## 2 Problem Formulation
In the proposed model, the Darcy-Brinkman momentum equations with buoyancy and advective inertia terms are used (see Equation (5)). The quantities that appear in the equations are the classical averaged quantities for flows in porous media. On the other hand, Equation (6), which is the transport equation for the microrotation vector, \(\mathbf{n}\), represents quantities which are not averaged directly. It is applied within the pores and, therefore, the correct equation to be used would be an averaged form of these equations. The microrotation vector is affected by the presence of particles and pore walls; however, a single-phase fluid equation in a porous setup was used, incorporating the effective microrotational properties of the particulate phase of the nanofluid, assuming volume-averaged values in porous representative elementary volumes (REVs) and avoiding the incorporation of the details of the pore structure in the expression of the angular momentum equation itself. The same concept applies for the presence of the magnetic field, which should affect the properties of the micropolar nanofluid due to their intrinsic nature; however, the magnetic field is not required to be implemented as a term in the transport equation for \(\mathbf{n}\), since its influence on the flow can be regarded solely through the particular term in the linear momentum equation. Thus, the governing equations of the problem, i.e., the conservation of mass, energy, and linear and angular momentum, as well as the induction equation, are written as Equations (4)-(8) [39; 40; 41]. \[\
abla\mathbf{\cdot}\mathbf{u}=0, \tag{4}\]  
\[=-\
abla p+\frac{\frac{\rho_{nf}}{\varepsilon}\big{(}\frac{\partial\mathbf{u}}{ \partial t}+\frac{1}{\varepsilon}(\mathbf{u}\cdot\
abla)\mathbf{u}\big{)}}{\varepsilon} \
abla^{2}\mathbf{u}-\frac{\mu_{nf}}{K_{D}}\mathbf{u}+\frac{k}{\varepsilon}\
abla \times\mathbf{n}-\mathbf{g}\big{(}\big{(}\rho\beta_{T}\big{)}_{nf}(T-T_{H})\big{)} \tag{5}\]  
\[+\frac{1}{\mu_{m}}(\
abla\times\mathbf{B})\times\mathbf{B},\]  
\[\rho_{nf}\bigg{/}\bigg{(}\frac{\partial\mathbf{n}}{\partial t}+\frac{1}{ \varepsilon}\mathbf{u}\cdot\
abla\mathbf{n}\bigg{)}=\gamma_{nf}\
abla^{2}\mathbf{n}+k \
abla\times\mathbf{u}-2 kn, \tag{6}\]  
\[\sigma_{ef}\frac{\partial\mathbf{B}}{\partial t}-\frac{1}{\mu_{m}}\
abla^{2}\mathbf{ B}-\sigma_{nf}\
abla\times\left(\mathbf{u}\times\mathbf{B}\right)=0, \tag{7}\]  
\[\Big{(}\rho C_{p}\Big{)}_{eff}\frac{\partial T}{\partial t}+\Big{(}\rho C_{p }\Big{)}_{nf}\mathbf{u}\cdot\
abla T=k_{eff}\
abla^{2}T. \tag{8}\]  
The boundary conditions for the natural convection problem studied are set to the following:  
\[ \begin{array}{llll}u=v=0,&T=T_{h},&\mathbf{n}=0\;\;for\;x=0,&0\leq y\leq 1,\\ u=v=0,&T=T_{c},&\mathbf{n}=0\;\;for\;x=1,&0\leq y\leq 1,\\ u=v=0,&T_{y}=0,&\mathbf{n}=0\;\;for\;y=0,1,&0\leq x\leq 1,\end{array}  \tag{9}\]  
\(p\) is the pressure, \(T=T(x,y)\) is the fluid temperature, \(\mathbf{n}\) is the superficial microrotation vector, \(\mathbf{B}=\big{(}B_{x}(x,y),B_{y}(x,y)\big{)}\) is the induced magnetic field, and \(\mathbf{g}\) is the acceleration due to gravity. \(\rho_{nf}\) is the density, \(\mu_{nf}\) is the dynamic viscosity, \(k\) is the vortex viscosity, \(\gamma_{nf}\) is the spin-gradient viscosity, and \(j\) is the micro-inertia density. For the present simulation study, it is further assumed that \(\gamma_{nf}\) has the following form, as proposed in [42; 43]. \[\gamma_{nf}=\left(\mu_{nf}+\frac{k}{2}\right)j. \tag{10}\]  
The effective density for the nanofluid is given by an Effective Medium Theory (EFM) approach [30]:  
\[\rho_{nf}=(1-\varphi)\rho_{f}+\varphi\rho_{p}, \tag{11}\]  
with \(\varphi\) being the volume fraction of the nanoparticles. The effective dynamic viscosity for the nanofluid is given by the following:  
\[\mu_{nf}=\mu_{f}\big{(}1+39.11\varphi+533.9\varphi^{2}\big{)}, \tag{12}\]  
which was measured experimentally by [PERSON] and [PERSON] [44].