}}{J^{*}}=\frac{X_{\eta}^{2}+Y_{\eta}^{2}}{J^{*}}= \frac{\beta_{21}^{2}+\beta_{11}^{2}}{J^{*}}, \tag{39}\]  
\[\alpha_{22}=\frac{\gamma^{*}}{J^{*}}=\frac{X_{\xi}^{2}+Y_{\xi}^{2}}{J^{*}}= \frac{\beta_{12}^{2}+\beta_{22}^{2}}{J^{*}}. \tag{40}\]  
The upwind scheme is used to discretize the transformed convective terms while the diffusive terms are treated using the central differences scheme. The alternating direction implicit method (ADI) is introduced to solve the obtained algebraic system. The iterative technique is employed until the convergence criterion (\(10^{-6}\)) is fullfiled. Additionally, after many grid tests being made (as it is presented in Table 2), the grid size \(101\times 101\) was found suitable for all computations. Further, Figure 2 shows the mesh used in all computations, which consists of \(101\times 101\) nodes.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Grid Size** & **(N\(\mathbf{u}\))\(\mathbf{u}\)** & **(N\(\mathbf{u_{s}}\))\(\mathbf{u_{w}}\)** \\ \hline \(31\times 31\) & 1.396400 & 2.751032 \\ \(41\times 41\) & 1.297873 & 2.877911 \\ \(61\times 61\) & 1.935634 & 3.034790 \\ \(81\times 81\) & 1.964531 & 3.125698 \\ \(101\times 101\) & 1.993429 & 3.196916 \\ \(121\times 121\) & 2.021942 & 3.259058 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Grid independency test at \(Ra=10^{5},\ H=0.1,\ Ha=10,\ \phi_{Al}=\phi_{Cu}=1\%,\ \Phi=\frac{\pi}{3},\gamma=0\).  
Figure 2: Mesh generation of the current geometry.  
Figure 3 illustrates a comparison between the current code results and the results obtained by [PERSON] et al. [47] for the case of a square cavity in which the left wall is kept at a hot temperature, the right wall is kept at a cold temperature while the lower and upper walls of the cavity are considered adiabatic. The comparison is implemented for the case where \(Ra=10^{5},\;Pr=0.71,\Theta=10^{-4}\). Contours for the streamlines, isotherms, local Bejan number and local entropy generation due to heat transfer are constructed. Comparison shows a very good agreement between contours resulted from the current code to those presented in [PERSON] et al. [47].  
Figure 3: Comparisons of the streamlines, isotherms, local Bejan number and local entropy generation due to heat transfer with those of [PERSON] et al. [47]\(Pr=0.71,\;Ra=10^{5},\;\varphi_{1}=10^{-4}\).