Using Eq. (7), Eqs. (3 – 6) are transformed to a set of coupled non-linear ordinary differential equations
\begin{equation} \left(1+\frac{1}{\beta}\right)\frac{d^{3}f}{d\eta^{3}}+f\frac{d^{2}f}{d\eta^{2}}+\left(\frac{\text{df}}{\text{dη}}\right)^{2}+Gr\theta-M\left(\frac{\text{df}}{\text{dη}}-1\right)+1=0\ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(8)\nonumber \\ \end{equation}\begin{equation} \frac{d^{2}\theta}{d\eta^{2}}+Prf\frac{\text{dθ}}{\text{dη}}+\left(1+\frac{1}{\beta}\right)\text{PrEc}\left(\frac{d^{2}f}{d\eta^{2}}\right)^{2}+PrEcM\left(\frac{\text{df}}{\text{dη}}-1\right)^{2}=0\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(9)\nonumber \\ \end{equation}
and the boundary conditions are
\begin{equation} \begin{matrix}f\left(0\right)=0,\text{\ \ }\ \frac{\text{df}}{\text{dη}}\left(0\right)=\lambda,\text{\ \ \ }\frac{\text{dθ}}{\text{dη}}\left(0\right)=Bi\left[\theta\left(0\right)-1\right]\\ \frac{\text{df}}{\text{dη}}\left(\infty\right)=1,\text{\ \ }\ \frac{\text{dθ}}{\text{dη}}\left(\infty\right)=0\ .\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\ \end{matrix}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(10)\nonumber \\ \end{equation}
From the above Eqs. (8), (9) and (10), prime denote differentiation with respect to \(\eta\), \(\lambda\) is the velocity ratio parameter,Gr is Grashof number, \(M\) is Magnetic field parameter, \(\Pr\) is Prandtl number, Ec is Eckert number andBi is the Biot number respectively defined as follows;
\begin{equation} M=\frac{\sigma B_{0}^{2}}{\text{ρa}};\ Gr=\frac{\text{gβ}\left(T_{w}-T_{\infty}\right)}{U_{\infty}a};\ Pr=\frac{\upsilon}{\alpha};\ Ec=\frac{U_{\infty}^{2}}{c_{p}\left(T_{w}-T_{\infty}\right)};\ Bi=\frac{h}{k}\sqrt{\frac{\upsilon}{a}};\ \lambda=\frac{b}{a}\ .\text{\ \ \ \ \ \ \ }(11)\nonumber \\ \end{equation}
The physical quantities of practical interest are the skin friction coefficient \(C_{f}\) and the local Nusselt number \(Nu_{x}\) defined as
\begin{equation} C_{f}=\frac{\tau_{w}}{\rho U_{w}^{2}},\ \ Nu_{x}=\frac{xq_{w}}{\alpha\left(T_{w}-T_{\infty}\right)}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (12)\nonumber \\ \end{equation}
where \(\tau_{w}\) is the shear stress or skin friction along the stretching sheet and \(q_{w}\) is the heat flux from the sheet and defined thus
\begin{equation} C_{f}\left(Re_{x}\right)^{\frac{1}{2}}=\left(1+\frac{1}{\beta}\right)\frac{d^{2}f}{d\eta^{2}}\left(0\right),\text{\ \ }\ \frac{Nu_{x}}{\left(Re_{x}\right)^{\frac{1}{2}}}=-\frac{\text{dθ}}{\text{dη}}\left(0\right),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (13)\nonumber \\ \end{equation}
where \(Re_{x}=U_{w}x/\upsilon\) is the local Reynolds number.