## 2 Formulation of the Problem
\begin{array}{l}\widetilde{f}(\eta;\zeta)\Big{|}_{\eta=0}=\widetilde{f}_{w}, \left.\frac{\partial\widetilde{f}(\eta;\zeta)}{\partial\eta}\right|_{\eta=0}=1,\left.\frac{\partial\widetilde{f}(\eta;\zeta)}{\partial\eta}\right|_{\eta \rightarrow\infty}=0,\\ \widetilde{\theta}(\eta;\zeta)\Big{|}_{\eta=0}=1,\left.\widetilde{\theta}(\eta; \zeta)\right|_{\eta\rightarrow\infty}=0,\left.\widetilde{\phi}(\eta;\zeta) \right|_{\eta=0}=1,\left.\widetilde{\phi}(\eta;\zeta)\right|_{\eta\rightarrow \infty}=0\end{array}  \tag{32}\]  
where \(\zeta\in[0,1]\) is the imbedding parameter, and \(\dot{h}_{f},\ \dot{h}_{\theta}\) and \(\dot{h}_{\phi}\) are used to control the convergence of the solution. When \(\zeta=0\) and \(\zeta=1\), we have:  
\[\widetilde{f}(\eta;1)=\widetilde{f}(\eta),\ \widetilde{\theta}(\eta;1)=\widetilde{ \theta}(\eta)\ \text{and}\ \widetilde{\phi}(\eta;1)=\widetilde{\phi}(\eta), \tag{33}\]  
Expanding \(\widetilde{f}(\eta;\zeta),\ \widetilde{\theta}(\eta;\zeta)\) and \(\widetilde{\phi}(\eta;\zeta)\) in [PERSON]'s series about \(\zeta=0\),  
\[ \begin{array}{l}\widetilde{f}(\eta;\zeta)=\widetilde{f}_{0}(\eta)+\sum \limits_{m=1}^{\infty}\widetilde{f}_{m}(\eta)\zeta^{m},\\ \widetilde{\theta}(\eta;\zeta)=\widetilde{\theta}_{0}(\eta)+\sum\limits_{m=1}^{ \infty}\widetilde{\theta}_{m}(\eta)\zeta^{m},\\ \widetilde{\phi}(\eta;\zeta)=\widetilde{\phi}_{0}(\eta)+\sum\limits_{m=1}^{ \infty}\widetilde{\phi}_{m}(\eta)\zeta^{m}.\end{array}  \tag{34}\]where  
\[\widetilde{f}_{m}(\eta)=\left.\frac{1}{m!}\frac{\partial\widetilde{f}(\eta;\zeta) }{\partial\eta}\right|_{\zeta=0},\left.\widetilde{\phi}_{m}(\eta)=\left.\frac{ 1}{m!}\frac{\partial\widetilde{\theta}(\eta;\zeta)}{\partial\eta}\right|_{ \zeta=0}\text{ and }\widetilde{\phi}_{m}(\eta)=\left.\frac{1}{m!}\frac{ \partial\widetilde{\phi}(\eta;\zeta)}{\partial\eta}\right|_{\zeta=0}. \tag{35}\]  
The secondary constraints \(\left.\hat{h}_{\theta},\left.\hat{h}_{\theta}\right.\right.\) and \(\left.\hat{h}_{f}\right.\) are taken in such form that the sequences in Equation (34) converge at \(\zeta=1\). Switching \(\zeta=1\) in Equation (35), we obtain:  
\[ \begin{split}&\widetilde{f}(\eta)=\widetilde{f}_{0}(\eta)+\sum \limits_{m=1}^{\infty}\widetilde{f}_{m}(\eta),\\ &\widetilde{\theta}(\eta)=\widetilde{\theta}_{0}(\eta)+\sum \limits_{m=1}^{\infty}\widetilde{\theta}_{m}(\eta),\\ &\widetilde{\phi}(\eta)=\widetilde{\phi}_{0}(\eta)+\sum \limits_{m=1}^{\infty}\widetilde{\phi}_{m}(\eta).\end{split}  \tag{36}\]  
The \(m^{\text{th}}\) -order problem, fulfills the following:  
\[ \begin{split}& L_{f}\Big{[}\widetilde{f}_{m}(\eta)-\chi_{m} \widetilde{f}_{m-1}(\eta)\Big{]}=\left.\hat{h}_{f}R_{m}^{f}(\eta),\\ & L_{\theta}\Big{[}\widetilde{\theta}_{m}(\eta)-\chi_{m} \widetilde{\theta}_{m-1}(\eta)\Big{]}=\left.\hat{h}_{\theta}R_{m}^{\theta}( \eta),\\ & L_{\varphi}\Big{[}\widetilde{\phi}_{m}(\eta)-\chi_{m} \widetilde{\phi}_{m-1}(\eta)\Big{]}=\left.\hat{h}_{\varphi}R_{m}^{\theta}( \eta).\right.\end{split}  \tag{37}\]  
The consistent boundary conditions are:  
\[ \begin{split}& f_{m}(0)=f_{m}^{\prime}(0)=\theta_{m}(0)=\phi_{m}(0 )=0\\ & f_{m}^{\prime}(\infty)=\theta_{m}(\infty)=\phi_{m}(\infty)=0 \end{split}  \tag{38}\]  
where  
\[ \begin{split}& R_{m}^{f}(\eta)=f_{m-1}^{\prime\prime\prime}+\sum \limits_{k=0}^{m-1}f_{m-1-k}f_{k}^{\prime\prime}-\sum\limits_{k=0}^{m-1}f_{m- 1-k}^{\prime}f_{k}^{\prime}\\ &-\lambda\big{[}f_{m-1}^{\prime}+\tfrac{\eta}{2}f_{m-1}^{\prime \prime}\big{]}-\beta\Big{[}\sum\limits_{k=0}^{m-1}f_{m-1-k}^{\prime}\sum \limits_{l=0}^{k}f_{k-l}^{\prime}f_{l}^{\prime\prime}-2\sum\limits_{k=0}^{m-1 }f_{m-1-k}\sum\limits_{l=0}^{k}f_{k-l}^{\prime}f_{l}^{\prime\prime\prime} \Big{]}\\ &-Mf_{m-1}^{\prime}-MEI,\end{split}  \tag{39}\]  
\[ \begin{split}& R_{m}^{\theta}(\eta)=\left(1+\tfrac{4}{3}Rd\right) \theta_{m-1}^{\prime\prime}+\\ &\Pr\Bigl{[}\sum\limits_{k=0}^{m-1}f_{m-1-k}\theta_{k}^{\prime}- \sum\limits_{k=0}^{m-1}\theta_{m-1-k}f_{k}^{\prime}-\lambda\big{(}\theta_{m-1} +\tfrac{\eta}{2}\theta_{m-1}^{\prime}\big{)}\Bigr{]}\\ &+Af_{m-1}^{\prime}+B\theta_{m-1},\end{split}  \tag{40}\]  
\[