## 5 Numerical Simulation  
Figures 5-7 depict the nanofluid velocity evolutions at several enclosure levels (\(X=0.25\), \(X=0.5\), \(Y=0.25\) and \(Y=0.5\)) and for different Rayleigh numbers (Ra = \(10^{3}\), Ra = \(10^{4}\) and Ra = \(10^{5}\)) and inclination angles (\(0^{\circ}\), \(15^{\circ}\), \(30^{\circ}\) and \(45^{\circ}\)). When the inclination angle was \(0^{\circ}\), the velocity evolutions were quite symmetric along the \(y\)-axis. The symmetry of the velocity evolution along the \(y\)-axis was more pronounced than the one along the \(x\)-axis. This gradually deteriorated going towards the highest inclination angles. For a relatively low Rayleigh number Ra = \(10^{3}\), and as the inclination angle increased, the higher extremum velocity globally decreased while the lower extremum globally increases at the level \(X=0.25\) and conversely at the level \(X=0.5\), along the \(y\)-axis. Along the \(x\)-axis, the higher and the lower extrema of velocity evolutions, for the different inclination angles, alternated around \(X=0.4\) at the level \(Y=0.25\) and globally triple alternated around \(X=0.2\) and \(X=0.75\) at the level \(Y=0.5\). As the Rayleigh number increased to a relatively moderate value Ra = \(10^{4}\), the evolutions of velocity along all the investigated directions (\(x\)-axis and \(y\)-axis) and at all the investigated levels (\(X=0.25\), \(X=0.5\), \(Y=0.25\), \(Y=0.5\)) kept the same tendency as for the case when the Rayleigh number was relatively low. It is to note here that the higher and the lower extrema are drastically amplified. For a relatively high Rayleigh number Ra = \(10^{5}\), and as the inclination angle increased, the higher and lower velocity extrema globally increased along the \(y\)-axis at the levels \(X=0.25\) and \(X=0.5\). Along the \(x\)-axis, and for the different inclination angles, the higher and lower velocity extrema alternated around \(X=0.35\) at the level \(Y=0.25\) and globally triple alternated around \(X=0.15\) and \(X=0.7\) at the level \(Y=0.5\). Figure 8 depicts the heat transfer behavior of the nanofluid in terms of temperature and Nusselt number related to a certain thermal conductivity ratio \(\kappa\). This \(\kappa\) ratio is defined by:  
\[\kappa=-k_{f}/k_{nf} \tag{24}\]  
Figure 5: Nanofluid velocity evolutions at the levels: \(X=0.25\) (**a**) \(X=0.5\) (**b**) \(Y=0.25\) (**c**) \(Y=0.5\) (**d**) for a Rayleigh number Ra = \(10^{3}\). Figure 6: Nanofluid velocity evolutions at the levels: \(X=0.25\) (**a**) \(X=0.5\) (**b**) \(Y=0.25\) (**c**) \(Y=0.5\) (**d**) for a Rayleigh number \(\text{Ra}=10^{4}\). Figure 7: Nanofluid velocity evolutions at the levels: \(X=0.25\) (**a**) \(X=0.5\) (**b**) \(Y=0.25\) (**c**) \(Y=0.5\) (**d**) for a Rayleigh number \(\text{Ra}=10^{5}\). The nanofluid temperature evolutions are depicted along the \(x\)-axis at the level \(Y=0.5\), for different Rayleigh numbers (Ra = \(10^{3}\), Ra = \(10^{4}\) and Ra = \(10^{5}\)) and inclination angles (\(0^{\circ}\), \(15^{\circ}\), \(30^{\circ}\), and \(45^{\circ}\)). This temperature evolution depended on the inclination angle only for relatively higher Rayleigh number (in the order of Ra = \(10^{5}\)). The nanofluid temperature globally increased as the inclination angle increases. The evolutions of the nanofluid Nusselt number Nu related to the thermal conductivity ratio \(\kappa\) are depicted along the \(x\)-axis at the level \(Y=1\), for different inclination angles (\(0^{\circ}\), \(15^{\circ}\), \(30^{\circ}\) and \(45^{\circ}\)). This evolution globally decreased as the inclination angle increased. It is to note that the symmetry of the heat transfer behavior of the nanofluid along the \(x\)-axis gradually deteriorated going towards the highest inclination angles. Figure 9 shows the combined effect of Rayleigh number and enclosure inclination angle on nanofluid streamlines, when there are no fins below the absorber plate (\(\delta~{}=~{}0\)).