## 4 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \(N\) & \(K\) & \(M\) & \(\lambda\) & **fil\({}_{1}\) (Upper Branch)** & **fil\({}_{1}\) (Lower Branch)** \\ \hline
1 & 1 & 1 & \(-9\) & 3.4985 & \(-0.2172\) \\  & & & \(-9.2\) & 1.8120 & \(-0.1051\) \\
1 & 1 & 0.8 & \(-6\) & 5.0945 & \(-0.1339\) \\  & & & \(-6.1\) & 3.3007 & \(-0.0133\) \\
1 & 0.5 & 1 & \(-5\) & 5.2992 & \(-0.4028\) \\  & & & \(-5.5\) & 1.8051 & \(-0.1048\) \\
0.9 & 1 & 1 & \(-5\) & 8.9037 & \(-0.4313\) \\  & & & \(-5.9\) & 5.1995 & \(-0.1139\) \\ \hline \end{tabular} Note: \(M\) is the magnetic parameter, \(n\) is the positive constant, \(N\) is the velocity slip parameter, \(K\) is the temperature slip parameter, \(\lambda\) is the mixed convection parameter.  
\end{table} 
Table 3: Minimum eigenvalue, \(\gamma_{1}\) at several values of \(N\), \(K\), \(M\), and \(\lambda\). Additionally, Figures 2-19 display the variations of the reduced skin friction coefficients \(f^{\prime\prime}\left(0\right)\), \(h^{\prime\prime}\left(0\right)\) and the reduced local Nusselt number \(-\theta^{\prime}(0)\) with \(\lambda\) for varied values of parameters \(N\), \(K\), and \(M\) for both Cu-water and Ag-water nanofluid. The solid lines are used throughout this paper in order to refer to the upper branch solution, whereas the dotted lines represent the lower branch solution. One can observe that the dual (upper and lower branch) solutions do exist for Equations (9)-(11), subjected to boundary conditions (12) within the range of \(\lambda c<\lambda\), where \(\lambda c\) is the critical value of parameter \(\lambda\), while no solution exists for \(\lambda<\lambda c\). Besides, interestingly, the unique (upper branch) solution is attained when \(\lambda>0\) (assisting flow), while the dual solutions are found only for \(\lambda<0\) (opposing flow). Furthermore, for \(\lambda<\lambda c\), both full Navier-Stokes and energy equations have to be solved. In addition, it is clear that all \(\left|\lambda c\right|\) values for Ag-water nanofluid are higher than those for Cu-water nanofluid. Hence, one can assume that the presence of Ag nanoparticles in the water-based nanofluid is more capable of delaying the boundary layer separation, in comparison to that for Cu nanoparticles. Other than that, Figures 2-7 illustrate the variations of \(f^{\prime\prime}\left(0\right)\), \(h^{\prime\prime}\left(0\right)\), and \(-\theta^{\prime}(0)\) against \(\lambda\) for varied values of velocity slip parameter \(N\) for both Cu-water and Ag-water nanofluids. As a result, it was observed that the value of \(\left|\lambda c\right|\) increased as \(N\) increased. This indicates that the impact of \(N\) escalates the range of existence of the solutions for the boundary value problem (9)-(12). On top of that, this hints that the increment of velocity slip could decelerate the separation of boundary layer in the opposing flow region. Meanwhile, for a fixed value of \(N\), the upper branch solution decreases with the decrease of \(\lambda\) until it reaches a critical value at \(\lambda c\), for both opposing and assisting flows. Besides, the corresponding lower branch solution decreases with the increase of \(\lambda\) for an opposing flow. Nonetheless, the values of \(-\theta^{\prime}(0)\) (see Figures 6 and 7) for the lower branch solution decreased rapidly with the increase of \(\lambda\). Moreover, the values of \(-\theta^{\prime}(0)\) for the upper branch solution are extremely large, when compared to the corresponding lower branch solution in the neighborhood of \(\lambda=0\). Furthermore, Figures 2-5 demonstrate that at any \(\lambda\) station within the range of \(\lambda c<\lambda\), the values of \(f^{\prime\prime}\left(0\right)\) and \(h^{\prime\prime}\left(0\right)\) for the dual solutions decrease as \(N\) increases. In contrast, Figures 6 and 7 display that at any \(\lambda\) station in the range of \(\lambda c<\lambda<0\), the values of \(-\theta^{\prime}(0)\) for the dual solutions escalate with the increase in \(N\).