## 6 Conclusions  
This paper was presented to study the characteristics and behaviour of unsteady stagnation-point flow and heat transfer over a permeable exponential stretching/shrinking sheet in nanofluids with slip effect, and also to find a solution which is stable and physically realizable for some parameters, namely, velocity slip, stretching/shrinking, suction, injection, and with different types of nanoparticles. The system of partial differential equations was transformed to ordinary differential equations using the similarity variables. The values of the skin friction coefficient and heat transfer coefficient were obtained by a set of initial guesses in the bvp4c program in Matlab, and were presented graphically. Figure 10: (**a**) Velocity profile and (**b**) Temperature profile for different \(s\) in Copper-water nanofluid when \(\sigma=0.1\), \(\varphi=0.1\), \(A=0.1\) and \(\varepsilon=0.5\) (stretching). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\varphi\) & \(\varepsilon\) & \(\gamma\) Upper & \(\gamma\) Lower \\ \hline  & \(-3.36287\) & \(0.014484\) & \(-0.014485\) \\
0.0 & \(-3.36286\) & \(0.021581\) & \(-0.021578\) \\  & \(-3.36285\) & \(0.026868\) & \(-0.026858\) \\ \hline  & \(-3.91902\) & \(0.04267\) & \(-0.042654\) \\
0.1 & \(-3.91901\) & \(0.045607\) & \(-0.04558\) \\  & \(-3.919\) & \(0.04836\) & \(-0.04834\) \\ \hline  & \(-4.07216\) & \(0.00934\) & \(-0.00934\) \\
0.2 & \(-4.07215\) & \(0.01862\) & \(-0.01862\) \\  & \(-4.0721\) & \(0.04056\) & \(-0.04054\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The smallest eigenvalues \(\gamma\) for different nanoparticle volume fraction \(\varphi\) with various values of \(\varepsilon\). The [PERSON] and [PERSON] model was applied in this study and the value of the Prandtl number of water was 6.2. In order to test the stability of the first and second solutions, a stability analysis was implemented. Hence, some conclusion based on the above findings were:  
1. Dual solutions were obtained when \(\varepsilon_{c}\leq\varepsilon\); no solution exists when the value of \(\varepsilon\) is less than \(\varepsilon_{c}\). Besides that, dual solutions were found at any value of \(s\), but a significant difference between the first and second solutions was apparent when \(s>1\), which implies that the dual solutions for the suction case were much easier to find compared to the injection case. 2. The skin friction coefficient was found to be higher for the shrinking surface than the stretching surface, and also with the presence of suction parameter compared to injection parameter. In addition, an increase in the nanoparticle volume fraction in the base fluid reduced the velocity slip between the fluid, Copper nanoparticles, and the surface, and increased the unsteadiness parameter, resulting in an increase in the skin friction coefficient. 3. The heat transfer coefficient was increased more for the stretching surface than for the shrinking surface; the same trend was observed with the existence of a suction parameter. Meanwhile, a decreasing amount of nanoparticles, an increasing value of velocity slip, the use of Copper nanoparticle, and an increase of the unsteadiness parameter also increased the heat transfer rate at the surface. 4. The velocity and the temperature of the fluid were increased when, with the stronger slip parameter, the sheet was stretched, and with the mass injection parameter. 5. The first solution was found to be stable and physically realizable, whereas the second was not stable based on the performed stability analysis. [PERSON] conceived and designed the problem; [PERSON] performed the derivation and numerical data; [PERSON], [PERSON] and [PERSON] analyzed the numerical data; [PERSON] and [PERSON] contributed reagents/materials/analysis tools; [PERSON] wrote the paper. This work is supported by Fundamental Research Grant Scheme (FRGS/1/2018/STG06/UPM/02/4), Fundamental Research Grant Scheme (FRGS/1/2017/STG06/UiTM/02/9) and the financial support received from Universiti Putra Malaysia.