## 5 Results and Discussion  
To discuss in detail the effects of the slip parameters (first-order slip \(\sigma\) and second-order slip \(\delta\)), the Soret and Dufour effects (\(Sr\) and \(Du\)), the Brownian motion \(Nb\), and also the thermophoresis parameter \(Nt\), a set of ordinary differential Equations (8)-(10) along with boundary conditions (11) was substituted into bvp4c codes in Matlab software to obtain the numerical results of the tested parameters. The results of the skin friction coefficient, heat and mass transfer coefficient, as well as the velocity, temperature, and concentration profiles were graphically presented (Figures 2-10). The effects of the first-order slip parameter \(\sigma\) and second-order slip parameter \(\delta\) were plotted (Figures 2-4). From the figures, it was clear that a unique solution existed for \(\varepsilon>-1\), dual solutions were found for \(\varepsilon_{c}\leq\varepsilon\leq-1\), and no solution was reported for \(\varepsilon<\varepsilon_{c}\). These three figures indicate that an increase in the slip parameters (\(\sigma\) and \(\delta\)) caused a decrement in the value of the skin friction coefficient. On the other hand, an increment of the heat transfer coefficient was observed for an increase of the slip parameters. A different result can be observed in Figures 3 and 4, in which the range of solutions for Figure 3 was widely expanded for the increment of the first-order slip parameter, while, in Figure 4, the expanded range of solutions was very small when the second-order slip parameter was taken into account. In these figures, the graph of the mass transfer coefficient is not shown graphically since it completely reflected the graph of the heat transfer coefficient, because of the value of \(Nb=Nt\). Figure 5 shows the effects of the Soret (\(Sr\)) and Dufour (\(Du\)) parameters on the heat transfer coefficient, where the increment of the \(Sr\) effect (decreased \(Du\) effect) led to a decrease in the value of the heat transfer coefficient. It could also be concluded that a higher \(Sr\) value decreased the heat transfer rate at the surface. Further, the increment of the \(Sr\) value did not expand the range of the solutions. The effect of different values of the Brownian motion (\(Nb\)) parameter is presented in Figure 6. The heat transfer coefficient decreased with an increase in \(Nb\), while the mass transfer coefficient increased with an increase in \(Nb\). Therefore, the heat transfer rate at the surface increased rapidly with the smallest \(Nb\), while largest values of \(Nb\) were required to increase the mass transfer rate at the surface. The effects of the thermophoresis parameter (\(Nt\)) on the heat and mass transfer coefficients could also be seen (Figure 7). The heat transfer coefficient increased as \(Nt\) increased, whereas the mass  
Figure 5: Effects of the Soret number \(Sr\) and Dufour number \(Du\) on the heat transfer coefficient \(-\theta^{\prime}(0)\). Figure 6: Effects of the Brownian motion \(Nb\) on (**a**) the heat transfer coefficient \(-\theta^{\prime}(0)\) and (**b**) the mass transfer coefficient \(-\phi^{\prime}(0)\). Figure 7: Effects of the thermophoresis parameter \(Nt\) on (**a**) the heat transfer coefficient \(-\theta^{\prime}(0)\) and (**b**) the mass transfer coefficient \(-\phi^{\prime}(0)\). The velocity, temperature, and concentration profiles are presented in Figures 8-10 for the effects of the slip parameters, Soret and Dufour parameters, and also for different values of \(\varepsilon\). It is clear from these profiles that the boundary condition (12) was converged asymptotically. Apart from that, the dual solution obtained in Figures 2-7 was graphically supported by the dual velocity, temperature, as well as concentration profiles presented in these figures. In addition, the boundary layer thickness for the second solution was always greater than for the first solution for each profile. The thermal and concentration boundary layer thickness increased as \(Sr\) increased (\(Du\) decreased).