## 2 Problem Formulation
\begin{array}{l}u\ =\ U_{w}\ =\ ax,\ v\ =\ V_{w}\ =\ by\\ \\ -k\frac{\partial T}{\partial z}\ =\ h_{f}\Big{(}T_{f}-T\Big{)},\ -D_{B}\frac{ \partial C}{\partial z}\ =\ h_{s}(C_{s}-C)\ at\ \eta\ =\ 0\\ \\ u\to 0,\ v\to 0,\ w\to 0,\ T\to T_{\infty},\ C\to C_{\infty}\ as\ \eta\to\infty.\end{array} \right\} \tag{14}\]  
In the preceding equation, \(U_{w}\) and \(V_{w}\) are the stretching velocities, \(a\) and \(b\) are positive constants, where \(c\ =\ \frac{a}{b}\), \(h_{f},h_{s}\) are the convective heat and mass transmission coefficients, \(T_{f}\) is the convective fluid temperature and \(C\) is the concentration below the moving sheet. In view of the above equations, the similarity variables are defined as  
\[\left. \begin{array}{l}u\ =\ axf^{\prime}(\eta),\ v\ =\ byg^{\prime}(\eta),\ w\ =\ -(uv)^{\frac{1}{2}}(f(\eta)+cg(\eta)),\\ \\ \theta(\eta)\ =\ \frac{T-T_{\infty}}{T_{f}-T_{\infty}},\ \phi(\eta)\ =\ \frac{C-C_{\infty}}{C_{f}-C_{\infty}},\ \eta\ =\ z\sqrt{\frac{a}{v}}.\end{array} \right. \tag{15}\]  
In view of Equation (15), Equation (7) is satisfied, Equations (8), (9), (11), and (13) are reduced to  
\[f^{\prime\prime\prime}\ -\big{(}f^{\prime}\big{)}^{2}+(f+cg)f^{\prime\prime} +Wef^{\prime\prime}f^{\prime\prime}\ +gf^{\prime\prime}\ -\gamma f^{\prime}\ =\ 0, \tag{16}\]  
\[g^{\prime\prime\prime}\ -\big{(}g^{\prime}\big{)}^{2}+(f+cg)g^{\prime\prime} +Weg^{\prime\prime}g^{\prime\prime\prime}+fg^{\prime\prime}\ -\gamma g^{\prime}\ =\ 0, \tag{17}\]  
\[\frac{1}{\Pr}\bigg{(}1+\frac{4}{3}Rd\bigg{)}\theta^{\prime\prime}\ +(f+cg)\theta^{\prime}+Nb\phi^{\prime}\theta^{\prime}+ Nt\big{(}\theta^{\prime}\big{)}^{2}\ =\ 0, \tag{18}\]  
\[\phi^{\prime\prime}(\eta)+Le(f+cg)\phi^{\prime}(\eta)+\bigg{(}\frac{ Nt}{Nb}\bigg{)}\theta^{\prime\prime}(\eta)\ =\ 0. \tag{19}\]  
With boundary conditions  
\[ \begin{array}{l}f(0)\ =\ 0,f^{\prime}(0)\ =\ 1,g^{\prime}(0)\ =\ c,\\ \\ g^{\prime}(0)\ =\ 0,f^{\prime}(\infty)\ =\ 0,g^{\prime}(\infty)\ =\ 0\\ \\ \theta^{\prime}(0)\ =\ -Bi_{1}(1-\theta(0)),\phi^{\prime}(0)\ =\ -Bi_{2}(1-\phi(0)), \\ \\ \theta(\infty)\ =\ \phi(\infty)\ =\ 0.\end{array}  \tag{20}\]  
In the above equations, \(We\ =\ \Gamma x\sqrt{2a^{3}/\
u}\) represents Williamson parameter, \(\gamma\ =\ \
u/ak\) indicates the porosity parameter, \(\Pr\ =\ \rho\
u c_{p}/k\) indicates the Prandtl number, \(Rd\ =\ 4\sigma T_{\infty}^{3}/kk^{*}\) represents the thermal radiation parameter,\(Nt\ =\ \tau_{1}D_{T}(T_{f}-T_{\infty})/\
u T_{\infty},Nb\ =\ \tau_{1}D_{T}(C_{f}-C_{\infty})/\
u\) represent the thermophoresis and Brownian motion parameters. \(Bi_{1}\ =\ \frac{h_{f}}{\xi}/\sqrt{\
u/a},Bi_{2}\ =\ \frac{h_{f}}{\xi}/\sqrt{\
u/a}\), where \(Bi_{1}\)and \(Bi_{2}\) the Biot numbers. Skin friction, local Nusselt number and Sherwood number in dimensionless form are defined as  
\[ \begin{array}{ll}\frac{1}{\sqrt{2}}\mathrm{C}_{f}\sqrt{\mathrm{Re}_{x}}&=& \Big{(}f^{\prime\prime}(0)+We(f^{\prime\prime})^{2}(0)\Big{)},\\ \frac{1}{\sqrt{2}}\mathrm{C}_{f}\sqrt{\mathrm{Re}_{x}}&=&\Big{(}g^{\prime \prime}(0)+We(g^{\prime\prime})^{2}(0)\Big{)},\\ u&=&-\bigg{(}1+\frac{4}{3}\mathrm{R}d\bigg{)}\Theta^{\prime}(0),\quad Sh~{}=~{}- \Phi^{\prime}(0).\end{array}  \tag{21}\]