## 4 Results and Discussions  
The system of the governing Equations (19)-(21) are solved numerically by using shooting method. We have taken \(\triangle(\eta)=0.02\) also the boundary condition \(\infty\) has been replaced with 5 due to the requirement of the shooting method. The convergence has been obtained at \(\mathrm{tol}\ \varepsilon=10^{-5}\), while the thickness of the boundary layer \(\eta_{\infty}\) is taken between 2 and 15. Morever the current result is compared with previusaly published data see in Table 1. The variation of different physical aspects are presented through graphs (as shown in Figures 2-23). The distribution of horizontal velocity on (thermal stratification \(St\)) shown in Figure 2. It is viewed that the velocity gradient reduces by enlarging values of \(St\). Figure 3 is sketches for the distribution temperature gradient for unlike values of \(St\). it is clear that temperature gradient increases for higher values of \(St\) Plotted Figure 4 is for the variation of velocity and temperature distribution for unlike values of \(M\). Decrease variation of velocity and temperature profiles increases by enlarge values of \(M\). According to the [PERSON]'s force theory, the behavior of magnetic field and velocity fraction is inverse. Therefore the velocity distribution is reduced by the higher values of \(M\). Plotted Figure 5 is plotted for the distribution of concentration and temperature profiles for the distinct values of (Prandtl number \(Pr\)). The temperature profile decreases and concentration profile increases by increasing values of \(Pr\). The physical reason behind that the fluids contain a little number of \(Pr\) have a higher thermal diffusivity, and this behaviour is inverse for larger values of Prandtl numbers. Figure 6 indicates that the significance of velocity for various values of material fluid parameter \(H\). By increasing \(H\) the velocity decreases, hence velocity and momentum boundary layer thickness are increased. Figure 4: Variation of \(f^{\prime}\) and \(\theta\) for distinct values of \(M\). Figure 3: The \(f^{\prime}\) and \(\theta\) for distinct values of \(St\). Figure 2: Distribution of \(f^{\prime}\) for distinct values of \(St\). Figure 6: The behaviour of \(H\) on \(f^{\prime}\). Figure 7: The behaviour of \(\lambda\) on \(f^{\prime}\). Figure 7 demonstrated for velocity profile for unlike values of \(\lambda\) parameter. It is seen that the variation is enhancing \(\lambda\) is to increase both the film thickness and the free surface velocity \(f^{\prime}(\eta)\). The distribution of velocity profile for power law index \(m\) is shown in Figure 8. Decreases momentum and velocity boundary layer thickness by improving values of \(m\). Here \(m=1\) linear represent the surface case and \(m\
eq 1\) corresponds to nonlinear stretching surface. Figure 9 exhibits the variation of \(m\) on the temperature field. It is noticed the temperature profile reduces by enhancing values of \(m\). Figure 10 designates for the distribution of the concentration profile for various values of \(Le\). The results shows that decreases in concentration profile is noticed for large valves of Lewis number \(Le\). This is because of the fact that there is a reduction in the nano-particle volume fraction boundary layer thickness with the raise in the Lewis number. The variation of the concentration profile for the distinct values of \(Nb\) shown in Figure 11. It is noticed that reduction in concentration profile noticed by incrementing value of Brownian motion \(Nb\). The effect of \(Nt\) on the concentration shown in Figure 12. It is seen that the concentration decreases as the thermosteresis parameter increases. Figure 13 is plotted to represents the variation (activation energy \(E\)) on concentration field \(\phi\). it is concluded that \(E\) is diminishing function of \(\phi\). The results shows that increases in concentration profile is noticed for large valves of \(E\).