## 4 Results and Discussion
Therefore, the thermal boundary layer thickness of SWCNT/Cu-water decreases compared with MWCNT/Cu-water and CNT-water. In light of the above fact, hybrid nanofluid shows better heat transfer enhancement than regular nanofluid. Overall, the results indicate that variation in parameter \(\lambda\) is able to obtain a critical value itself. Based on results presented in Figures 2-5, the variation shows that different values of \(\lambda\) increase shrinking strength and cause a significant increase in the value of \(Re_{x}^{1/2}C_{f}\) but significantly reduce the value of \(Re_{x}^{1/2}Nu_{x}\). Next, the dual solutions of the boundary layer velocity and temperature profiles are presented for various governing parameters. For the dual solutions of a boundary value problem, two different solutions are obtained under the same conditions by assuming different values of the missing initial conditions. From a physical point of view, it is important to know which solution is physically relevant. Unlike the second solution, we expect that the first solution is physically stable and frequently occurs in practice. In the present paper, we highlight the viability of the first solution only. Based on recent studies, stability analysis shows that the second solution is not stable due to the negative eigenvalues against the governing parameters where an initial growth of disturbance occurs, which means that the flow is unstable. Based on the volume fraction of hybrid nanoparticles in the existing CNT-water nanofluid, the first solution in Figure 6a shows an increasing value in velocity when the \(\varphi_{2}\) of copper increases. This situation can refer to the thermophysical properties of Cu in Table 2, where the high-density value of Cu increases the skin friction on the surface and hence makes it less viscous. Thus, a decrease in value in the momentum boundary layer thickness coincides with an increase in the value of \(\varphi_{2}\), as shown in Figure 6a. Therefore, the effect of shear stress on the exponentially permeable shrinking surface increases. The numerical solution shown in Figure 6b expresses the increase in temperature profile \(\theta(\eta)\) when the value of \(\varphi_{2}\) increases from 0 to 0.04 towards a shrinking surface. This is due to the synergistic effect because the strength of the shrinking condition is not large enough. The heat conduction effect of hybrid nanofluids increases the heat flux on the exponentially shrinking surface in the region \(\lambda<-1\). Thus, the temperature gradient increases and the thickness of the temperature boundary layer becomes thinner when the value of \(\varphi_{2}\) increases, as shown in Figure 6b, which ultimately increases the exponentially shrinking surface heat transfer rate. Additional parameters in the problem of this study are the effect of heat absorption/generation and the effect of thermal radiation aimed at diversifying the MHD flow antics of hybrid nanofluids against exponentially shrinking surfaces. Generally, the parameters' heat absorption/generation, \(\epsilon\), and thermal radiation, \(Rd\), only affect the heat transfer because the value of the local skin friction coefficient is unchanged, and this proves that it does not depend on the \(\epsilon\) and \(Rd\). Based on Figure 7a, a decrease in the value of temperature occurs with the increase in the value of \(\epsilon\). This is because when heat generation occurs in hybrid nanofluids, a low heat flux occurs on the shrinking permeable exponential surface, and this will slow down the heat exchange process from that surface. In fact, when heat absorption occurs on hybrid nanofluids, a high heat flux occurs on the exponentially shrinking permeable surface, and this will accelerate the heat exchange process from that surface. As a result, the temperature gradient increases as the thickness of the temperature boundary layer becomes thinner when the effect of heat absorption, \(\epsilon<0\), occurs, as shown in Figure 7a.