## 2 Mathematical Model
\begin{array}{c}\mathrm{u}=\mathrm{u}_{\mathrm{w}}(\mathrm{x})= \mathrm{CxP},\;\mathrm{v}=0\;,\;\frac{\partial\mathrm{H}_{1}}{\partial y}= \mathrm{H}_{2}=0,\\ \mathrm{T}=\mathrm{T}_{\mathrm{w}}=\mathrm{T}_{\infty}+\mathrm{Ax}^{\lambda} \;\mathrm{at}\;\mathrm{y}=0\\ \mathrm{u}=\mathrm{u}_{\mathrm{e}}(\mathrm{x})=\mathrm{DxP}\;,\mathrm{H}_{1}= \mathrm{H}_{\mathrm{e}}(\mathrm{x})=\mathrm{H}_{0}\;\mathrm{xP}\\ \mathrm{T}=\mathrm{T}_{\infty}\;\mathrm{as}\;\;y\rightarrow\infty\end{array} \right. \tag{10}\]  
In this case \(C,\;D\) and \(A\) are constants (positive), \(H_{0}\) denotes consistent magnetic field at infinity (free stream), \(p\) and \(\lambda\) are velocity exponent parameter and temperature exponent parameter, respectively, \(T_{w}\) and \(T_{\infty}\) are the wall temperature and the ambient temperature, respectively. The system of major equations has been transformed into a system of dimensionless equations by the introduction of the dimensionless variables listed below:  
\[ \begin{array}{c}\eta=\frac{y}{x}(R_{e_{x}})^{\frac{1}{2}},\;\;\xi=\frac{G_{ tr}C_{0}S^{\gamma}}{(R_{e_{x}})^{2}},\;\;R_{e_{x}}=\frac{u_{w}}{\
u}\,x,\;\psi(x,y)= \
u(R_{e_{x}})^{\frac{1}{2}}\;\;f(\xi,\eta)\\ \phi=H_{e}\Big{(}\frac{y\chi}{u_{w}}\Big{)}^{\frac{1}{2}}g(\xi,\eta),\;\;Gr_{ x}=\frac{g(\Omega_{W}-T_{\infty})x^{3}}{\
u^{2}},\;\;\;\;\;\theta(\xi,\eta)= \frac{T-T_{\infty}}{T_{w}-T_{\infty}}\end{array}  \tag{11}\]  
where \(\psi\) is the dimensional stream function, \(f\) is the dimensionless stream flow function, and \(\theta\) is the dimensionless fluid temperature, \(\xi\) is the upthrust force variable, and \(\eta\) is the dimensionless interval perpendicular to the sheet. The continuity Equations (1) and (2) are now found to be identically satisfied by Equation (11). After some simplification, by substituting Equation (11) into Equations (3), (4) and (9), we obtain:  
_Momentum boundary layer equation_  
\[\left\{ \begin{array}{c}f^{\prime\prime}-{p}{f^{\prime}}^{2}+\frac{p+1}{2}ff ^{\prime\prime}+\delta^{2}p+\beta\Big{[}{g^{\prime}}^{2}p-\Big{(}\frac{p+1}{ 2}\Big{)}g\delta^{\prime\prime}-p\Big{]}=\\ (\lambda-2p+1)\xi[\frac{\partial f^{\prime}}{\partial\xi}f^{\prime}\;-\frac{ \partial f}{\partial\xi}f^{\prime\prime}+\beta\Big{\{}g^{\prime\prime}\frac{ \partial g}{\partial\xi}-g^{\prime}\frac{\partial g^{\prime}}{\partial\xi} \Big{\}}]\end{array} \right. \tag{12}\]  
_Magnetic induction boundary layer equation_  
\[\gamma g^{\prime\prime}+\frac{p+1}{2}g^{\prime\prime}f-\frac{p+1}{2}f^{ \prime\prime}g=(\lambda-2p+1)\xi\bigg{[}\frac{\partial g^{\prime}}{\partial \xi}f^{\prime}\;-\frac{\partial f^{\prime}}{\partial\xi}g^{\prime}-\frac{ \partial f}{\partial\xi}\;g^{\prime\prime}+\frac{\partial g}{\partial\xi}f^{ \prime\prime}\bigg{]} \tag{13}\]  
_Thermal boundary layer (heat) equation_  
\[\frac{1}{P_{r}}\bigg{(}1+\frac{4}{3}N\bigg{)}\theta^{\prime\prime}-\lambda f^{ \prime}\theta+\frac{p+1}{2}f\theta^{\prime}=(\lambda-2p+1)\xi\bigg{[}f^{\prime} \;\frac{\partial\theta}{\partial\xi}-\theta^{\prime}\frac{\partial f}{\partial \xi}\bigg{]} \tag{14}\]where \(\beta=\frac{\mu}{4\pi\rho}\left(\frac{H_{\rm r}}{\rm C}\right)^{2}\) is body force magnetic parameter, a constant term \(\delta=\frac{\rm D}{\zeta}\), \(\gamma=\frac{\mu_{\rm c}}{\rm P}\) is the reciprocal of the magnetic Prandtl number, \(\rm P_{r}=\frac{\
u}{\alpha}\) is the Prandtl number and \(N=\left(\frac{4\pi T_{\rm so}^{3}}{k_{\rm B}}\right)\) is the radiation variable. The following boundary conditions are required after transformation:  
\[\left\{ \begin{array}{cc}f(\xi,0)=0,\ f^{\prime}(\xi,0)=1,&g(\xi,0)=0,\\ g^{\prime\prime}(\xi,0)=0,&\theta(\xi,0)=1&at\ y=0\\ f^{\prime}(\xi,\infty)=\delta,\ g^{\prime}(\xi,\infty)=1,&\theta(\xi,\infty)=0 \ as\ y\rightarrow\infty\end{array} \right.