## 2 Physical Model  
The 2D MHD steady incompressible flow and heat transfer of HNF over a porous medium are here considered. As shown in Figure 1, the transverse magnetic field with strength \(B_{0}\) is applied along the \(y\)-axis. In addition, the \(Cu\) and \(Al_{2}O_{3}\) nanoparticles are added to the water base fluid. This shows that the velocity boundary layer thickness is more than the thermal boundary layer thickness. The ambient temperature of the HNF is kept constant at \(T_{\infty}\). In adopting the standard boundary layer approximation, the leading equations are as follows [27],  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]  
The radiative heat flux is calculated by applying the Rosseland approximation for radiation as follows [12,28],  
\[q_{r}=-\frac{4\sigma^{*}}{3k^{*}}\frac{\partial T^{4}}{\partial y} \tag{6}\]  
It is implicit that the temperature varies within the flow, where the term \(T^{4}\) is the linear function of the temperature. Therefore, using Taylor series expansion to the term \(T^{4}\) about \(T_{\infty}\) and ignoring the higher order terms, we acquire  
\[T^{4}\cong 4T_{\infty}^{3}T-3T_{\infty}^{4} \tag{7}\]  
Equation (3) reduces to  
\[u\frac{\partial T}{\partial x}+\mathrm{v}\frac{\partial T}{\partial y}=\frac{ \kappa_{lnf}}{\left(\rho C_{P}\right)_{lnf}}\frac{\partial^{2}T}{\partial y^{2 }}+\frac{16\sigma^{*}T_{\infty}^{3}}{3k^{*}(\rho C_{P})_{lnf}}\frac{\partial^{ 2}T}{\partial y^{2}} \tag{8}\]  
Consider the suitable similarity transformations as follows [27]:  
\[\begin{array}{l}\psi(\eta)=\xi_{2}xf(\eta),\eta=\xi_{1}y\\ u=\xi_{1}\xi_{2}xf_{\eta}(\eta),\,\mathrm{v}=-\xi_{2}f(\eta)\,,\,\theta(\eta)= \frac{T-T_{\infty}}{ax^{2}}\end{array} \tag{9}\]  
On using (9), Equations (2) to (4) are converted as  
\[\Lambda f_{\eta\eta\eta}+C_{1}\Big{(}ff_{\eta\eta}-f_{\eta}^{2}\Big{)}-\Big{(} C_{3}M+C_{2}Da^{-1}\Big{)}f_{\eta}=0 \tag{10}\]  
\[\frac{1}{C_{4}}\bigg{(}\frac{C_{5}}{\mathrm{Pr}}+N_{R}\bigg{)}\theta_{\eta\eta }+f\theta_{\eta}-2f_{\eta}\theta=0 \tag{11}\]  
With the imposed BCs as  
\[\begin{array}{l}f(0)=S\,\ f_{\eta\eta}(0)=-2(1-\phi_{1})^{2.5}(1-\phi_{2})^{2.5 }\,\ \theta(0)=1,\\ f_{\eta}(\eta)\to 0\,\ \theta(\eta)\to 0\ \ as\ \ \eta\rightarrow\infty\end{array} \tag{12}\]  
where \(Da^{-1}=\frac{v_{f}}{K\xi_{1}\xi_{2}}\) is the inverse Darcy number; \(\Lambda=\frac{\mu_{eff}}{\mu_{f}}\) is the Brinkman ratio; \(M=\frac{\sigma_{f}B_{0}^{2}}{\xi_{1}\xi_{2}v_{f}}\) is the magnetic field; \(N_{R}=\frac{16\sigma^{*}T_{\infty}^{3}}{3k_{f}k^{*}}\) is thermal radiation; and \(C_{i}\), where \(i=1\) to \(5\), is taken as  
\[C_{1}=\frac{\rho_{lnf}}{\rho_{f}}\,\ C_{2}=\frac{\mu_{lnf}}{\mu_{f}}\,\ C_{3}=\frac{\sigma_{lnf}}{\sigma_{f}}\,\ C_{4}=\frac{(\rho C_{P})_{lnf}}{( \rho C_{P})_{f}}\ \ \mathrm{and}\ \ C_{5}=\frac{k_{lnf}}{k_{f}} \tag{13}\]  
The interested physical local Nusselt number \(Nu_{x}\) is given by  
\[Nu_{x}=\frac{xq_{w}}{k_{f}(T-T_{\infty})} \tag{14}\]  
where \(q_{w}\) is the heat flux given as  
\[q_{w}=-k_{lnf}\bigg{(}\frac{\partial T}{\partial y}+q_{r}\bigg{)}_{y=0} \tag{15}\]  
Equations (14) and (15) lead to  
\[Nu_{x}=-\Bigg{(}\frac{k_{lnf}}{k_{f}}-k_{lnf}\mathrm{Pr}N_{R}\bigg{)}\xi_{1} x\theta_{\eta}(0) \tag{16}\]