## 3 Solution by Homotopy Analysis Method  
For the solution of obtained nonlinear model Equations (16)-(19) with modeled boundary Conditions (20), we have used the Homotopy Analysis Method (HAM). The preliminary suppositions are chosen as follows:  
\[f_{0}(\eta)~{}=~{}1-e^{-\eta},~{}\mathrm{g}_{0}(\eta)~{}=~{}c(1-e^{-\eta}), \theta_{0}(\eta)~{}=~{}\bigg{(}\frac{Bi_{1}}{1+Bi_{1}}\bigg{)}e^{-\eta},\phi_{ 0}(\eta)~{}=~{}\bigg{(}\frac{Bi_{2}}{1+Bi_{2}}\bigg{)}e^{-\eta}. \tag{23}\]  
The \(\mathrm{L}_{f}\), \(\mathrm{L}_{g}\), \(\mathrm{L}_{\theta}\) and \(\mathrm{L}_{\phi}\) are linear operators, which are taken as  
\[L_{f}(f)~{}=~{}f^{\prime\prime}-f^{\prime},~{}\mathrm{L}_{g}(\mathrm{g})~{}= ~{}\mathrm{g}^{\prime\prime}-g^{\prime},\mathrm{L}_{\theta}(\theta)~{}=~{} \theta^{\prime\prime}-\theta,\mathrm{L}_{\phi}(\phi)~{}=~{}\phi^{\prime\prime }-\phi, \tag{24}\]  
which have the succeeding properties:  
\[ \begin{array}{ll}L_{f}(\lambda_{1}+\lambda_{2}e^{-\eta}+\lambda_{3}e^{\eta })~{}=~{}0,~{}\mathrm{L}_{g}(\lambda_{4}+\lambda_{5}e^{-\eta}+\lambda_{6}e^{ \eta})~{}=~{}0,\\ \mathrm{L}_{\theta}(\lambda_{7}e^{-\eta}+\lambda_{8}e^{\eta})~{}=~{}0,~{} \mathrm{L}_{\phi}(\lambda_{9}e^{-\eta}+\lambda_{10}e^{\eta})~{}=~{}0.\end{array}  \tag{25}\]  
where \(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\lambda_{6},\lambda _{7},\lambda_{8},\lambda_{9},\lambda_{10}\) are constant coefficients. The non-linear operators \(N_{f},N_{g},N_{\theta}\) and \(N_{\phi}\) are indicated as  
\[ \begin{array}{ll}N_{f}[f(\eta;\Re),g(\eta;\Re)]~{}=~{}\frac{\partial^{3}f( \eta;\Re)}{\partial\eta^{3}}-\Big{(}\frac{\partial f(\eta;\Re)}{\partial\eta} \Big{)}^{2}+(f(\eta;\Re)+g(\eta;\Re))\frac{\partial^{2}f(\eta;\Re)}{\partial \eta^{2}}\\ +(We)\frac{\partial^{2}f(\eta;\Re)}{\partial\eta^{2}}\frac{\partial^{3}f(\eta; \Re)}{\partial\eta^{3}}+g(\eta;\Re)\frac{\partial^{2}f(\eta;\Re)}{\partial \eta^{2}}-(\gamma)\frac{\partial f(\eta;\Re)}{\partial\eta},\end{array}  \tag{26}\]  
\[ \begin{array}{ll}N_{g}[f(\eta;\Re),g(\eta;\Re)]~{}=~{}\frac{\partial^{3}g( \eta;\Re)}{\partial\eta^{3}}-\Big{(}\frac{\partial g(\eta;\Re)}{\partial\eta} \Big{)}^{2}\\ +(f(\eta;\Re)+g(\eta;\Re))\frac{\partial^{2}g(\eta;\Re)}{\partial\eta^{2}}+( We)\frac{\partial^{2}g(\eta;\Re)}{\partial\eta^{2}}\frac{\partial^{3}g(\eta;\Re)}{ \partial\eta^{3}}\\ +f(\eta;\Re)\frac{\partial^{2}g(\eta;\Re)}{\partial\eta^{2}}-(\gamma)\frac{ \partial g(\eta;\Re)}{\partial\eta},\end{array}  \tag{27}\]  
\[ \begin{array}{ll}N_{\theta}[\theta(\eta;\Re),\phi(\eta;\Re)]~{}=~{}\frac{1}{ \mathrm{F}}\Big{(}1+\frac{4}{3}Rd\Big{)}\frac{\partial^{2}\theta(\eta;\Re)}{ \partial\eta^{2}}\\ +(f+cg)\frac{\partial\theta(\eta;\Re)}{\partial\eta}+(Nb)\frac{\partial\phi( \eta;\Re)}{\partial\eta}\frac{\partial(\eta;\Re)}{\partial\eta}+(Nt)\Big{(} \frac{\partial\theta(\eta;\Re)}{\partial\eta}\Big{)}^{2},\end{array}  \tag{28}\]  
\[ \begin{array}{ll}N_{\phi}[\theta(\eta;\Re),\phi(\eta;\Re)]~{}=~{}\frac{ \partial^{2}\phi(\eta;\Re)}{\partial\eta^{2}}+Le(f+cg)\bigg{(}\frac{\partial \phi\phi(\eta;\Re)}{\partial\eta}\bigg{)}+\frac{Nt}{Nb}\bigg{(}\frac{\partial^{2} \phi(\eta;\Re)}{\partial\eta^{2}}\bigg{)}.\end{array}  \tag{29}\]  
the zeroth-order problems are defined as  
\[(1-\Re)L_{f}[f(\eta;\Re)-f_{0}(\eta)]~{}=~{}\Re\dot{h}_{f}N_{f}[f(\eta;\Re),g( \eta;\Re)], \tag{30}\]  
\[(1-\Re)L_{g}[g(\eta;\Re)-g_{0}(\eta)]~{}=~{}\Re\dot{h}_{g}N_{g}[f(\eta;\Re),g( \eta;\Re)], \tag{31}\]  
\[(1-\Re)L_{g}[\theta(\eta;\Re)-\theta_{0}(\eta)]~{}=~{}\Re\dot{h}_{\theta}N_{g}[f (\eta;\Re),\theta(\eta;\Re)], \tag{32}\]\[(1-\Re)L_{\phi}[\phi(\eta;\Re)-\phi_{0}(\eta)] = \Re\hslash_{\phi}N_{\phi}[\theta(\eta;\Re),\phi(\eta;\Re)]. \tag{33}\]  
The equivalent boundary conditions are  
\[