## 4 Results and Discussion
Figure 2: Variation of \(f^{\prime\prime}\left(0\right)\) with \(\lambda\) for several values of \(N\) for Cu-water nanofluid. Figure 4: Variation of \(h^{r}(0)\) with \(\lambda\) for several values of \(N\) for Cu-water nanofluid. Figure 5: Variation of \(h^{r}(0)\) with \(\lambda\) for several values of \(N\) for Ag-water nanofluid. Figure 3: Variation of \(f^{r}(0)\) with \(\lambda\) for several values of \(N\) for Ag-water nanofluid. Meanwhile, in Figures 8-13, the variations of \(f^{\prime\prime}(0)\), \(h^{\prime\prime}(0)\), and \(-\theta^{\prime}(0)\) are plotted with \(\lambda\) for several values of temperature slip parameter \(K\) for both nanoparticles in water-based nanofluids. As noted from the figures, the critical value \(|\lambda c|\) escalates as the parameter \(K\) is increased. Thus, it is possible that the increment of temperature slip can delay the separation of the boundary layers in the opposing flow region. When seen together, these results indicate that the range of existence of the solutions for the boundary value problem (9)-(12) increases with increment in \(K\). In addition, Figures 8-11 point out that the values of \(f^{\prime\prime}(0)\) and \(h^{\prime\prime}(0)\) for the upper branch solution increase with increment in \(K\) within the domain of \(\lambda c\,<\lambda<0\). On the other hand, the opposite trend is observed when \(\lambda>0\). Additionally, Figures 12 and 13 portray that at any \(\lambda\) within the range of \(\lambda c<\lambda<0\), the values of \(-\theta^{\prime}(0)\) for both solution branches increase as \(K\) increases. Figure 6: Variation of \(-\theta^{\prime}(0)\) with \(\lambda\) for several values of \(N\) for Cu-water nanofluid. Figure 7: Variation of \(-\theta^{\prime}(0)\) with \(\lambda\) for several values of \(N\) for Ag-water nanofluid. Figure 8: Variation of \(f^{\prime\prime}(0)\) with \(\lambda\) for several values of \(K\) for Cu-water nanofluid. Figure 10: Variation of \(h^{\prime\prime}(0)\) with \(\lambda\) for several values of \(K\) for Cu-water nanofluid. Figure 9: Variation of \(f^{\prime\prime}(0)\) with \(\lambda\) for several values of \(K\) for Ag-water nanofluid. Figure 2:0 Variation of \(h^{r}\left(0\right)\) with \(\lambda\) for several values of \(K\) for Ag-water nanofluid. Figure 2:1 Variation of \(-\theta^{\prime}(0)\) with \(\lambda\) for several values of \(K\) for Cu-water nanofluid. Figure 2:2 Variation of \(-\theta^{\prime}(0)\) with \(\lambda\) for several values of \(K\) for Ag-water nanofluid. Figure 2:3 Variation of \(h^{\pi}\left(0\right)\) with \(\lambda\) for several values of \(M\) for Cu-water nanofluid. Figure 2:4 Variation of \(h^{\pi}\left(0\right)\) with \(\lambda\) for several values of \(M\) for Ag-water nanofluid. Finally, Figures 20-22 offer sample of dimensionless velocity \(f^{\prime}(\eta)\), \(h^{\prime}(\eta)\), and temperature \(\theta(\eta)\) profiles against \(\eta\), for both Cu-water and Ag-water nanofluids. In addition, it is vivid from these figures that the aforementioned profiles do satisfy the far field boundary conditions (12) asymptotically. Hence, it validates and supports the numerical results retrieved for Equations (9)-(11), which are subject to boundary conditions (12), and admits the existence of dual nature of the solutions illustrated in Figures 2-19. Furthermore, the upper branch solution for these profiles portray a thinner boundary layer, in comparison to that of the lower branch solution. Additionally, Figures 20 and 21 present for a fixed values of \(n\), \(\phi\), \(c\), \(s\), \(\lambda\), \(N\), \(K\), and \(M\), the upper branch solution for \(f^{\prime}(\eta)\) and \(h^{\prime}(\eta)\) are higher for Ag-water nanofluid. Other than that, Figure 22 shows that the upper branch solution of \(\theta(\eta)\) for Cu-water and Ag-water nanofluids are almost identical. Nonetheless, the temperature for Ag-water nanofluid is found to be lower.