## 4 Results  
A shooting technique method is performed in this section to obtain the numerical solutions for different values of involving parameters in our governing system of reduced ordinary differential equations in Equation (13), subjected to the boundary conditions (14). The respective results are given to carry out the influences of several kind of parameters on the parametric study, such as mixed convection parameter \(\lambda\), radiation parameter \(Rd\), suction parameter \(S\), curvature parameter \(\xi\), as well as nanoparticle volume fraction \(\varphi\), on three nanoparticles types namely Cu (copper), Al\({}_{2}\)O\({}_{3}\) (aluminium) and TiO\({}_{2}\) (titanium). It is worth mentioning that Table 1 lists the physical properties of base fluid and selected types of nanoparticle, while Table 2 shows the values of constants \(A\) and \(B\) as defined in Equations (8) and (9) by using the thermophysical properties of water and Cu-nanoparticles. In this present study, the reduced skin friction coefficient values, \(\left(2 Pe_{x}\right)^{1/2}C_{f}\), for selected numbers of nanoparticle volume fraction parameter \(\varphi\), against \(\lambda\) when \(Rd=S=0.1\) and \(\xi=0.5\) for Cu-water nanoparticles are shown in Figure 2. The analysis shows that there are dual solutions obtained for certain range of \(\lambda\) because of the uniqueness and existence of dual solutions depend on the mixed convection parameter \(\lambda\). These regions of unique solutions are known as first solution (upper branch) and second solution (lower branch). The terms of upper and lower branches correspond to the bifurcation presences in the curves when \(\lambda\) is in critical value. From Figure 2, we can see that the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Physical Properties** & **Fluid** & **Copper** & **Aluminium** & **Titanium** \\ \hline \(C_{p}\) (K/kg K) & 4179 & 385 & 765 & 686.2 \\ \(\rho\) (kg/m\({}^{3}\)) & 997.1 & 8933 & 3970 & 4250 \\ \(k\) (W/m K) & 0.613 & 400 & 40 & 8.9538 \\ \(\beta\times 10^{-5}\) (1/K) & 21 & 1.67 & 0.85 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of constants \(A\) and \(B\) as defined in Equations (8) and (9) by using the physical properties of water and Cu-nanoparticles. solution is unique for \(\varphi=0.05\) where the range of \(\lambda\) is \(-1.93591\leq\lambda<0\), followed by \(\varphi=0.1\) where the range of \(\lambda\) is \(-2.25488\leq\lambda<0\) and \(\varphi=0.2\) with the range of \(\lambda\) is \(-3.14057\leq\lambda<0\). Plots of the reduced skin friction coefficient \(\left(2 Pe_{x}\right)^{1/2}C_{f}\), for selected values of radiation parameter \(Rd\), against \(\lambda\) when other parameters are constant for Cu-nanoparticles are depicted in Figure 3. Meanwhile, Figure 4 illustrates the plots of skin friction coefficient \(\left(2 Pe_{x}\right)^{1/2}C_{f}\), for selected values of suction parameter \(S\), in the case of Cu-water. Similar with those \(\left(2 Pe_{x}\right)^{1/2}C_{f}\) in Figure 2, dual solutions are obtained for certain range of \(\lambda\) when \(Rd\) and \(S\) are increasing. From these two figures, we noticed that the range of \(\lambda\) is increasing when the parameters \(Rd\) and \(S\) increase. Figure 3: Skin friction coefficient \(\left(2 Pe_{x}\right)^{1/2}C_{f}\), for various values of \(Rd\) against \(\lambda\) when \(\varphi=S=0.1\) and \(\xi=0.5\) for Cu-nanoparticles. Figure 2: Skin friction coefficient \(\left(2 Pe_{x}\right)^{1/2}C_{f}\), for various values of \(\varphi\) against \(\lambda\) when \(Rd=S=0.1\) and \(\xi=0.5\) for Cu-nanoparticles. We now consider velocity profiles \(f^{\prime}(\eta)\), as illustrated in Figure 5, for selected values of mixed convection parameter \(\lambda\), when \(\varphi=Rd=S=0.1\) and \(\xi=0.5\) for three types of nanoparticles and it is clearly seen that both solutions increase when \(\lambda\) increases.