## 2 Problem Description  
The natural convection of a magnetohydrodynamic nanofluid in an enclosure subject to the effects of thermal radiation and the shape factor of nanoparticles is analyzed. A sample grid distribution and a physical model of this work are displayed in Figure 1. We assume the nanofluid to be Newtonian and incompressible, while we assume that the flow is laminar. Under the above-mentioned suppositions and the Boussinesq approximation, the energy, momentum, and continuity equations for the current problem can be defined as:  
\[\frac{\partial v}{\partial y}+\frac{\partial u}{\partial x}=0 \tag{1}\]  
\[v\frac{\partial u}{\partial y}+u\frac{\partial u}{\partial x}=-\frac{1}{\rho_ {nf}}\frac{\partial p}{\partial x}+\frac{\mu_{nf}}{\rho_{nf}}\bigg{(}\frac{ \partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial x^{2}}\bigg{)} \tag{2}\]  
\[v\frac{\partial v}{\partial y}+u\frac{\partial v}{\partial x}=-\frac{1}{\rho_ {nf}}\frac{\partial p}{\partial y}+\frac{\mu_{nf}}{\rho_{nf}}\bigg{(}\frac{ \partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}v}{\partial x^{2}}\bigg{)}+ \beta_{nf}g(T-T_{c})-\frac{\sigma_{f}B_{0}^{2}}{\rho_{nf}}v \tag{3}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{nf}}{ \big{(}\rho_{C}\big{)}_{nf}}\bigg{(}\frac{\partial^{2}T}{\partial x^{2}}+ \frac{\partial^{2}T}{\partial y^{2}}\bigg{)}-\frac{1}{\big{(}\rho_{C}\big{)}_ {nf}}\bigg{(}\frac{\partial q_{x,rad.}}{\partial y}+\frac{\partial q_{y,rad.}} {\partial x}\bigg{)} \tag{4}\]  
In these equations \(v\), \(u\), \(B_{0}\), \(P\), \(\sigma_{f}\), \(T\), and \(q_{md}\) denote the velocity in the \(y\)-direction, velocity in the \(x\)-direction, magnetic field, pressure, electric conductivity, and temperature, respectively, with the last variable representing radiative heat flux. For radiation, by applying the Rosseland approximation, we have:  
\[ \begin{split} q_{x,rad.}=-\Big{(}4\sigma^{*}/3k_{nf}^{*}\Big{)} \frac{\partial T^{4}}{\partial x},\ q_{y,rad.}=-\Big{(}4\sigma^{*}/3k_{nf}^{*} \Big{)}\frac{\partial T^{4}}{\partial y}\\ T^{4}\cong-3T_{\infty}^{4}+4T_{\infty}^{3}T\end{split}  \tag{5}\]  
So, Equation (4) has been altered to  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{nf}}{ \big{(}\rho C_{p}\big{)}_{nf}}\bigg{(}\frac{\partial^{2}T}{\partial x^{2}}+ \frac{\partial^{2}T}{\partial y^{2}}\bigg{)}+\frac{16\sigma^{*}T_{\infty}^{3} }{3k_{nf}^{*}\big{(}\rho C_{p}\big{)}_{nf}}\bigg{(}\frac{\partial^{2}T}{ \partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}\bigg{)} \tag{6}\]  
\(\rho_{nf}\), \(\mu_{nf}\), \((\rho Cp)_{nf}\) are defined as follows:  
\[\rho_{nf}=\phi\rho_{s}+(1-\phi)\rho_{f},\ \mu_{nf}=(1-\phi)^{-2.5}\mu_{f},\ \big{(}\rho C_{p}\big{)}_{nf}=\phi\big{(}\rho C_{p}\big{)}_{s}+(1-\phi)\big{(} \rho C_{p}\big{)}_{f} \tag{7}\]  
Considering the influences of nanoparticle shape, the nanofluid's thermal conductivity (\(k_{nf}\)) is characterized as follows:  
\[\frac{k_{nf}}{k_{f}}=\frac{(m-1)\phi\Big{(}k_{s}-k_{f}\Big{)}+k_{s}-(1-m)k_{f} }{k_{s}-(1-m)k_{f}-\phi\Big{(}k_{s}-k_{f}\Big{)}} \tag{8}\]  
The shape factor is represented here as \(m\). The shape factor for diverse particle shapes is illustrated in Table 1 [32]. Further, the thermophysical features of the nanofluid are listed in Table 2[15]. \begin{table}
\begin{tabular}{c c c c} \hline
**Particle Shapes** & **Sphere** & **Column** & **Lamina** \\ \hline \(m\) & 3 & 6.3698 & 16.1576 \\ \hline \end{tabular}
\end{table} 
Table 1: Shape factor values for various nanoparticle shapes [32]. Figure 1: (a) Physical model and coordinate system (b) grid distribution.