## 5 Results and Discussion  
The effects of the nanoparticle volume fraction \(\varphi\), velocity slip parameter \(\sigma\), unsteadiness parameter \(A\), stretching/shrinking parameter \(\varepsilon\), suction/injection parameter \(s\), and different types of nanoparticles (Copper, Alumina, and Titania) are studied, and have been displayed in Figures 1-10. As suggested in the [PERSON] and [PERSON] model, water with a Prandtl number of 6.2 (\(\mathrm{Pr}=6.2\)) was chosen as the base fluid, and the value of the nanoparticle volume fraction was between 0 and 0.2 (\(0\leq\varphi\leq 0.2\)), where the absence of nanoparticles in the base fluid can be known when \(\varphi=0\). The thermo physical properties of water, Copper, Alumina, And Titania are tabulated in Table 1, for which the value of each substance is taken from [PERSON] and Abu-Nadda [42]. Validation has been made from previous work by [PERSON] et al. [7], whereby the steady problem (\(A=0\)) over a permeable surface (\(s=0\)) with no-slip condition (\(\sigma=0\)) was considered, as shown in Table 2; we observed excellent agreement. The effects of the nanoparticle volume fraction \(\varphi\) on the skin friction coefficient \(f^{\prime\prime}\left(0\right)\) and heat transfer coefficient \(-\theta^{\prime}(0)\) with stretching/shrinking parameter \(\varepsilon\) in Copper-water nanofluid are depicted in Figure 2. Both figures show that dual solutions can be obtained when \(\varepsilon_{c}\leq\varepsilon\), while no solution can be found when \(\varepsilon<\varepsilon_{c}\), i.e., where the critical point \(\varepsilon_{c}\) represents the end point of the existed solution. The range of solutions was shown to widen when the amount of Copper nanoparticles in the water increases. Apart from that, the skin friction coefficient \(f^{\prime\prime}\left(0\right)\) for both solutions in the shrinking surface was found to be higher than in the stretching surface, while the heat transfer coefficient shows the opposite trend. Moreover, increasing the volume fraction of Copper nanoparticles in \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Substance** & **Density, \(\rho\) (kg/m\({}^{3}\))** & **Specific Heat, \(C_{p}\) (J/kg-K)** & **Conductivity, \(K\) (W/m\(\cdot\)K)** \\ \hline Water & 997.1 & 4179 & 0.613 \\ Copper & 8933 & 385 & 400 \\ Alumina & 3970 & 765 & 40 \\ Titania & 4250 & 686.2 & 8.9538 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermo physical properties of water, Copper, Alumina, and Titania (see [PERSON] and [PERSON] [42]). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\varepsilon\)} & \multirow{2}{*}{\(\varphi\)} & \multicolumn{2}{c}{**[PERSON] et al. [7]**} & \multicolumn{2}{c}{**Present Results**} \\ \cline{3-6}  & & \(f^{\prime}\left(0\right)\) & \(-\theta^{\prime}\left(0\right)\) & \(f^{\prime}\left(0\right)\) & \(-\theta^{\prime}\left(0\right)\) \\ \hline \multirow{3}{*}{\(-0.5\)} & 0 & 2.1182 & 0.687 & 2.1182 & 0.6870 \\  & 0.1 & 3.2381 & 1.1432 & 3.2330 & 1.1400 \\  & 0.2 & 4.5071 & 1.5184 & 4.5009 & 1.5151 \\ \hline \multirow{3}{*}{0} & 0 & 1.6872 & 1.7148 & 1.6872 & 1.7148 \\  & 0.1 & 2.5794 & 2.1358 & 2.5793 & 2.1358 \\  & 0.2 & 3.5901 & 2.5400 & 3.5901 & 2.5400 \\ \hline \multirow{3}{*}{0.5} & 0 & 0.9600 & 2.4874 & 0.9604 & 2.4874 \\  & 0.1 & 1.4682 & 2.915 & 1.4682 & 2.9150 \\ \cline{1-1}  & 0.2 & 2.0436 & 3.3565 & 2.0436 & 3.3565 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of \(f^{\prime\prime}\left(0\right)\) and \(-\theta^{\prime}\left(0\right)\) for different values of \(\varepsilon\) and \(\varphi\) for Cu-water. water increases the skin friction coefficient for the first solution, but decreases the second solution. Meanwhile, for the heat transfer coefficient, increasing \(\varphi\) shows a decreasing trend for both solutions.