## 4 Results and Discussion
### Effect of Magnetic Field
Figure 13: Variations of U\({}_{3\max}\) with Hartmann number for different Rayleigh number with \(\varphi\) = 4%, K = 1, N = \(-0.5\) and N = \(-2\). produces a heat and mass transfer enhancement. However, in contradiction, both \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) decrease with increase in the Hartmann number. Thus, the magnetic field suppress the heat and mass transfer within the cavity by reducing the average Nusselt and Sherwood numbers. Also, it can be noticed that the magnetic field effect is more significant for a high Ra number. Moreover, it should be mentioned that, the maximum values of \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) occur when the magnetic field is removed (Ha = 0) for both thermally dominated flow (N = \(-\)0.5) and solutal dominated flow (N = \(-\)2). A closer examination of the average Nusselt and Sherwood ratios reveal the impacts of magnetic field on the cavity heat and mass transfer performances. Figure 16 present respectively the variations of the average Nusselt number ratio \(\overline{\mathrm{Nu/Nu}}(\mathrm{Ha}=0)\) and the average Sherwood ratio \(\overline{\mathrm{Sh/Sh}}(\mathrm{Ha}=0)\) versus Rayleigh number at different values of the Hartmann number for both thermally-dominated regime (N = \(-\)0.5) and solutal-dominated one (N = \(-\)2). The average Nusselt and Sherwood number ratios at Ha = 0 are considered as reference values. The nanoparticles volume fraction is kept constant (\(\varphi\) = 0.04). The results show that a critical Rayleigh number minimizes both the average Nusselt number ratio and the average Sherwood number ratio regardless of the Hartmann number value. These minimums move towards higher Ra number as the Ha number increases. Figure 17 presents the variations of \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) with nanoparticles volume fraction at \(\mathrm{Ra}=10^{5}\) for various Hartmann numbers. As seen from the figure, for lower value of Ha, both \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) are decreased with increase in the nanoparticles volume fraction. In fact, adding nanoparticles to base fluid increase not only the thermal conductivity but also the viscosity of the fluid which results in the reduction of heat transfer rate. However, addition of nanoparticles enhances the heat and mass transfer for Ha more than 30. Indeed, by increasing the Hartmann number, the rotation of the particles remains without intensive mixing in the fluid. At higher nanoparticles volume fraction, the effect of thermal conductivity exceeds the opposite effect of the viscosity. So, an increase in Hartmann number may lessen the deterioration in the heat transfer brought about by the effect of the viscosity. These results demonstrate the effectiveness of using magnetic field in enhancing the heat transfer by adding nanoparticles. In fact, the heat transfer is more sensitive to the heat transfer, and the heat transfer is more sensitive to the heat transfer. Figure 16: Variations of the average Nusselt number ratio \(\overline{\text{Nu}/\text{Nu}}(\text{Ha}=0)\) and the average Sherwood number ratio \(\overline{\text{Sh}/\text{Sh}}(\text{Ha}=0)\) according to the Rayleigh number for different Hartman numbers with \(\varphi=4\%\), \(\text{K}=1\), \(\text{N}=-0.5\) and \(\text{N}=-2\), \(\text{Pr}=6.2\) and \(\text{Le}=1\). Figure 9:0 Variations of \(\overline{\text{Nu}}\) and \(\overline{\text{Sh}}\) according to the nanoparticles volume fraction for different Hartmann numbers for \(\text{Ra}=10^{5}\), \(\text{K}=1\) and \(\text{N}=-0.5\). one can control the magnetic field intensity and nanoparticles volume fraction to find an optimum heat transfer enhancement using nanoparticles. Figure 18a,b are plotted to explore the effect the Rayleigh number Ra for different vortex viscosity parameter K on the average Nusselt and Sherwood numbers without (Ha = 0) and with (Ha = 40) magnetic field.