## 2 Basic Equations
\begin{array}{l}u\frac{\partial u}{\partial\xi}+v\frac{\partial u}{\partial \eta}=u_{e}\frac{\partial u_{e}}{\partial\xi}+\frac{\rho_{f}}{\rho_{nf}}\Big{(} \frac{\mu_{nf}}{\mu_{f}}+\Delta\Big{)}\frac{\partial^{2}u}{\partial\eta^{2}}+ \frac{\rho_{f}}{\rho_{nf}}\frac{(\rho\beta)_{nf}}{(\rho\beta)_{f}}\lambda \theta\sin\xi\\ +\frac{\rho_{f}}{\rho_{nf}}\Big{(}\frac{\mu_{nf}}{\mu_{f}}+\Delta\Big{)}\xi(u _{e}-u)+\frac{\rho_{f}}{\rho_{nf}}\Delta\frac{\partial N}{\partial\eta}\end{array}  \tag{9}\]  
Figure 1: Geometry of the flow process. \[u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}=\frac{\rho_{f}}{\rho_{ nf}}\bigg{(}\frac{\mu_{nf}}{\mu_{f}}+\frac{\Delta}{2}\bigg{)}\frac{\partial^{2}N}{ \partial y^{2}}-\frac{\rho_{nf}}{\rho_{nf}}\Delta B\bigg{(}2N+\frac{\partial u }{\partial y}\bigg{)} \tag{10}\]  
\[u\frac{\partial\theta}{\partial\xi}+v\frac{\partial\theta}{\partial\eta}= \frac{1}{\Pr}\frac{\alpha_{nf}}{\alpha_{f}}\frac{\partial^{2}\theta}{\partial \eta^{2}} \tag{11}\]  
The dimensionless boundary conditions become:  
\[u=v=0,\;\frac{k_{nf}}{k_{f}}\frac{\partial\theta}{\partial\eta}=-Bi[1-\theta], \;S=1\;\text{on}\;\eta=0\]  
\[u\to u_{e}\,,\theta\to 0\,,\;S\to 0\;\;\text{as}\;\eta\rightarrow\infty \tag{12}\]  
where \(\lambda\) is the mixed convection or buoyancy parameter, \(\text{Gr}\) is the Grashof number, \(\varepsilon\) is the permeability parameter (porous medium parameter), \(\Delta\) and \(B\) are the material parameters, Bi is the Biot number, and \(\Pr\) is the Prandtl number. These dimensionless numbers can be written as;  
\[\lambda=\frac{g\beta_{f}(T_{f}-T_{\infty})a}{U_{\infty}^{2}}=\pm\frac{Gr}{ \text{Re}^{2}},Gr=\frac{g\beta_{f}(T_{f}-T_{\infty})a^{3}}{v_{f}^{2}}, \varepsilon=(v_{f}a)/(U_{\infty}K),B=\frac{av_{f}}{jU_{\infty}},\Delta=\frac{ \kappa}{\mu_{f}}\]  
\[\Pr=\frac{v_{f}}{\alpha_{f}},Bi=\frac{h_{ft}a}{\text{Re}^{1/2}k_{f}} \tag{13}\]  
It is important to note that \(\lambda>0\) and \(\lambda<0\) show the assisting and opposing flows, respectively. According to [25], the stream function \(\psi\) can be defined as \(u=\partial\psi/\partial\eta,v=-\partial\psi/\partial\xi\) and then, Equation (8) is satisfied. Using the following transformations. \[\psi=\xi f(\xi,\eta),\;\theta=\theta(\xi,\eta) \tag{14}\]  
Into Equations (8)-(10), we get:  
\[ \begin{array}{l}\frac{\rho_{f}}{\rho_{nf}}\bigg{(}\frac{\mu_{nf}}{\mu_{f}}+ \Delta\bigg{)}f^{\prime\prime\prime}+ff^{\prime\prime}-f^{\prime 2}+\frac{ \sin\xi\cos\xi}{\xi}+\lambda\frac{\sin\xi}{\xi}\frac{\rho_{f}}{\rho_{nf}} \frac{(\rho\beta)_{nf}}{(\rho\beta)_{f}}\theta\\ +\frac{\rho_{f}}{\rho_{nf}}\bigg{(}\frac{\mu_{nf}}{\mu_{f}}+\Delta\bigg{)} \xi\bigg{(}\frac{\sin\xi}{\xi}-f^{\prime}\bigg{)}+\frac{\rho_{f}}{\rho_{nf}} \Delta g^{\prime}=\xi\bigg{(}f^{\prime}\frac{\partial f^{\prime}}{\partial \xi}-f^{\prime}\frac{\partial f}{\partial\xi}\bigg{)}\end{array}  \tag{15}\]  
\[\frac{\rho_{f}}{\rho_{nf}}\bigg{(}\frac{\mu_{nf}}{\mu_{f}}+\frac{\Delta}{2} \bigg{)}g^{\prime\prime}+fg^{\prime}-f^{\prime}g-\frac{\rho_{nf}}{\rho_{nf}} \Delta B(2g+f^{\prime\prime})=\xi\bigg{(}f^{\prime}\frac{\partial g}{\partial \xi}-g^{\prime}\frac{\partial f}{\partial\xi}\bigg{)} \tag{16}\]  
\[\frac{1}{\Pr}\frac{\alpha_{nf}}{\alpha_{f}}\theta^{\prime\prime}+f\theta^{ \prime}=\xi\bigg{(}f^{\prime}\frac{\partial\theta}{\partial\xi}-\theta^{ \prime}\frac{\partial f}{\partial\xi}\bigg{)} \tag{17}\]  
The accompanying non-dimensional boundary conditions become:  
\[ \begin{array}{l}f=f^{\prime}=0,g=0,\frac{k_{nf}}{k_{f}}\theta^{\prime}=-Bi[1- \theta]\quad\text{at}\;\eta=0\\ f^{\prime}\rightarrow\frac{\sin\xi}{\xi},g=0,\theta\to 0\quad\text{as}\quad\eta \rightarrow\infty\end{array}