## 3 Steady-State Solution (\(\partial\)/\(\partial\)t=0)  
The following similarity transformation was introduced  
\[\psi=\sqrt{cv}xf(\eta),\quad\theta(\eta)=\frac{T-T_{\infty}}{T_{w}-T_{\infty}}, \quad\phi(\eta)=\frac{C-C_{\infty}}{C_{w}-C_{\infty}},\quad\eta=\sqrt{\frac{c} {v}}y, \tag{7}\]  
where \(\eta\) is the similarity variable, \(\psi\) is the stream function defined as \(u=\partial\psi/\partial y\), and \(v=-\ \partial\psi/\partial x\), which automatically satisfies Equation (1). The similarity variables (7) were substituted by Equations (2)-(4) to obtain the following ordinary (similarity) differential equations  
\[f^{\prime\prime\prime}+ff^{\prime\prime}-{f^{\prime}}^{2}+1=0 \tag{8}\]  
\[\frac{1}{\Pr}\theta^{\prime\prime}+f\theta^{\prime}+Nb\theta^{\prime}\phi^{ \prime}+Nt\theta^{\prime 2}+Du\phi^{\prime\prime}=0 \tag{9}\]  
\[\phi^{\prime\prime}+Lef\phi^{\prime}+\frac{Nt}{Nb}\theta^{\prime\prime}+LeSr \theta^{\prime\prime}=0 \tag{10}\]  
subject to the boundary conditions  
\[\begin{array}{ll}f(0)=0,&f^{\prime}(0)=\varepsilon+\sigma f^{\prime\prime}( 0)+\delta f^{\prime\prime\prime}(0),\quad\theta(0)=1,\quad Nb\phi^{\prime}(0) +Nt\theta^{\prime}(0)=0,\\ f^{\prime}(\infty)\to 1,&\theta(\infty)\to 0,\quad\phi(\infty)\to 0\end{array} \tag{11}\]  
In the above equations, primes denote the differentiation with respect to \(\eta\). According to [PERSON] and [PERSON] [57], \(A=\sigma\sqrt{v/c}\) and \(B=\delta(v/c)\), with \(\sigma>0\) being the first velocity slip parameter, and \(\delta<0\) the second velocity slip parameter (see [PERSON] et al. [24]). Here, \(\Pr\) is the Prandtl number, \(Le\) is the Lewis number, \(Sr\) is the Soret number, \(Du\) is the Dufour number, \(Nb\) is the Brownian motion, \(Nt\) is the thermophoresis parameter, and \(\varepsilon\) is the velocity ratio parameter, which are defined as:  
\[\begin{array}{ll}\Pr=\frac{v}{\alpha},\quad\mathrm{Le}=\frac{v}{D_{B}}, \quad Sr=\frac{D_{B}K_{T}}{vT_{m}}\frac{T_{m}-T_{\infty}}{C_{w}-C_{\infty}}, \quad Du=\frac{D_{B}K_{T}}{v_{\varepsilon}c_{p}}\frac{C_{w}-C_{\infty}}{T_{w}- T_{\infty}},\\ \varepsilon=\frac{b}{\varepsilon},\quad Nb=\frac{\Omega D_{B}}{v}(C_{w}-C_{ \infty}),\quad Nt=\frac{\Omega D_{T}}{T_{\infty}v}(T_{w}-T_{\infty})\end{array} \tag{12}\]  
where \(\varepsilon>0\) for stretching and \(\varepsilon<0\) for shrinking.  
The physical quantities of practical interest are the local skin friction coefficient \(C_{f}\), the local Nusselt number \(Nu_{x}\), and the local Sherwood number \(Sh_{x}\) which are defined as:  
\[C_{f}=\frac{\tau_{w}}{\rho U_{\infty}{}^{2}},\quad Nu_{x}=\frac{xq_{w}}{k(T_{w }-T_{\infty})},\quad Sh_{x}=\frac{xq_{m}}{D_{B}(C_{w}-C_{\infty})}, \tag{13}\]  
Figure 1: Physical model and coordinate system: (**a**) stretching sheet and (**b**) shrinking sheet.  
where \(\tau_{w}\) is the skin friction or the shear stress on the stretching/shrinking sheet, \(q_{w}\) is the heat flux from the surface of the plate, and \(q_{m}\) is the mass flux from the surface of the plate, which are given by  
\[\tau_{w}=\mu\left(\frac{\partial u}{\partial y}\right)_{y=0},\quad q_{w}=-k \left(\frac{\partial T}{\partial y}\right)_{y=0},\quad q_{m}=-D_{B}\left(\frac{ \partial C}{\partial y}\right)_{y=0}, \tag{14}\]  
Using (7) in (13) and (14), we obtained  
\[\text{Re}_{x}^{1/2}C_{f}=f^{\prime\prime}(0),\quad\text{Re}_{x}^{-1/2}Nu_{x}=- \theta^{\prime}(0),\quad\text{Re}_{x}^{-1/2}Sh_{x}=-\phi^{\prime}(0) \tag{15}\]  
where \(\text{Re}_{x}=cx^{2}/v\) is the local Reynolds number.