## 4 Results and Discussion  
In this research, a numerical study was carried out to analyze the influences of the Hartmann number (Ha = 0-50), Rayleigh number (Ra = 10\({}^{3}\), 10\({}^{4}\), and 10\({}^{5}\)), radiation parameter (\(N\) = 1-3), volume fraction of the nanofluid (\(\phi\) = 0-0.04), and shape factor (\(m\) = 3, 6.3698, and 16.1576) on magnetohydrodynamic nanofluid natural convection in a cavity. The influences of thermal radiation and the shape factor of nanoparticles were determined. The impacts of the Ra and Ha on isotherms and streamlines are illustrated in Figures 2 and 3, respectively. As shown by the increasing Ra, the stream function's maximum absolute value increases, hence, the flow moves more quickly in the cavity. This phenomenon is due to the intensity of the natural convection, which has a direct relationship with the stream function's maximum absolute value. On the other hand, a higher Ra leads to a distortion of the isotherms and consequently causes the vertical stratification of isotherms, particularly at lower Ha and higher Ra. That is, the thermal boundary layer develops along two side walls. This is because, at a higher Ra and a lower Ha, the prevailing mode of heat transfer is convection. Moreover, the power of natural convection and, consequently, the stream function's maximum absolute value both diminish as Ha goes up. In addition, as Ha goes up, the central streamline expands, and the core's temperature stratification diminishes. So, the two walls' thermal boundary layers will disappear. In fact, in this case, the isotherms are parallel to each other. Since an increasing Ha leads to a decreased buoyancy force, the conduction mode of heat transfer dominates over the convection mode, especially at higher Ha. Figure 4 demonstrates the influences of N and \(\phi\) on the local and average Nusselt number. Based on Equation (15), the multiplication of three terms, i.e., \((k_{n\!/\!k}/k)\), \(((4/3)N+1)\), and \(\partial\theta/\partial X\), will create a non-dimensional number called the local Nusselt number. By increasing \(\phi\), the growth of the ratio \((k_{n\!/\!k}/k_{\!/\!j})\) will overcome a smaller \(\partial\theta/\partial X\) for a specified \(N\), so the local Nusselt number will rise as \(\phi\) is increased. Conversely, by increasing \(N\), the increased \(((4/3)N+1)\) term will counteract the reduced value of \(\partial\theta/\partial X\) for a given \(\phi\), which will result in a higher \(Nu_{loc}\). However, the average Nu number behaves similarly to the local Nu number. Figures 5 and 6 illustrate the impacts of Ra, Ha, and \(m\) (sphere, column, and lamina) on the local and average Nusselt number, respectively. According to Equation (8) and Table 1, for a specified nanofluid (for example, \(Cu\)-_water_), as the shape factor increases, \((k_{n\!/\!k}/k_{\!/\!j})\) also increases. For instance, for a \(Cu\)-_water_ nanofluid with \(\phi\) = 2%, the spherical nanoparticle (\(m\) = 3) results in the lowest thermal conductivity rate \((k_{n\!/\!k}/k_{\!/\!j}=1.0609)\), while the laminar nanoparticle (\(m\) = 16.1576) yields the highest thermal conductivity rate \((k_{n\!/\!k}/k_{\!/\!j}=1.3216)\) of the studied various nanoparticle shapes in this work. Hence, based on Equation (15), by increasing the shape factor, the increased ratio \((k_{n\!/\!k}/k_{\!/\!j})\) will overcome the reduction in \(\partial\theta/\partial X\) for a specified \(\phi\), so the local Nusselt number increases with an increased \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ra** & **Present Work** & **[PERSON] et al. [37]** & **[PERSON] [38]** \\ \hline \(10^{3}\) & 1.1307 & 1.118 & 1.118 \\ \(10^{4}\) & 2.2674 & 2.245 & 2.243 \\ \(10^{5}\) & 4.5851 & 4.522 & 4.519 \\ \(10^{6}\) & 8.8341 & 8.826 & 8.799 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between present results and other works for \(Nu_{avg}\).