## 4 Results and Discussion  
We examined the 2D MHD steady incompressible flow and heat transfer due to a porous medium containing \(Cu-Al_{2}O_{3}\) nanoparticles in the base fluid by applying a magnetic field of strength \(B_{0}\) to the fluid flow. The addition of nanoparticles enhances the thermal efficiency of the flow system. The leading equations form the system of PDEs and are then converted into the system of ODEs by adopting suitable similarity transformations. The system is analytically solved to obtain the solutions for the velocity profile and temperature distribution in terms of exponential and Gamma functions, respectively. In all plots, the dotted lines refer to the behavior of the base fluid, while the solid lines refer to the behavior of HNF for \(Cu-Al_{2}O_{3}\). Figure 2 demonstrates the axial velocity \(f(\eta)\) for various \(Da^{-1}\). We found that the velocity declines as \(Da^{-1}\) increases. Panels (a)-(c), where the velocity is examined for \(S=0\) show that there is no permeability. For suction \(S=1\) and injection \(S=-1\), we observe that as \(S\) increases from injection to suction, the axial velocity for HNF very quickly coincides with the base fluid as \(Da^{-1}\) increases. In all cases, the profile of \(f_{\eta}(\eta)\) has a decreasing nature, and it becomes constant to zero at a certain point of \(\eta\). Figure 3 depicts the velocity \(f_{\eta}(\eta)\) as a function of \(\eta\) for various values of \(\Lambda\). The velocity increases with an increase in \(\Lambda\). Panels (a)-(c), where the velocity is examined for \(S=0\), show that there is no permeability. For suction \(S=1\) and injection \(S=-1\), we observe that the difference between the axial velocity of the base fluid and HNF is larger in the case of suction and smaller in the case of injection. In all cases, the axial velocity is less for an HNF than the base fluid. Figure 4 shows the profile of \(f_{\eta}(\eta)\) for various \(M\). It can be seen that \(f_{\eta}(\eta)\) is smaller for larger values of \(M\). Panels (a)-(c), where the velocity is examined for \(S=0\), show that there is no permeability. For suction \(S=1\) and injection \(S=-1\), we observe that the  
difference between the axial velocity of the base fluid and the HNF is more in the case of suction and less in the case of injection. Figure 2: _Cont._  
Figure 3: _Cont._  
Figure 2: The axial velocity as a function of \(\eta\) for various values of \(Da^{-1}\) and for three different regimes of \(S\): (**a**) \(S=0\), (**b**) \(S=1\), and (**c**) \(S=-1\). The other fixed parameters are: \(\Lambda=M=1\), \(\phi_{1}=0.1\), \(\phi_{2}=0.04\). Figure 3: The axial velocity as a function of \(\eta\) for various values of \(\Lambda\) and for three different regimes of \(S\): (**a**) \(S=0\), (**b**) \(S=1\) and (**c**) \(S=-1\). The other fixed parameters are: \(Da^{-1}=M=1\), \(\phi_{1}=0.1\), \(\phi_{2}=0.04\). The effect of \(S\) is shown in Figure 5 for both the suction and injection cases. Clearly, the velocity profile \(f_{\eta}(\eta)\) decreases for a larger \(S\). Panel (a) shows that the difference between the axial velocity of the base fluid and the HNF is larger in the case of suction and smaller in the case of injection as in panel (b). At \(\eta=0\), the axial velocity is different for various values of each parameter. Figure 4: _Cont._  
Figure 4: The axial velocity as a function of \(\eta\) for various values of \(M\) and for three different regimes of \(S\): (**a**) \(S=0\), (**b**) \(S=1\) and (**c**) \(S=-1\). The other fixed parameters are: \(\Lambda=Da^{-1}=1\), \(\phi_{1}=0.1\), \(\phi_{2}=0.04\). Figure 5: _Cont._  
Figure 6 displays the temperature profile \(\theta(\eta)\) for various values of \(Da^{-1}\). We observe that \(\theta(\eta)\) increases as \(Da^{-1}\) increases. Panels (a)-(c), where the temperature is shown for \(S=0\), show that there is no permeability. For suction \(S=1\) and injection \(S=-1\), we observe that \(\theta(\eta)\) for HNF coincides with the base fluid as \(Da^{-1}\) increases.