## 5 Numerical Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(\kappa^{*}\) & \(A\) & \(\phi\) & \(\epsilon\) & \(S\) & \(\delta\) & \(\Delta\) & \(-f^{\prime\prime}(0)\) & \(-\theta^{\prime}(0)\) \\ \hline \multicolumn{2}{c}{\(a=1\)} & & & & & & & \\ \hline
0.2 & 0.2 & 0.2 & 0.3 & 0.2 & 0.1 & 0.1 & 0.5844 & 0.2513 \\
0.8 & & & & & & & 0.9158 & 0.2678 \\ \hline
0.1 & 0.2 & 0.2 & 0.3 & 0.1 & 0.1 & 0.1 & 0.6139 & 0.6594 \\
0.4 & & & & & & & 0.8145 & 0.6951 \\
0.9 & & & & & & & 1.0470 & 0.7290 \\ \hline
0.4 & 0.1 & 0.2 & 0.3 & 0.1 & 0.1 & 0.1 & 0.7858 & 0.7996 \\  & 0.6 & & & & & & 0.8780 & 0.8098 \\  & 1.5 & & & & & & 1.0186 & 0.8239 \\ \hline
0.4 & 0.2 & 0.0 & 0.3 & 0.1 & 0.1 & 0.1 & 0.7815 & 0.9284 \\  & & 0.05 & & & & & 0.7880 & 0.8621 \\  & & 0.2 & & & & & 0.8145 & 0.6951 \\ \hline
0.4 & 0.2 & 0.2 & 0.1 & 0.1 & 0.1 & 0.1 & 0.8158 & 0.7709 \\  & & & 0.4 & & & & 0.8139 & 0.6639 \\  & & & 0.8 & & & & 0.8115 & 0.5693 \\ \hline
0.4 & 0.2 & 0.2 & 0.3 & 0.1 & 0.1 & 0.1 & 0.7742 & 0.6135 \\  & & & & 0.2 & & & 0.8145 & 0.6951 \\  & & & & & 0.3 & & 0.8553 & 0.7794 \\ \hline
0.4 & 0.2 & 0.2 & 0.3 & 0.2 & 0.1 & 0.1 & 0.8145 & 0.6951 \\  & & & & & 0.3 & & 0.7163 & 0.7404 \\  & & & & & 0.6 & & 0.6007 & 0.7869 \\ \hline
0.4 & 0.2 & 0.2 & 0.3 & 0.2 & 0.1 & 0.1 & 0.8145 & 0.6951 \\  & & & & & & 0.3 & 0.8052 & 0.6188 \\  & & & & & & 0.6 & 0.7949 & 0.5292 \\ \hline \end{tabular}
\end{table} 
Table 2: Values of skin friction \(=-f^{\prime\prime}(0)\) and Nusselt number \(=-\theta^{\prime}(0)\). The effect of nanoparticle volume concentration parameter \(\phi\) on the velocity and temperature profiles of the Cu-water nanofluid is shown in Figures 5 and 6. It is observed that the velocity of the fluid decreases, whereas the temperature of the nanofluid increases with the increase in volume  
Figure 4: Temperature \(\theta(\eta)\) profiles for different values of permeability parameter \(k^{*}\). Figure 3: Velocity \(f^{\prime}(\eta)\) profiles for different values of permeability parameter \(k^{*}\). parameter \(A\) after the crossing-over point. This corresponds to an increase in the thickness of the boundary layer. It is observed from Figure 8 that the increase in the viscosity parameter enhances the heat transfer rate and decreases the thickness of the thermal boundary layer. The impact of \(A\) on the velocity profiles is more pronounced than on the temperature profiles. Figures 9 and 10 depict the velocity and temperature profiles for the specified values of thermal conductivity parameter \(\epsilon\). It is noticed that the variation in \(\epsilon\) greatly affects the temperature profiles, as compared to the velocity profiles. The variation in velocity profiles show an opposite effect as  
Figure 8: Temperature \(\theta(\eta)\) profiles for different values of viscosity parameter \(A\). Figure 7: Velocity \(f^{\prime}(\eta)\) profiles for different values of viscosity parameter \(A\). the variation in viscosity parameter \(A\), i.e., the increase in thermal conductivity initially causes the decrease in fluid velocity and shows the opposite behavior after the cross-over point; whereas an increase in \(\epsilon\) results in an increase in thermal conductivity, thereby raising the fluid temperature across the boundary layer. It would also increase the thermal boundary layer thickness. In Figure 11, the effect of velocity slip parameter \(\delta\) on the velocity profile of the Cu-water nanofluid is presented. The comparison of the curves shows that the increase in the velocity slip at the boundary increases the fluid velocity within the boundary layer. This is due to the positive value of the fluid velocity adjacent to the surface of the plate and results in a reduction of the momentum boundary  
Figure 10: Temperature \(\theta(\eta)\) profiles for different values of \(\epsilon\). Figure 9: Velocity \(f^{\prime}(\eta)\) profiles for different values of thermal conductivity parameter \(\epsilon\). layer thickness.