## 4 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Pr**} & \multirow{2}{*}{**M**} & \multirow{2}{*}{\(\mathbf{R}d\)} & \multirow{2}{*}{\(\mathbf{\varepsilon}\)} & \multicolumn{4}{c}{\(-\theta^{\prime}(0)\)} \\ \cline{5-8}  & & & & **[PERSON] and [PERSON] [32]** & & **[PERSON] [33]** & **[PERSON] and [PERSON] [34]** & **[PERSON] [35]** & **[PERSON] [31]** & **Present** \\ \hline
0.5 & 0 & 0 & 0 & 0.594338 & 0.594493 & & & 0.594339 & 0.596687601 \\
1 & 0 & 0 & 0 & 0.954782 & 0.954785 & 0.9548 & 0.9548 & 0.954783 & 0.954810637 \\
2 & 0 & 0 & 0 & & 1.4714 & 1.4715 & 1.47146 & 1.471454035 \\
3 & 0 & 0 & 0 & 1.869075 & 1.869074 & 1.8691 & 1.8691 & 1.869074 & 1.869068789 \\
5 & 0 & 0 & 0 & 2.500135 & 2.500132 & & 2.500132 & 2.500127789 \\
10 & 0 & 0 & 0 & 3.660379 & 3.660372 & & 3.6604 & 3.660372 & 3.660369292 \\
1 & 1 & 0 & & & & 0.8611 & 0.861094 & 0.861508689 \\
1 & 0 & 1 & 0 & & & 0.5315 & 0.5312 & 0.531158 & 0.535301194 \\
1 & 1 & 1 & 0 & & & 0.4505 & 0.450536 & 0.461965494 \\
1 & 1 & 1 & \(-1\) & & & & 0.873201623 \\
25 & 1 & 1 & 1 & & & & 1.201645626 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison several of numerical results of \(-\theta^{\prime}(0)\) for various parameters (\(\lambda=1\), \(\varphi_{1}=\varphi_{2}=0\), \(G=0\)). tolerance, which is limited. The first solution in Figure 2a shows that the increasing local skin friction coefficient, \(Re_{x}^{1/2}C_{f}\), refers to \(f^{\prime\prime}(0)\) on the permeable exponentially shrinking surface when the volume fraction nanoparticle of copper \(\varphi_{2}\) increases. Thus, the reduction in the momentum boundary layer thickness, \(\delta\), coincides with the increase in value of \(\varphi_{2}\), causing an increase in \(Re_{x}^{1/2}C_{f}\) when the \(\varphi_{2}\) value is increased. Therefore, the effect of shear stress on the permeable exponentially shrinking surface increases. In addition, the numerical solution that is shown in Figure 2b expresses the increase in the value of the local Nusselt number, \(Re_{x}^{1/2}Nu_{x}\), referred to as \(-\theta^{\prime}(0)\) when the value of \(\varphi_{2}\) increases from 0 to 0.04 on a shrinking surface in region \(\lambda<-1\). This happens because of the synergy effect. In region \(-1<\lambda<0\), \(Re_{x}^{1/2}Nu_{x}\) values decrease when the \(\varphi_{2}\) value increases. This is because the shrinking strength \(\lambda\) is not large enough. The thermal conductivity effect of hybrid nanofluid increases the heat flux on the exponentially shrinking surface in region \(\lambda<-1\), whereas the heat flux decreases in region \(-1<\lambda<0\). Thus, as the temperature gradient increases, the thermal boundary layer thickness \(\delta_{T}\) becomes thin at region \(\lambda<-1\) when the value of \(\varphi_{2}\) increases, which ultimately increases the heat transfer rate of the exponentially shrinking surface. Next, the upper and lower branches in Figure 3a also describe the increasing \(Re_{x}^{1/2}C_{f}\) on the shrinking exponential surface when magnetic parameter \(M\) increases. This happened because the influence of higher Lorentz force \(J\times B\) acts in same direction as the drift force or viscous force on the surface when in opposition to fluid flow. Moving the current density, \(J\), is equivalent to fluid flow and the induced magnetic field moves normally or perpendicular to fluid flow according to the right-hand rule that was used to determine the direction of the physical quantity. This causes the skin friction on the surface to increase and the shear stress effect in the \(x\) axis to also increase. Therefore, the momentum boundary layer thickness over the permeable shrinking exponential surface becomes thin. Figure 3b describes the trend of the increasing value of \(Re_{x}^{1/2}Nu_{x}\) when the value of M is higher. Because of that, the higher heat flux on the permeable shrinking exponential surface that speeds up the heat exchange process with support from the Lorentz force contributes to the increasing value of \(Re_{x}^{1/2}Nu_{x}\).