## 2 Problem Formulation
\begin{array}{ll}\frac{1}{\left(1-\phi\right)^{25}}f^{\prime\prime}+\Big{(} 1-\phi+\phi\frac{\rho_{s}}{\rho_{f}}\Big{)}\Big{[}\frac{n+1}{2}(f+h)f^{\prime \prime}-n(f^{\prime}+h^{\prime})f^{\prime}\Big{]}\\ -\bigg{[}1+\frac{3\left(\sigma_{s}/\sigma_{f}-1\right)\phi}{\sigma_{s}/\sigma_ {f}+2-\big{(}\sigma_{s}/\sigma_{f}-1\big{)}\phi}\Big{]}Mf^{\prime}+\bigg{[}1- \phi+\phi\frac{\left(\rho\beta\right)_{s}}{\left(\rho\beta\right)_{f}}\bigg{]} \lambda\theta=0,\\ \\ \frac{1}{\left(1-\phi\right)^{25}}h^{\prime\prime}+\Big{(}1-\phi+\phi\frac{ \rho_{s}}{\rho_{f}}\Big{)}\Big{[}\frac{n+1}{2}\ (f+h)h^{\prime\prime}-n(f^{\prime}+h^{\prime})h^{\prime}\Big{]}\\ -\bigg{[}1+\frac{3\left(\sigma_{s}/\sigma_{f}-1\right)\phi}{\sigma_{s}/\sigma_ {f}+2-\big{(}\sigma_{s}/\sigma_{f}-1\big{)}\phi}\bigg{]}Mh^{\prime}=0,\\ \\ \frac{k_{nf}}{k_{f}}\bigg{(}\frac{1}{\Pr}\bigg{)}\theta^{\prime\prime}+\bigg{[} 1-\phi+\phi\ \frac{\left(\rho C_{p}\right)_{s}}{\left(\rho C_{p}\right)_{f}}\bigg{]} \bigg{[}\frac{n+1}{2}(f+h)\theta^{\prime}-\big{(}2n-1\big{)}(f^{\prime}+h^{ \prime})\theta\bigg{]}=0,\end{array}  \tag{11}\]with the following boundary conditions:  
\[ \begin{array}{l}f(0)=s,\quad h(0)=0,\quad\theta(0)=1+K\theta^{\prime}(0),\\ f^{\prime}(0)=c+Nf^{\prime\prime}(0),\quad h^{\prime}(0)=c+Nh^{\prime\prime}(0), \\ f^{\prime}(\eta)=0,\quad h^{\prime}(\eta)=0,\quad\theta(\eta)=0\text{ as }\eta\to\infty. \end{array}  \tag{12}\]  
In fact, throughout this paper, the governing parameters are as follows: Pr refers to the Prandtl number, \(\lambda\) denotes the mixed convection parameter with \(\lambda<0\) corresponding to opposing flow and \(\lambda>0\) corresponding to assisting flow, \(M\) denotes the magnetic parameter, \(N\) is the velocity slip parameter and \(K\) is the temperature slip parameter, which are defined as:  
\[\text{Pr}=\frac{\
u_{f}}{\alpha_{f}},\quad\lambda=\frac{Gr_{x}}{\text{Re}_{x} ^{2}},\quad M=\frac{\sigma_{f}B_{0}^{2}}{a\rho_{f}},\quad N=N_{0}\sqrt{\frac{a }{\
u_{f}}},\quad K=K_{0}\sqrt{\frac{a}{\
u_{f}}}. \tag{13}\]  
Here, the local Grashof number, \(Gr_{x}\) and the local [PERSON] number, \(\text{Re}_{x}\) along \(x\)-direction, given by the following:  
\[Gr_{x}=\frac{g\beta_{f}(T_{w}-T_{\infty})(x+y)^{3}}{\
u_{f}^{2}},\quad\text{ Re}_{x}=\frac{u_{w}(x+y)}{\
u_{f}}. \tag{14}\]  
The important physical quantities in this study are the skin friction coefficients, \(C_{fx}\) and \(C_{ty}\) and the local Nusselt number, \(Nu_{x}\) which are defined as follows:  
\[C_{fx}=\frac{\tau_{zx}}{\rho_{f}u_{w}^{2}},\quad C_{ty}=\frac{\tau_{zy}}{\rho_ {f}v_{w}^{2}},\quad Nu_{x}=\frac{(x+y)q_{w}}{k_{f}(T_{w}-T_{\infty})}, \tag{15}\]  
where \(\tau_{zx}\) and \(\tau_{zy}\) are shear stresses at the walls denoted as \(zx\) and \(zy\), respectively, while \(q_{w}\) is the heat flux at surface \(z=0\). The shear stresses are defined as:  
\[\tau_{zx}=\mu_{nf}\bigg{(}\frac{\partial u}{\partial z}+\frac{\partial w}{ \partial x}\bigg{)}_{z=0},\quad\tau_{zy}=\mu_{nf}\bigg{(}\frac{\partial v}{ \partial z}+\frac{\partial w}{\partial y}\bigg{)}_{z=0},\quad q_{w}=-k_{nf} \bigg{(}\frac{\partial T}{\partial z}\bigg{)}_{z=0}. \tag{16}\]  
Using (7) in (15) and (16), the following are obtained:  
\[\text{Re}_{x}^{1/2}C_{fx}=\frac{1}{(1-\phi)^{2.5}}f^{\prime\prime}(0),\quad \text{Re}_{y}^{1/2}C_{ty}=\frac{1}{(1-\phi)^{2.5}}h^{\prime\prime}(0),\quad \text{Re}_{x}^{-1/2}Nu_{x}=-\frac{k_{nf}}{k_{f}}\phi^{\prime}(0), \tag{17}\]  
where \(\text{Re}_{x}=u_{w}(x+y)/\
u_{f}\) is the local Reynolds number along \(x\)-direction and \(\text{Re}_{y}=v_{w}(x+y)/\
u_{f}\) is the local Reynolds number along \(y\)-direction.