## 4 Results and Discussion
### Effect of the Volume Fraction (\(\varphi\))  
The influence of the volume fraction of the \(Al_{2}O_{3}\) nanoparticles on the flow regime and the heat-transfer properties of the nanofluid was under examination. The Hartmann number was set equal to \(Ha=100\), the Rayleigh number to \(Ra=5\times 10^{8}\), and the porosity to \(\varepsilon=0.7\). In the present simulations, the nanoparticles volume fraction, \(\varphi\), ranges from \(\varphi=0\) to \(\varphi=0.03\).  
The volume fraction of the nanoparticles added in the base fluid alters the heat-transfer properties of the nanofluid. When nanoparticles were added, the thermal and electrical conductivity of the base fluid (water) increased. Even with low (1%-5% volumetric) concentration of the nanoparticles, the thermal conductivity increased by 10%-20%.  
As the nanoparticles volume fraction increased, the intensity of the flow regime became weaker, and that of the flow field decreased, Figure 10. This is attributed to the increase of the electrical conductivity, as the effect of the applied magnetic field on the nanoparticles becomes stronger. The temperature contours follow the alteration of the flow field, and the microrotation isolines form a vortex at the center of the cavity.  
Figure 9: Average Nusselt number as a function of the Rayleigh number, for characteristic Hartmann numbers; \(\varphi=0.03\).  
The dependence of \(Nu_{\rm{ave}}\) on the solid volume fraction is shown in Figure 11. At low-flow conditions, \(Ra\leq 5\times 10^{6}\), conduction is the dominant mechanism for heat transfer, and the Nusselt number increases as the volume fraction of the particles increases, as expected by the increased thermal conductivity offered by the nanoparticles with respect to the base fluid thermal properties. As the flow conditions intensify, convection is at play, and \(Nu_{\rm{ave}}\) increases, as well, provided the volumetric concentration of nanoparticles is constant. However, at elevated flow conditions, \(Ra=5\times 10^{8}\), even in the absence of a magnetic field, the Nusselt number is reduced as the volume fraction of the nanoparticles increases, and this slope declines as the Rayleigh number increases. As the Hartmann number increases, the Nusselt number recedes further.  
Figure 10: Stream function (\(\Psi\)), temperature (\(\theta\) ), microrotation (\(N\) ), and vorticity (\(\Omega\) ) contour plots for characteristic volume fractions of the nanoparticles.  
The decrease in the heat transfer with the addition of nanoparticles under elevated flow conditions was not an anticipated result, given the intended purpose of nanofluids to increase the efficiency of thermal processes. However, similar behavior has been observed and reported in previous experimental [36] and numerical [21] investigations of the thermal behavior of buoyancy-driven nanofluids in rectangular enclosures. In [36], the Nusselt number of the base fluid in a porous medium was found to be higher than that of the nanofluid for elevated Rayleigh numbers. In [21], the Nusselt number decreased (\(\sim\)10-30%) with the addition of nanoparticles for high Rayleigh numbers. In the present case, the phenomenon was more intense (\(\sim\)50%) due to the additional presence of the porous medium. The nanofluid moved with additional hindrance through the pores, due to the quadratic increase of the effective fluid viscosity (Equation (11)), and an increase of the volume fraction reduced the fluid flow and, thus, the convection. Therefore, even though the thermal conductivity increased, the convective heat transfer was reduced, and the Nusselt number decreased. Moreover, in the case of an applied magnetic field, the decrease was steeper, since the magnetic field had a negative effect on the intensity of the flow field.