## 5 Results Discussion
### Effect of Hartmann Number  
Hartmann number is a direct indication of the amount of electromagnetic loading impacting the nanofluid. To investigate its impact, the streamlines, isotherms, and entropy generations were plotted for various Ha, ranging from 0 to 100 (Figure 16). First, symmetry about the vertical centerline is apparent in all cases of Ha. Nonetheless, comparing case to case, it can be clearly seen that Ha extensively changes all the aforementioned; especially for Ha values higher than 25. That is, while there is a similarity between a zero magnetic field and a small magnetic field, once the electromagnetic force becomes dominant as compared with the viscous force, the changes are clearly observed. For the streamlines, the plume centers are pushed downward under the effect of the Lorentz force, bringing vortex centers closer to the hot surface, but much slower (\(\sim\)64% drop in the velocity of the center of Ha = 0 to Ha = 100), which, in turn, impacts the isothermal lines by making them more uniform (closer to stagnant fluid heat conduction), with diminished \"thermal void/jump\" in the center of the upper part. The result is a much-smoothened thermal gradient which reduces heat transfer. Consequently, the entropy generation, especially near the center of the top, is impacted.  
Figures 17-20 provide more insight about the variation of Ha at the higher \(Ra\) value of \(10^{6}\) and with Da = 0.01. The general conclusion, which was previously drawn earlier, that increasing the value of Ha will dampen the heat transfer, and thus the entropy generation is still applicable. However, it is of interest to know that varying the nano-solid particles volume fraction \(\phi\) between 0.02 and 0.08 is not impacting the heat transfer for such Ra and Da at all Ha values (evidenced by Figure 17). In addition, the entropy generation is minimally affected by \(\phi\), where the greatest is at low Ha values, and disappears with higher Ha (Figure 18). Contrary to \(\phi\), the porosity \((\varepsilon_{p})\) (Figures 19 and 20) affects the heat transfer and total entropy, as concluded before from Figures 6 and 10, respectively. Nonetheless, the new revelation here is the dampening effect of the Ha on the improvement \(\varepsilon_{p}\) on both Nu and entropy. In a summary, the main conclusion drawn from these figures is that at high values of Ra where heat transfer is usually governed by convection and fluid motion and vortices, and hence generation of large amounts of entropy, all of that can be damped down by increasing the magnetic field which, in turn, reduces the motion of the fluid, and hence reduces the efficiency of the heat transfer.  
Figure 15: Total entropy as a function of solid volume fraction (\(\phi\)) at various values of porosity \(\varepsilon_{p}\). Ra = 10\({}^{6}\), Ha = 0, and Da = 0.01.  
Figure 16: Variations of the streamlines (**left**), isotherms (**middle**), and general entropy (**right**) with various values of Hartmann number (Ha). \(\phi=0.02\), Ra \(=10^{6}\), Da \(=0.01\), and \(\varepsilon=0.4\).  
Figure 19: Average Nusselt number (Nu\({}_{\rm{ave}}\)) as a function of Hartmann number (Ha) at various values of porosity \(\varepsilon_{p}\). Ra = 10\({}^{6}\), \(\phi\) = 0.02, and Da = 0.01.  
Figure 17: Average Nusselt number (Nu\({}_{\rm{ave}}\)) as a function of Hartmann number (Ha) at various values of \(\phi\). Ra = 10\({}^{6}\), Da = 0.01, and \(\varepsilon_{p}\) = 0.4.