## 5 HAM Solution  
To solve Equations (12) and (13) with (14) by using HAM, it is assumed that \(f=V\) and \(\theta=H\). Initial guesses and linear operators are  
\[V_{0}(\eta)=\frac{1}{4}\Big{(}4\eta-6\eta^{2}+3S\eta^{2}+2\eta^{3}-S\eta^{3} \Big{)},H_{0}(\eta)=1+2\eta-\eta^{2}, \tag{25}\]  
\[L_{V}=V^{\prime\prime\prime},L_{H}=H^{\prime\prime}, \tag{26}\]  
\[L_{V}\Big{[}\Omega_{1}+\Omega_{2}\eta+\Omega_{3}\eta^{2}+\Omega_{4}\eta^{3} \Big{]}=0,L_{H}\Big{[}\Omega_{5}+\Omega_{6}\eta+\Omega_{7}\eta^{2}\Big{]}=0, \tag{27}\]  
where \(\Omega_{i}(i=1,2,3,\ldots,7)\) are arbitrary constants. 0\({}^{th}-\) order problems  
Assume that \(W\in[0,1]\) be the rooted factor and \(\hbar_{V}\) and \(\hbar_{H}\) are the non-zero supporting factors, then  
\[(1-W)L_{V}[V(\eta;W)-V_{0}(\eta)]=W\hbar_{V}\mathbf{N}_{V}[V(\eta;W)], \tag{28}\]  
\[(1-W)L_{H}[H(\eta;W)-H_{0}(\eta)]=W\hbar_{H}\mathbf{N}_{H}[H(\eta;W)], \tag{29}\]  
\[V(0;W)=0,V^{\prime}(0;W)=1,H(0;W)=1,V(1;W)=\frac{S}{2},V^{\prime\prime}(1;W)=0,H^{\prime}(1;W)=0. \tag{30}\]  
\[\mathbf{N}_{V}[V(\eta;W)]=\frac{\partial^{3}V(\eta W)}{\partial\eta^{3}}+(1- \phi)^{2.5}\Big{(}1-\phi+\phi\frac{P_{CNT}}{\rho_{f}}\Big{)}\lambda\Big{[}V( \eta;p)\frac{\partial^{2}V(\eta;W)}{\partial\eta^{2}}-\Big{(}\frac{\partial V (\eta;W)}{\partial\eta}\Big{)}^{2}\]  
\[-S\Big{(}\frac{\partial V(\eta;W)}{\partial\eta}+\frac{1}{2}\eta\frac{ \partial^{2}V(\eta;W)}{\partial\eta^{2}}\Big{)}\Big{]}+\alpha(1-\phi)^{2.5} \Big{[}2V(\eta;W)\frac{\partial^{3}V(\eta W)}{\partial\eta^{3}}+S\Big{(}2 \frac{\partial^{3}V(\eta;W)}{\partial\eta^{3}}+ \tag{31}\]  
\[\mathbf{N}_{H}[H(\eta;W);V(\eta;W)]=\frac{\left[\frac{k_{nf}}{k_{f}}+\frac{4}{ 3}Rd\right]}{\Pr\left[1-\phi+\phi\frac{(\rho_{G})_{CNT}}{(\rho_{G})_{f}} \right]}\frac{\partial^{2}H(\eta;W)}{\partial\eta^{2}}+\frac{1}{\Pr\left[1- \phi+\phi\frac{(\rho_{G})_{CNT}}{(\rho_{G})_{f}}\right]}\]  
\[ \begin{array}{l}\left(\begin{array}{c}A^{*}\frac{\partial V(\eta;W)}{ \partial\eta}\\ +B^{*}H(\eta;W)\end{array}\right)-\lambda\Big{[}2\frac{\partial V(\eta;W)}{ \partial\eta}H(\eta;W)-V(\eta;W)\frac{\partial H(\eta;W)}{\partial\eta}+\frac {5}{2}\Big{(}3H(\eta;W)+\eta\frac{\partial H(\eta;W)}{\partial\eta}\Big{)} \Big{]}\\ +\gamma\Big{[}-\frac{15}{2}S^{2}H(\eta;W)-\frac{7}{2}S^{2}\eta\frac{\partial H( \eta;W)}{\partial\eta}-8 SH(\eta;W)\frac{\partial V(\eta;W)}{\partial\eta}- \eta SH(\eta;W)\frac{\partial^{2}V(\eta;W)}{\partial\eta^{2}}\\ -\frac{3}{2}\eta S\frac{\partial V(\eta;W)}{\partial\eta}\frac{\partial H(\eta;W) }{\partial\eta}-\frac{1}{4}S^{2}\eta^{2}\frac{\partial^{2}H(\eta;W)}{\partial \eta^{2}}+\frac{9}{2}SV(\eta;W)\frac{\partial H(\eta;W)}{\partial\eta}+\\ \eta SV(\eta;W)\frac{\partial^{2}H(\eta;W)}{\partial\eta^{2}}-4H(\eta;W)\Big{(} \frac{\partial V(\eta;W)}{\partial\eta}\Big{)}^{2}-\Big{(}\frac{\partial V(\eta ;W)}{\partial\eta}\Big{)}^{2}\frac{\partial^{2}H(\eta;W)}{\partial\eta^{2}}\\ +3V(\eta;W)\frac{\partial V(\eta;W)}{\partial\eta}\frac{\partial H(\eta;W)}{ \partial\eta}+2V(\eta;W)H(\eta;W)\frac{\partial^{2}V(\eta;W)}{\partial\eta^{2}} \Big{]}+\\ \alpha E\bigg{[}\frac{5}{2}\bigg{(}\Big{(}\partial^{2}V(\eta;W)\Big{)}^{2}+ \eta\frac{\partial^{3}V(\eta;W)}{\partial\eta^{2}}\frac{\partial^{3}V(\eta;W)}{ \partial\eta^{3}}\bigg{)}\\ +\frac{\partial V(\eta;W)}{\partial\eta}\bigg{(}\frac{\partial^{2}V(\eta;W)}{ \partial\eta^{2}}\bigg{)}^{2}-V(\eta;W)\frac{\partial^{2}V(\eta;W)}{\partial \eta^{2}}\frac{\partial^{3}V(\eta;W)}{\partial\eta^{3}}\Bigg{]},\end{array}  \tag{32}\]\(k^{th}-\) order problems  
\(k^{th}-\) order problems are  
\[L_{V}[V_{k}(\eta)-\chi_{k}V_{k-1}(\eta)] =h_{k}\Re_{k}^{V}(\eta), \tag{33}\] \[L_{H}[H_{k}(\eta)-\chi_{k}H_{k-1}(\eta)] =h_{H}\Re_{k}^{H}(\eta),\] (34) \[V_{k}(0)=V_{k}^{\prime}(0)=V_{k}(1)=V_{k}^{\prime\prime}(1)=H_{k}(0)= H_{k}^{\prime}(1)=0. \tag{35}\]  
\[\Re_{k}^{V}(\eta)=V_{k-1}^{\prime\prime\prime}+(1-\phi)^{2.5}\Big{(}1-\phi+ \phi\frac{p_{CNT}}{\rho_{f}}\Big{)}\lambda\Bigg{[}