## 3 Formulation of the Mathematical Model  
### Governing Equations and Boundary Conditions  
The stationary natural Magnetohydrodynamics (MHD) convection flow is analyzed through a novel shape of porous cavity for hybrid nanofluid. For modeling a porous medium, the model of Brinkman-Forchheimeris used [36]. Navier-Stokes and heat equa  
Figure 1: The geometry of the physical model. Figure 2: Mesh of the model. tions, expressed in Cartesian coordinates for the problem taking into account the above assumptions, can be formulated as follows:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\ =\ 0 \tag{1}\]  
\[\frac{1}{\varepsilon_{p}^{2}}\left(u\frac{\partial u}{\partial x}+v\frac{ \partial u}{\partial y}\right)\ =\ -\frac{1}{\rho_{nf}}\frac{\partial p}{\partial x}+\frac{v_{nf}}{ \varepsilon_{p}}\bigg{(}\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^ {2}u}{\partial y^{2}}\bigg{)}-v_{nf}\frac{u}{K}-\frac{F_{c}}{\sqrt{K}}u|u| \tag{2}\]  
\[\frac{1}{\varepsilon_{p}^{2}}\left(u\frac{\partial v}{\partial x}+v\frac{ \partial v}{\partial y}\right)\ =\ -\frac{1}{\rho_{nf}}\frac{\partial p}{\partial y}+\frac{v_{nf}}{ \varepsilon_{p}}\bigg{(}\frac{\partial^{2}v}{\partial x^{2}}+\frac{\partial^ {2}v}{\partial y^{2}}\bigg{)}-v_{nf}\frac{v}{K}-\frac{F_{c}}{\sqrt{K}}v|u|+ \beta_{nf}g(T-T_{avg})+\frac{\sigma_{nf}}{\rho_{nf}}B_{0}^{2}v \tag{3}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\ =\ \alpha_{nf} \bigg{(}\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y ^{2}}\bigg{)} \tag{4}\]  
where \(\alpha_{nf}\ =\ \frac{k_{nf}}{\left(\rho\varepsilon_{p}\right)_{nf}}\) is the thermal diffusivity of the nanofluid, \(|u|\ =\ \sqrt{u^{2}+v^{2}}\) amplitude velocity. \(F_{c}=\frac{b}{\sqrt{a}e^{3/2}}\) denotes the Forchheimer coefficient (where a = 150 and b = 1.75) and demonstrates the operative thermal conductivity of porous medium saturated with nanofluid, where the porous medium K is the permeability and \(\epsilon_{\rm p}\) is the porosity of the medium, that can be described as follows [37]:  
\[K=\frac{\varepsilon_{p}{}^{3}d_{m}^{2}}{150\big{(}1-\varepsilon_{p}\big{)}^{2}} \tag{5}\]  
For the base fluid and nanoparticles, Table 1 exhibits the thermophysical properties. The following nondimensional variables are employed to transform these governing equations:  
\[X\ =\ \frac{x}{L},Y\ =\ \frac{y}{L},U\ =\ \frac{u.L}{\alpha_{bf}},V\ =\ \frac{v.L}{\alpha_{bf}},\ \theta\ =\ \frac{T-T_{f}}{T_{h}-T_{f}},P\ =\ \frac{\left(p+\rho_{0 bf}/g_{y}\right)L^{2}}{ \rho_{bf}\alpha_{bf}^{2}} \tag{6}\]  
Dimensionless numbers are written as:  
\[Pr\ =\ \frac{v_{bf}}{\alpha_{bf}},Da=\frac{K}{L^{2}},Ha=LB_{0}\sqrt{\frac{ \sigma_{bf}}{\mu_{bf}}},Ra\ =\ \frac{\beta_{bf}g(T_{h}-T_{c})L^{3}}{\alpha_{bf} \gamma_{bf}} \tag{7}\]  
The nondimensional governing equations form as follows:  
\[\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}\ =\ 0 \tag{8}\]  
Figure 2:0  
Figure 2:1 \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Physical Properties** & \(C_{p}\)**(/kg\(\cdot\)K)** & \(k\)**(W/m\(\cdot\)k)** & \(\rho\)**(kg/m\({}^{3}\))** & \(\beta\)**\(\cdot\)\(10^{-5}\)**(K\({}^{-1}\))** & \(\sigma\)**(s/m)** & \(\sigma\)**(m\({}^{2}\)/s)** \\ \hline Water & 4179 & 0.613 & 997.1 & 21 & 5.5\(\times 10^{-6}\) & 1.47 \(\times 10^{-7}\) \\ \hline Ag & 235 & 429 & 10500 & 5.4 & 8.1\(\times 10^{-4}\) & 147 \(\times 10^{-3}\) \\ \hline MgO & 879 & 30 & 3580 & 3.36 & 8 \(\times 10^{-4}\) & 95.3 \(\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermophysical properties of the nanoparticles and the fluid (Ag/MgO/water (50:50)) [37].