## 4 Results and Discussion
### Effect of Magnetic Field
The variations of U\({}_{3\rm max}\) according to the Rayleigh number without (Ha = 0) and with (Ha = 40) magnetic field for both thermally-dominated flow (N = \(-\)0.5) and solutal-dominated flow (N = \(-\)2) are presented in Figure 12. When the magnetic field is absent, U\({}_{3\rm max}\) increases slightly with increase in Rayleigh number for both value of N. However, U\({}_{3\rm max}\) enhanced considerably due to the influence of the magnetic field and is more prominent for the thermally dominated flow (N = \(-\)0.5). Figure 13 shows the effects of the Hartmann number on U\({}_{3\rm max}\) with the Rayleigh number for both thermally-dominated flow (N = \(-\)0.5) and solutal-dominated flow (N = \(-\)2). As seen from the figure, the velocity is augmented for Hartmann numbers lower than the critical value of Ha and decreased by increasing the Hartmann number once more. Indeed, by increasing the Ha numbers, Lorentz force becomes stronger and overcomes the buoyancy force, which causes a considerable reduction in the three-dimensional character of the flow. The 3-D flow behavior against the Hartmann number is already discussed by [PERSON] and [PERSON] [43] and by [PERSON] et al. [50] for three-dimensional natural convection of pure fluid. It is interesting to detect the existence of peak at which U\({}_{3\rm max}\) is maximized regardless of the value of Ra. This peak value moves towards the highest value of Ha as an increase  
Figure 11: Variation of U\({}_{3\rm max}\) according to buoyancy ratio without (Ha = 0) and with magnetic field (Ha = 40) for Pr = 6.2, Le = 1, Ra = \(10^{5}\) and \(\varphi\) = 0.04. Figure 12: Variation of U\({}_{3\rm max}\) according to the Rayleigh number in the plane X = 0.5 without (Ha = 0) and with magnetic field (Ha = 40) for \(\varphi\) = 4% and K = 1 for N = \(-\)0.5 and N = \(-\)2.  
in Ra. In fact, for high Rayleigh number, convection is the dominant mechanism of heat transfer and the three-dimensional character of the flow is more prominent. Thus, to act against the convection mechanism, a stronger magnetic field must be applied and hence the Hartmann number should be higher. Therefore, the critical value of Ha is influenced by the values of Ra. Also, the magnitude of U\({}_{3\max}\) is greater for thermally-dominated flow (N = \(-0.5\)) than for solutal-dominated flow (N = \(-2\)). Besides, it should be mentioned that the minimum values of U\({}_{3\max}\) occur when magnetic field is removed (Ha = 0) for both thermally-dominated flow (N = \(-0.5\)) and solutal-dominated flow (N = \(-2\)). To highlight the effect of magnetic field on heat and mass transfers, the variations of \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) versus the buoyancy ratio for both cases, Ha = 0 and Ha = 40 are shown in Figure 14. It is clear that both \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) have a quasi-similar profile for both Ha = 0 and Ha = 40. It is noticed that for both cases \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) have a decreasing trend upon increasing N in the negative sense. Beyond N = \(-1\), \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) increase as N increases. The presence of a magnetic field decreased both \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) regardless of the buoyancy ratio. The magnetic field effect is more prominent for thermally-dominated regimes compared to solutal-dominated regimes. Figure 15 illustrates the variations of \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) with the Hartmann number for different Rayleigh numbers for both thermally-dominated regime (N = \(-0.5\)) and compositionally dominated regime. (N = \(-2\)). As revealed from the figure for both cases N = \(-0.5\) and N = \(-2\), increase in Ra number  
Figure 14: Variations of \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) according to buoyancy ratio without (Ha = 0) and with magnetic field (Ha = 40) for Pr = 6.2, Le = 1, Ra = \(10^{5}\) and \(\varphi\) = 0.04.