where, prime denotes differentiation with respect to \(\eta\) and\(Re_{x}\) is the local Reynolds number\(\left(Re_{x}=\frac{\text{Ux}}{\upsilon}\right)\). Applying Eqs.
(8a) – (8b) to Eqs. (2) – (7), the coupled nonlinear ordinary
differential equations are obtained as
\begin{equation}
\frac{d^{3}f}{d\eta^{3}}+\frac{1}{2}f\frac{d^{2}f}{d\eta^{2}}-M\frac{\text{df}}{\text{dη}}+Gr\theta+Gc\phi=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (9)\nonumber \\
\end{equation}\begin{equation}
\frac{d^{2}\theta}{d\eta^{2}}+k_{0}P\operatorname{r}\left[\frac{1}{2}f\frac{\text{dθ}}{\text{dη}}+Ec\left(\frac{d^{2}f}{d\eta^{2}}\right)^{2}+Q\theta\right]+k_{0}\left[\varepsilon\left(\frac{\text{dθ}}{\text{dη}}\right)^{2}+D_{f}\frac{d^{2}\theta}{d\eta^{2}}\right]=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (10)\nonumber \\
\end{equation}\begin{equation}
\frac{d^{2}\phi}{d\eta^{2}}+Sc\left(\frac{1}{2}f\frac{\text{dϕ}}{\text{dη}}+Sr\frac{d^{2}\theta}{d\eta^{2}}\right)=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (11)\nonumber \\
\end{equation}with Blasius boundary condition
\begin{equation}
\frac{f=0\ ;\ \ f^{{}^{\prime}}=0\ \ ;\ \ \theta^{{}^{\prime}}=-\varepsilon\left[1-\theta\left(0\right)\right]\ \ \ \ at\ \ \ \ \ \eta=0}{f^{{}^{\prime}}=1\ \ ;\ \ \ \theta=0;\ \text{\ ϕ}=1\text{\ \ \ \ \ \ }as\ \ \ \eta\rightarrow\infty,}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (12)\nonumber \\
\end{equation}and Sakiadis boundary condition
\begin{equation}
\frac{f=0\ \ ;\ \ f^{{}^{\prime}}=1\ \ ;\ \ \ \theta^{{}^{\prime}}=-\varepsilon\left[1-\theta\left(0\right)\right]\ \ \ \ \ at\ \ \ \eta=0}{f^{{}^{\prime}}=0\ \ ;\ \ \ \theta=0;\ \ \phi=1\ \ \ \ as\ \ \ \ \eta\rightarrow\infty.}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (13)\nonumber \\
\end{equation}where,
\begin{equation}
M_{x}=\frac{\sigma B_{0}^{2}x}{\text{ρU}};\ \ Gr_{x}=\frac{\text{gβx}\left(T_{w}-T_{\infty}\right)}{U^{2}};\ \ Gc_{x}=\frac{g\beta^{*}x\left(C_{w}-C_{\infty}\right)}{U^{2}};\ \ Pr=\frac{\upsilon}{\alpha};\ \ Q_{x}=\frac{\text{qx}}{\rho c_{p}U};\ \ \nonumber \\
\end{equation}\begin{equation}
Ec=\frac{U^{2}}{c_{p}\left(T_{w}-T_{\infty}\right)};\ \ D_{f}=\frac{Dk_{t}}{c_{p}c_{s}\alpha}\frac{\left(C_{w}-C_{\infty}\right)}{\left(T_{w}-T_{\infty}\right)};\ \ k_{0}=\frac{3N_{R}}{3N_{R}\left[1+\theta\varepsilon\right]+4};\ \ \varepsilon=\gamma\left[T_{w}-T_{\infty}\right];\nonumber \\
\end{equation}\begin{equation}
Sc=\frac{\upsilon}{D};\ \ Sr=\frac{Dk_{t}}{\upsilon T_{m}}\frac{T_{w}-T_{\infty}}{C_{w}-C_{\infty}}\nonumber \\
\end{equation}are the local magnetic field parameter \(\left(M_{x}\right)\), local
Grashof number \(\left(Gr_{x}\right)\), local Solutal number\(\left(Gc_{x}\right)\), Prandtl number \((Pr)\), local Heat release
parameter \(\left(Q_{x}\right)\), Eckert number\(\left(\text{Ec}\right)\), Dufour number \(\left(D_{f}\right)\),
thermal radiation parameter \(\left(k_{0}\right)\), variable thermal
conductivity parameter \(\left(\varepsilon\right)\), Schmidt number\(\left(\text{Sc}\right)\) and Soret number\(\left(\text{Sr}\right)\) respectively. The local parameters\(M_{x}\), \(Gr_{x}\), \(Gc_{x}\) and \(Q_{x}\) in Eqs. (9) – (12) are
functions of \(x\).