## 4 Results
The reason behind these increasing/decreasing patterns is because the buoyancy force behaves like a pressure gradient which causes the fluid to accelerate or decelerate within the boundary layer. In addition, Figure 6 shows the various values of nanoparticle volume fraction parameter \(\varphi\) on velocity profiles \(f^{\prime}(\eta)\), when \(\lambda=-1.6\), \(Rd=S=0.1\) and \(\xi=0.5\) for Cu, Al\({}_{2}\)O\({}_{3}\) and TiO\({}_{2}\) nanoparticles. For these three types of nanoparticles, both the first and second solutions decrease when values of \(\varphi\) increasing. Figure 4: Skin friction coefficient \((2 Pe_{x})^{1/2}C_{f}\), for selected values of \(S\) against \(\lambda\) when \(\varphi=0.05\), \(Rd=0.1\) and \(\xi=0.5\) for Cu-nanoparticles. Figure 5: Various values of \(\lambda\) on velocity profiles \(f^{\prime}(\eta)\), when \(\varphi=Rd=S=0.1\) and \(\xi=0.5\) for three types of nanoparticles. In order to explore the thermal radiation parameter \(Rd\), influences on \(f^{\prime}(\eta)\) for all three types of nanoparticles, we set a constant value of \(\varphi=S=0.1\), \(\lambda=-1.5\) and \(\xi=1.0\) as depicted in Figure 7, while Figure 8 shows the numerous values of suction parameter \(S\) on velocity profiles \(f^{\prime}(\eta)\), for copper, aluminium and titanium nanoparticles, and taking this account into are \(\varphi=0.05\), \(Rd=0.1\), \(\lambda=-1.6\) and \(\xi=0.5\). From Figure 7, we noticed that both solutions are decreasing when parameter \(Rd\) increases. It is worth to know that the radiation effect is to escalate the rate of heat transfer, thus radiation should be at its minimum value in order to ease the cooling process, or else the higher number of radiation might taking a slow process of heat transfer. On the other hand, we confirmed that the first solution is increasing, while the second solution is conversely decreasing when parameter \(S\) increasing as in Figure 8. This is due to the fact that the flow in the first solution of boundary layer is taken up by suction, and thus sparks an enhancing energy on the same matter. Not stopping here, we consider the influences of curvature parameter \(\xi\), on velocity profiles \(f^{\prime}(\eta)\), for three kinds of nanoparticles when \(\lambda=-1.6\) and \(\varphi=Rd=S=0.1\) as illustrated in Figure 9. From this result, we can clearly observed that the thickness of the boundary layer is increasing in both the first and second solutions when the value of \(\xi\) increasing. Figure 6: Various values of \(\varphi\) on velocity profiles \(f^{\prime}(\eta)\), when \(\lambda=-1.6\), \(Rd=S=0.1\) and \(\xi=0.5\) for three types of nanoparticles. Figure 7: Various values of \(Rd\) on velocity profiles \(f^{\prime}(\eta)\), when \(\varphi=S=0.1\), \(\lambda=-1.5\) and \(\xi=1.0\) for three types of nanoparticles. Due to the existence of dual solutions in a selected range of parameters, as shown in our respective numerical results, an analysis of stability is performed in order to determine the most stable solution between these two solutions by finding the smallest eigen value \(\gamma\). The eigenvalue solutions as in Equation (27), subjected to the boundary conditions (28), were numerically programmed in MATLAB software (Matlab R2017a, MathWorks, Natick, MA, USA, 1984) by using a bvp4c function. The selected values of \(\lambda\) together with the smallest eigenvalue \(\gamma\) for three types of nanoparticles are properly listed in Table 3, when other parameters such as \(\varphi=Rd=S=0.1\) and \(\xi=0.5\). Meanwhile, Table 4 listed the smallest eigenvalue \(\gamma\) for various values of radiation parameter \(Rd\), when \(\varphi=S=0.1\), \(\lambda=-1.6\) and \(\xi=0.5\) for Cu, Al\({}_{2}\)O\({}_{3}\) and TiO\({}_{2}\) nanoparticles. From these two tables, it is observed that the first solutions (upper branches) show positive values, while second solutions (lower branches) show negative values.