## 2 Problem Formulation  
In the present study, we consider the viscous, laminar, incompressible, natural convective flow in a square porous enclosure of length (\(L\)) filled with a micropolar Al\({}_{2}\)O\({}_{3}\)/water nanofluid in the presence of a magnetic field. The geometry and the configuration of the system are schematically shown in Figure 1.  
The top and bottom walls are insulated, and the fluid is isothermally heated and cooled at the left and right vertical walls at uniform temperatures, \(T_{h}\) and \(T_{c}\), respectively. In the case of natural convection, the buoyancy, which is a result of local fluid density variations due to temperature differences, is the dominant driving force, and the flow is permeated by an applied uniform external magnetic field, \(\mathbf{B}\), in the \(-y\) direction of strength, \(B_{0}\). The fluid is in thermal equilibrium with the porous medium; the physical properties of the fluid are assumed constant, except of the density in the buoyancy force, which is calculated by using the Boussinesq approximation. We consider a granular porous medium of glass spheres. The thermophysical properties for the nanofluid and the glass spheres are listed in Table 1.  
The fluid is hydrodynamically, thermally, and electrically isotropic. No-slip and impermeable boundary conditions are imposed on all boundaries. Joule heating, viscous dissipation, pressure work, and Hall effects are neglected, as they are assumed negligible. From [PERSON]'s law, the current density, \(\mathbf{J}\), relates to the electric field, \(\mathbf{E}\), as follows:  
\[\mathbf{J}=\sigma(\mathbf{E}+\mathbf{u}\times\mathbf{B}), \tag{1}\]  
where \(\sigma\) is the electrical conductivity, and \(\mathbf{u}=(u(x,y),\sigma(x,y))\) is the superficial velocity vector, with \(u\) and \(v\) being the velocity components along \(x\) and \(y\) axes, respectively. Using [PERSON]'s law, \(\
abla\times\mathbf{B}=\mu_{m}\mathbf{J}\), we get the following:  
\[\frac{1}{\mu_{m}}(\
abla\times\mathbf{B})=\sigma(\mathbf{E}+\mathbf{u}\times\mathbf{B}), \tag{2}\]  
omitting the inclusion of electromagnetic waves propagation and neglecting displacement currents, \(\mu_{m}\) is the magnetic permeability (the nanofluid is assumed to have the same magnetic permeability with \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\rho\) & \(C_{p}\) & \(\mathbf{k}\) & \(\beta\times 10^{-5}\) & \(\sigma\times 10^{-4}\) \\  & (\(\mathbf{kg}/\text{m}^{3}\)) & (\(\mathbf{j}/\text{Kg}\mathbf{K}\)) & (\(\mathbf{W}/\text{m}\mathbf{K}\)) & \(\left(\mathbf{K}^{-1}\right)\) & \(\left(\mathbf{S}/\text{m}\right)\) \\ \hline Pure Water & 997.1 & 4179 & 0.613 & 21 & 3 \\ Alumina (Al\({}_{2}\)O\({}_{3}\)) & 3970 & 765 & 40 & 0.85 & \(10^{-4}\) \\ Glass spheres & 2800 & 13.96 & 0.7 & - & \(10^{-11}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermophysical properties of water, alumina nanoparticles, and glass spheres, namely, the density (\(\rho\)), the heat capacity (\(C_{p}\)), the thermal conductivity (\(k\)), the thermal expansion coefficient (\(\beta\)), and the electric conductivity (\(\sigma\)) [34; 35; 36]. Figure 1: Geometry and boundary conditions of the problem. that of the base fluid). Using [PERSON]'s law and after algebraic manipulation, the magnetic induction equation is written as follows [37]:  
\[\frac{\partial\mathbf{B}}{\partial t}-\
abla\times\left(\mathbf{u}\times\mathbf{B}\right)= \frac{1}{\sigma\mu_{m}}\
abla^{2}\mathbf{B}. \tag{3}\]  
Micropolar fluid theory was introduced by [PERSON] [38] as a generalization of the Navier-Stokes equations. Micropolar theory examines the microstructure of the fluid, along with the inertial characteristics of the particles, which can rotate under the influence of local forces. Microrotation, which is a new kinematic variable introduced in [PERSON]'s model, is related the rotation of the particles. The micropolar vector describes the rotation of the particles contained in a volume element [22; 23]. A detailed analysis of the micropolar fluid flow theory is presented in [24].