## 4 Results and Discussion
** [37]** & **[PERSON] and [PERSON]** & **FEM (Our Results)** \\ \hline
0 & \(-1.000008\) & \(-1.0000084\) & \(-1.0000082\) & 0.2 & - & - & 1.068027 \\
1 & 1.4142135 & 1.41421356 & 1.41421353 & 0.4 & - & - & 1.134687 \\
5 & 2.4494897 & 2.44948974 & 2.44948963 & 0.6 & - & - & 1.199118 \\
10 & 3.31662479 & 3.31662463 & 0.8 & 1.261512 & 1.261042 Figure 2 shows the effect of \(M\) on the velocity function with no hydro-dynamic slip and with hydro-dynamic slip. The results show that in both cases the velocity component decreases as the value of the magnetic parameter \(M\) increases. Physically, the magnetic parameter \(M\) produced Lorentz force like a drag force, which causes the motion of the fluid to decelerate. Figure 2 also shows that in both cases the suction \(f_{w}\) decelerates the momentum boundary layer thickness. The boundary layer thickness can be controlled with the help of suction \(f_{w}\). Figures 3 and 4 exemplify that the velocity boundary layer thickness increases with enhancing the thermal buoyancy parameter \(\lambda_{1}\) and the nano-particles concentration buoyancy parameter \(\lambda_{3}\) in the non-existence of hydro-dynamic slip and the existence of hydro-dynamic slip. In these two cases, we have to perceive that the momentum boundary layer is prolonged with the increasing values of buoyancy \(\lambda_{1}\) and \(\lambda_{3}\). Also in Figures 3 and 4, the velocity profile is increasing in the case of steady and unsteady flow and alike behavior of solutal buoyancy has been observed in Figure 5. It is observed that increasing the value of radiation parameter \(R\) is the reason to improve the velocity profile and incentive of instability in parameter \(k_{p}\), and the velocity profile is explained in Figure 6.  
Figure 4: Effects of \(\lambda_{3}\) and \(\sigma\) on \(f^{\prime}\). (**a**) \(S_{f}=0\) (**b**) \(S_{f}=0.1\). Also, we perceive in both cases that as the permeability of \(k_{p}\) increases, this causes a decline in the velocity profile with no hydro-dynamic slip and with hydro-dynamic slip. Furthermore, we also perceive in Figure 6 that the suction decreases the momentum boundary layer thickness. Similar behavior of the Deborah number \(\beta\) has been perceived on the velocity profile in Figure 7. Figure 8 indicates the impact of \(M\) on the temperature profile with the non-existence of thermal slip and with existence thermal slip. In the figure, it is clear that with an increasing value of \(M\) the temperature profile also increases in both cases. From Figure 8 it is also clear that suction \(f_{w}\) decreases the thermal boundary layer thickness in both cases and the thickness of the thermal boundary layer can be controlled with the help of suction \(f_{w}\). Figure 9 depicts the similar behavior of \(M\). It is perceived that the temperature increases with the discrepancy in parameter \(R\). Figure 10 demonstrates the influence of \(\sigma\) and \(Pr\) on the temperature profile with and without the existence of thermal slip, with increasing \(Pr\), \(\sigma\), and \(S_{\theta}\) causing the thermal boundary layer to decrease. The description of the effect of \(N_{b}\), \(f_{w}\), and \(S_{\theta}\) with the temperature profile is explained in Figure 11. It is clear that with the increase in \(N_{b}\), the thermal boundary layer of fluid flow increases in both cases. It can also be perceived that \(f_{w}\) decreases the thermal boundary layer thickness in both cases. In Figure 12, there is the influence of \(M\) and \(Sc\) on the solutal profile with the non-existence of solutal slip and with the existence of solutal slip. From the figure, it is clear that with the increasing value of \(M\), the solutal profile also increases in both cases. Furthermore, the effect of \(S_{c}\) over the solutal profile declines the profile. The impact of the existence and non-existence of solutal slip and increasing values of \(S_{r}\) is depicted in Figure 13. It is seen that increasing the value of the Soret parameter increases the solutal profile in both the steady and unsteady cases.