## 2 Mathematical Methods  
Consider a steady two-dimensional MHD flow and heat transfer of hybrid nanofluid along a permeable exponentially shrinking surface. As shown in Figure 1, the \(x\) direction is along the surface and the \(y\) direction is normal to it. The velocity wall is expressed as \(u_{w}(x)=U_{0}\ e^{xL}\), where \(U_{0}\) is a constant velocity and \(L\) is the characteristic length of the surface. The induced magnetic field variable \(B(x)=B_{0}\ e^{x/2L}\) acts on the \(y\) direction, where \(B_{0}\) is represented as an induced magnetic field constant. The dimensional heat sink/sourceis given as \(Q(x)=Q_{0}\ e^{x/L}\), where \(Q_{0}\) is the heat sink/source constant. The viscous dissipation effect is also ignored. Assume that the wall temperature of the exponentially shrinking/stretching surface is given as \(T_{w}(x)=T_{\infty}+T_{0}e^{x/2L}\), where \(T_{\infty}\) is the ambient temperature, and \(T_{0}\) is the reference temperature when \(T_{0}>0\), referred to as assisting flow, and \(T_{0}<0\) is an opposing flow. Additionally, assume that the size of nanoparticles is uniform and small, with a nanosize of \(10^{-9}\). The agglomeration of nanoparticles is neglected due to the synthesis of hybrid nanofluid as a stable compound. The impact of thermal radiation along the \(y\) axis with the convection heat transfer is integrated in the energy equation. Hence, the governing equations for continuity, momentum, and energy are given in Equations (1)-(3), respectively. All nomenclature in Equations (1)-(3) are illustrated in Nomenclature. \[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{\mu_{lmf} }{\rho_{lmf}}\left[\frac{\partial^{2}u}{\partial y^{2}}\right]-\frac{\sigma_{ lnnf}B^{2}(x)u}{\rho_{lmf}}+gB(T-T_{\infty}) \tag{2}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{lmf}} {\left(\rho C_{p}\right)_{lnf}}\frac{\partial^{2}T}{\partial y^{2}}-\frac{1} {\left(\rho C_{p}\right)_{lnf}}\frac{\partial q_{r}}{\partial y}+\frac{Q(x)}{ \left(\rho C_{p}\right)_{lnf}}(T-T_{\infty}) \tag{3}\]  
where the heat flux of radiation, \(q_{r}\), is described by using the Rosseland approximation:  
\[q_{r}=-\frac{4\sigma^{*}}{3k^{*}}\frac{\partial T^{4}}{\partial y} \tag{4}\]  
where \(\sigma^{*}\) is the Stefan-Boltzman constant, and \(k^{*}\) represents the mean absorption coefficient. The complication with this equation is that temperature appears to the fourth power, which makes the problem nonlinear in \(T\) and eliminates most hope of finding an analytical solution to the conduction problem. One way to deal with this is to linearize the radiation rate law via a first-order Taylor series expansion, assuming that the temperature difference is small enough in the flow. Thus, the expansion term \(T^{4}\) in the form of a Taylor series with  
Figure 1: MHD flow and heat transfer of hybrid nanofluid over permeable exponentially shrinking sheet. ambient temperature, \(T_{\infty}\), by neglecting a higher order to obtain the term in linear function is as follows:  
\[T^{4}\cong 4T_{\infty}^{3}T-3T_{\infty}^{4} \tag{5}\]  
From Equations (4) and (5):  
\[\frac{\partial q_{r}}{\partial y}=-\frac{16\sigma^{*}T_{\infty}^{3}}{3k^{*}} \frac{\partial^{2}T}{\partial y^{2}} \tag{6}\]  
Using Equation (6), the new energy equation will be:  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\left(\frac{k_{ lnf}}{\left(\rho C_{p}\right)_{lnf}}+\frac{1}{\left(\rho C_{p}\right)_{lnf}} \frac{16\sigma^{*}T_{\infty}^{3}}{3k^{*}}\right)\frac{\partial^{2}T}{\partial y ^{2}}+\frac{Q(x)}{\left(\rho C_{p}\right)_{lnf}}(T-T_{\infty}) \tag{7}\]  
Hence, a boundary condition for governing Equations (2), (3) and (7) is:  
\[ \begin{array}{l}u=\lambda u_{w}(x)=\lambda U_{0}e^{x/L},v=v_{w}(x),T=T_{w}(x )=T_{\infty}+T_{0}e^{x/2L}\;\mbox{as}\;y=0\\ u=0,T=T_{\infty}\;\mbox{as}\;y\rightarrow\infty\end{array}  \tag{8}\]  
where \(v_{w}(x)<0\) is velocity suction, whereas \(v_{w}(x)>0\) is velocity blowing.