## 2 Formulation of the Problem
\begin{array}{ll}\varepsilon\big{(}\rho c_{p}\big{)}_{\text{ln}f}\frac{ \partial T_{fp}}{\partial t}+&\left(\rho c_{p}\right)_{\text{ln}f}\Big{(}u_{p }\frac{\partial T_{fp}}{\partial x}+v_{p}\frac{\partial T_{fp}}{\partial y} \Big{)}\\ &=\varepsilon k_{\text{ln}f}\Big{(}\frac{\partial^{2}T_{fp}}{\partial x^{2}} +\frac{\partial^{2}T_{fp}}{\partial y^{2}}\Big{)}+h\Big{(}T_{p}-T_{fp}\Big{)} \end{array}  \tag{7}\]  
\[ \begin{array}{ll}(1-\varepsilon)\big{(}\rho c_{p}\big{)}_{p}\frac{ \partial T_{p}}{\partial t}=(1-\varepsilon)k_{p}\Bigg{(}\frac{\partial^{2}T_{ p}}{\partial x^{2}}+\frac{\partial^{2}T_{p}}{\partial y^{2}}\Bigg{)}+h \Big{(}T_{fp}-T_{p}\Big{)}\end{array}  \tag{8}\]  
The symbols in Equations (4)-(8) are the velocity fields \(u_{p}\), \(v_{p}\), the temperature for the hybrid nanofluid/solid phase \(T_{fp}\), \(T_{p}\), the porosity \(\varepsilon\), the permeability \(K\), the electrical conductivity \(\sigma_{\text{ln}f}\), the thermal conductivity \(k_{\text{ln}f}\), the independent variables (Cartesian coordinates and time) \(x,y,\ t\) and the gravity acceleration \(g\). In addition, the following dimensionless quantities are submitted:  
\[(U_{p},V_{p})=\frac{(u_{p},\ v_{p})L}{\alpha_{f}},\ \Big{(}\theta_{fp},\ \theta_{p} \Big{)}=\frac{\Big{(}\ T_{fp},\ T_{p}\Big{)}-T_{c}}{\Delta T},\ \Delta T=\Big{(}\ T_{fp},\ T_{p}\Big{)}_{h}-T_{c} ),(XL,YL)=(x,y)\,\ P=\frac{L^{2}}{\rho_{\text{ln}f}\ \alpha_{f}}^{2}p,\ \ \tau=\frac{\alpha_{f}}{L^{2}}\ t \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **H\({}_{2}\)O** & **Cu** & **Al\({}_{2}\)O\({}_{3}\)** \\ \hline \(\rho\) & 997.1 & 8933 & 3970 \\ \(C_{\text{P}}\) & 4179 & 385 & 765 \\ \(k\) & 0.613 & 401 & 40 \\ \(\beta\) & \(21\times 10^{-5}\) & \(1.67\times 10^{-5}\) & \(0.85\times 10^{-5}\) \\ \(\sigma\) & 0.05 & \(5.96\times 10^{7}\) & \(1\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the components of the hybrid nanofluid. Substituting Equation (9) into Equations (4)-(8), the following combined dimensionless system is given:  
\[\frac{\partial U_{p}}{\partial X}+\frac{\partial V_{p}}{\partial Y}=0 \tag{10}\]  
\[ \begin{array}{ll}\frac{1}{\epsilon}\frac{\partial U_{p}}{\partial\tau}+\frac{ 1}{\epsilon}&\left[U_{p}\frac{\partial\left(\frac{U_{p}}{\epsilon}\right)}{ \partial X}+V_{p}\frac{\partial\left(\frac{U_{p}}{\epsilon}\right)}{\partial Y }\right]\\ &=-\frac{\partial p}{\partial X}+Pr\frac{1}{\epsilon}\frac{\mu_{Inf}}{\mu_{f}} \frac{\rho_{f}}{\rho_{inf}}\left[\frac{\partial^{2}U_{p}}{\partial X^{2}}+\frac {\partial^{2}U_{p}}{\partial Y^{2}}\right]\\ &-\left[\frac{p_{r}}{D_{u}(Y)}\frac{\mu_{Inf}}{f},\frac{\rho_{f}}{\rho_{inf}} +\frac{C_{f}}{\partial\left(\
abla_{H}(Y)}\sqrt{U_{p}^{2}+V_{p}^{2}}\right]U_ {p}\right.\\ &\left.\qquad+RaPr\left(\frac{\rho_{Inf}}{\rho_{Inf}}\right)\frac{\rho_{f}}{ \rho_{inf}}\frac{\rho_{f}}{\rho_{inf}}\theta_{fp}\sin\gamma\\ &-\frac{\sigma_{inf}}{\sigma_{f}\epsilon}\frac{\rho_{f}}{\rho_{Inf}}Ha^{2}Pr \big{(}V_{p}\sin\gamma\cos\gamma+U_{p}\sin^{2}\gamma\big{)}\end{array}  \tag{11}\]  
\[ \begin{array}{ll}\frac{1}{\epsilon}\frac{\partial V_{p}}{\partial\tau}+\frac {1}{\epsilon^{2}}&\left[U_{p}\frac{\partial\left(\frac{V_{p}}{\epsilon}\right) }{\partial X}+V_{p}\frac{\partial\left(\frac{V_{p}}{\epsilon}\right)}{ \partial Y}\right]\\ &=-\frac{\partial p}{\partial Y}+Pr\frac{1}{\epsilon}\frac{\mu_{Inf}}{\mu_{f} }\frac{\rho_{Inf}}{\rho_{inf}}\left[\frac{\partial^{2}V_{p}}{\partial X^{2}}+ \frac{\partial^{2}V_{p}}{\partial Y^{2}}\right]\\ &-\left[\frac{p_{r}}{D_{u}(Y)}\frac{\mu_{Inf}}{\mu_{f}}\frac{\rho_{f}}{\rho_{ inf}}+\frac{C_{f}}{\sqrt{Da(Y)}}\sqrt{U_{p}^{2}+V_{p}^{2}}\right]V_{p}\\ &\qquad\qquad+RaPr\left(\frac{\rho_{Inf}}{\rho\rho_{Inf}}\frac{\rho_{f}}{ \rho_{Inf}}\frac{\rho_{f}}{\rho_{Inf}}\theta_{fp}\cos\gamma\right.\\ &\left.\qquad-\frac{\sigma_{inf}}{\sigma_{f}\epsilon}\frac{\rho_{f}}{\rho_{ inf}}Ha^{2}Pr(U_{p}\sin\gamma\cos\gamma+V_{p}\cos^{2}\gamma)\right.\end{array}  \tag{12}\]  
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