## 2 Problem Formulation
### Steady Flow Formulation
The resulting transformed ordinary differential equations (ODEs) align with the initial and boundary conditions are:  
\[f^{\prime\prime\prime}+ff^{\prime\prime}-\left(f^{\prime}\right)^{2}-Mf^{ \prime}+\lambda\theta+N\phi=0, \tag{15}\]  
\[\frac{1}{\Pr}\theta^{\prime\prime}+f\theta^{\prime}-f^{\prime}\left(\theta+ \delta_{1}\right)+Nb\theta^{\prime}\phi^{\prime}+Nt\theta^{\prime 2}-\gamma_{T} \theta_{T}=0, \tag{16}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=v\frac{\partial^ {2}u}{\partial y^{2}}-\frac{\sigma_{M}{B_{0}}^{2}}{\rho_{f}}u+\frac{g}{\rho_{ f}}\left[\left(1-C_{\infty}\right)\rho_{f_{\infty}}\beta_{T}\left(T-T_{\infty} \right)+\left(\rho_{p}-\rho_{f_{\infty}}\right)\beta_{C}\left(C-C_{\infty} \right)\right], \tag{8}\]0  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=v\frac{\partial^ {2}u}{\partial y^{2}}-\frac{\sigma_{M}{B_{0}}^{2}}{\rho_{f}}u+\frac{g}{\rho_{ f}}\left[\left(1-C_{\infty}\right)\rho_{f_{\infty}}\beta_{T}\left(T-T_{\infty} \right)+\left(\rho_{p}-\rho_{f_{\infty}}\right)\beta_{C}\left(C-C_{\infty} \right)\right], \tag{8}\]1\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=v\frac{\partial^ {2}u}{\partial y^{2}}-\frac{\sigma_{M}{B_{0}}^{2}}{\rho_{f}}u+\frac{g}{\rho_{ f}}\left[\left(1-C_{\infty}\right)\rho_{f_{\infty}}\beta_{T}\left(T-T_{\infty} \right)+\left(\rho_{p}-\rho_{f_{\infty}}\right)\beta_{C}\left(C-C_{\infty} \right)\right], \tag{8}\]2  
where  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=v\frac{\partial^ {2}u}{\partial y^{2}}-\frac{\sigma_{M}{B_{0}}^{2}}{\rho_{f}}u+\frac{g}{\rho_{ f}}\left[\left(1-C_{\infty}\right)\rho_{f_{\infty}}\beta_{T}\left(T-T_{\infty} \right)+\left(\rho_{p}-\rho_{f_{\infty}}\right)\beta_{C}\left(C-C_{\infty} \right)\right], \tag{8}\]3  
and prime implies the derivative with respect to the similarity variable \(\eta\). Furthermore, \(f\) is the dimensionless stream function along the \(x-\)direction, \(\theta\) is the dimensionless temperature, \(\phi\) is the dimensionless nanoparticle volume fraction, \(M\) is the magnetic parameter, \(\lambda\) is the thermal buoyancy parameter, \(N\) is the solutal buoyancy parameter, Pr is the Prandtl number, \(Le\) is the Lewis number, \(Nb\) is the Brownian motion parameter, \(Nt\) is the thermophoresis parameter, \(\delta_{1}\) is the thermal stratification parameter, \(\delta_{2}\) is the solutal stratification parameter, \(\gamma_{T}\) is the thermal relaxation parameter and \(S\) is the suction parameter, which are defined as  
\[ \begin{split}& M=\frac{\sigma_{M}{B_{0}}^{2}}{\rho a},\quad \lambda=\frac{Gr}{{\rm Re}_{x}^{2}},\quad N=\frac{Gc}{{\rm Re}_{x}^{2}},\quad Gr =\frac{gx^{3}\rho_{f_{\infty}}\beta_{T}\left(T_{w}-T_{0}\right)\left(1-C_{ \infty}\right)}{\rho_{f}\
u^{2}},\\ & Gc=\frac{gx^{3}\left(\rho_{p}-\rho_{f_{\infty}}\right)\beta_{C }\left(C_{w}-C_{0}\right)}{\rho_{f}\
u^{2}},\quad{\rm Re}_{x}=\frac{xU_{w}}{ \
u},\quad Pr=\frac{\
u}{\alpha},\quad Le=\frac{\alpha}{D_{B}},\\ Nb&=\frac{\tau_{1}D_{B}(C_{w}-C_{0})}{\
u},\quad Nt =\frac{\tau_{1}D_{T}(T_{w}-T_{0})}{\
u T_{\infty}},\quad\delta_{1}=\frac{B}{A },\quad\delta_{2}=\frac{F}{E},\quad\gamma_{T}=a\lambda_{2}.\end{split}  \tag{20}\]  
It is worthwhile to mention here that \(\lambda>0\) and \(\lambda<0\) corresponds to the assisting and opposing flow, respectively. The system of ODEs in Equations (15)-(17) is reduced to the model by [PERSON] et al. [24] for the problem of unstratified nanofluid. The physical quantities of interest in the present study are the dimensionless skin friction coefficient, local Nusselt number (heat transfer rate) and local Sherwood number (mass transfer rate) which are in the form of:  
\[{\rm Re}_{x}^{\frac{1}{2}}C_{f_{x}}=f^{\prime\prime}\left(0\right),\qquad{\rm Re }_{x}^{-\frac{1}{2}}Nu_{x}=-\theta^{\prime}\left(0\right),\quad{\rm Re}_{x}^{ -\frac{1}{2}}Sh_{x}=-\phi^{\prime}\left(0\right), \tag{21}\]  
respectively.