## 3 Results and Discussion
### Velocity Profile  
Figures 2-9 are depicted to show the physical behavior of \(M\), \(\lambda\), \(\delta\), \(\zeta\), \(\beta_{1}\), and \(K\) on velocity profile. The variations of \(M\) on the velocity is exhibited in the Figure 2. As expected, the velocity of the fluid flow become lower with the strengthening of the magnetic field. It is an agreement with the fact that growth in \(M\) causes Lorentz force that creates resistance to the flow, due to which the momentum boundary layer thins across the boundary. A similar effect is shown in Figure 3, contradicting the variation in \(M\). The angular velocity shows inverse relation with the magnetic field strength. The variations in material parameter \(K\) show the velocity profile upturn (see Figure 4). The variation in angular velocity contradicts the material parameter portrayed in Figure 5. Clearly, \(h(\theta)\) increases with growing magnitudes of \(K\) on matching with the Newtonian case (\(K=0\)); additionally, the boundary layer thickness reduces with growth of \(K\). The relation between \(\lambda\) and the velocity field is depicted in Figure 6. Here, \(\lambda\) presents the characteristics of the buoyancy forces, which offer a direct relation with the velocity field. Physically, the growing magnitude of the buoyancy forces causes a decline in the viscous force, which improves the fluid flow, which causes faster motion. Moreover, the relationship between the solutal buoyancy forces and velocity field is shown in Figure 7. Physically, the length, concentration difference, and kinematic viscosity of the fluid affect parameter \(\delta\). On the other hand, there is an inverse relationship between the viscosity and velocity of the fluid. Therefore, the viscosity of the fluid declines once it has increased the magnitude of \(\delta\) (which as a result causes faster the motion), and the concentration upsurges directly, due to which the velocity field rises. Finally, a direct relationship between factor \(\delta\) and the velocity outline is shown. Figure 8 presents the inclination variation in the velocity field. It can be observed that there is an inverse relation between \(\zeta\) and \(f^{\prime}(\eta)\). This can be ascribed to the fact that the maximum gravitational force acts on flow when the inclination factor \(\zeta=0\), because in this state the sheet will be vertical. However, for \(\zeta=~{}90^{\circ}\) the sheet will be horizontal, which causes the reduction in the velocity profile as the strength of the bouncy forces decline. The variation in Brinkman factor causes a decline in the velocity profile (see Figure 9). Parameter \(\beta_{1}\) is the ratio between the drag force and density; therefore, by increasing \(\beta_{1}\) the drag force improves and the fluid velocity reduces [40].  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \(Nb\) & \(Nt\) & \(Pr\) & \(Le\) & \(M\) & \(K\) & \(\lambda\) & \(\delta\) & \(N\) & \(S\) & \(\beta_{1}\) & \(\zeta\) & \(-\theta^{\prime}(0)\) & \(-\phi^{\prime}(0)\) & \(C_{fx}(0)\) \\ \hline
0.1 & 0.1 & 7.0 & 5.0 Figure 4: Variations in velocity profile for several values of \(K\).  
Figure 3: Variations in angular velocity for several values of \(M\).  
Figure 2: Variations in velocity profile for several values of \(M\).  
Figure 5: Variations in angular velocity profile for several values of \(K\).  
Figure 6: Variations in velocity profile for several values of \(\lambda\).  
Figure 7: Variations in velocity profile for several values of \(\delta\).