## 3 Stability Analysis
\begin{array}{l}\frac{1}{(1-\phi)^{2S}}\frac{\partial^{3}H}{\partial\eta^{3}}+ \left(1-\phi+\phi\frac{\rho_{5}}{\rho_{f}}\right)\left[\frac{n+1}{2}\left(f_{0} \frac{\partial^{2}H}{\partial\eta^{2}}+h_{0}\frac{\partial^{2}H}{\partial\eta^ {2}}+h_{0}^{\prime\prime}\ F+h_{0}^{\prime\prime}\ H\right)\right.\\ \left.-n\left(f_{0}^{\prime}\frac{\partial H}{\partial\eta}+2h_{0}^{\prime} \frac{\partial H}{\partial\eta}+h_{0}^{\prime}\frac{\partial F}{\partial\eta} \right)+\left(n-1\right)\tau\left(\gamma f_{0}^{\prime}\ \frac{\partial H}{\partial\eta}+\gamma H _{0}^{\prime}\frac{\partial H}{\partial\eta}-f_{0}^{\prime}\ \frac{\partial^{2}H}{\partial\eta\sigma}-h_{0}^{\prime}\frac{\partial^{2}H}{ \partial\eta\sigma}\right)\right.\\ \left.-(n-1)\tau\left(\gamma h_{0}^{\prime\prime}\ F+\gamma h_{0}^{\prime\prime }H-h_{0}^{\prime\prime}\ \frac{\partial F}{\partial\tau}-h_{0}^{\prime\prime}\ \frac{\partial H}{\partial\tau}\right)+\gamma\frac{ \partial H}{\partial\eta}-\frac{\partial^{2}H}{\partial\eta\sigma}\right]\\ \quad-\left[1+\frac{3\left(\sigma_{s}/\sigma_{f}-1\right)\phi}{\sigma_{s}/ \sigma_{f}+2-\left(\sigma_{s}/\sigma_{f}-1\right)\phi}\right]M\frac{\partial H }{\partial\eta}=0,\end{array}  \tag{28}\]  
\[ \begin{array}{l}\frac{1}{\mathrm{Pr}}\frac{k_{nf}}{k_{f}}\frac{\partial^{2}G }{\partial\eta^{2}}+\left(1-\phi+\phi\frac{\left(\rho C_{p}\right)_{nf}}{ \left(\rho C_{p}\right)_{f}}\right)\left[\frac{n+1}{2}\left(f_{0}\frac{ \partial G}{\partial\eta}+h_{0}\frac{\partial G}{\partial\eta}+\theta_{0}^{ \prime}F+\theta_{0}^{\prime}H\right)\right.\\ \left.-(2n-1)\left(f_{0}^{\prime}G+h_{0}^{\prime}G+\theta_{0}\frac{\partial H }{\partial\eta}+\theta_{0}\frac{\partial H}{\partial\eta}\right)+(n-1)\tau \left(\gamma f_{0}^{\prime}\ G+\gamma h_{0}^{\prime}G-f_{0}^{\prime}\frac{ \partial G}{\partial\tau}-h_{0}^{\prime}\frac{\partial G}{\partial\tau}\right) \right.\\ \left.-(n-1)\tau\left(\gamma\theta_{0}^{\prime}F+\gamma\theta_{0}^{\prime}H- \theta_{0}^{\prime}\frac{\partial F}{\partial\tau}-\theta_{0}^{\prime}\frac {\partial H}{\partial\tau}\right)+\gamma G-\frac{\partial G}{\partial\tau} \right]=0,\end{array}  \tag{29}\]  
subject to the boundary conditions:  
\[ \begin{array}{l}F(0,\tau)=0,\quad H(0,\tau)=0,\quad G(0,\tau)=K\frac{ \partial G}{\partial\eta}(0,\tau),\\ \frac{\partial F}{\partial\eta}(0,\tau)=N\frac{\partial^{2}F}{\partial\eta^{2}} (0,\tau),\quad\frac{\partial H}{\partial\eta}(0,\tau)=N\frac{\partial^{2}H}{ \partial\eta^{2}}(0,\tau),\\ \frac{\partial F}{\partial\eta}(\infty,\tau)\to 0,\quad\frac{\partial H}{ \partial\eta}(\infty,\tau)\to 0,\quad G(\infty,\tau)\to 0.\end{array}  \tag{30}\]  
Furthermore, as depicted by [PERSON] et al. [46], in order to identify the initial growth or the decay of solution (26), \(\tau=0\), is applied and hence, the following are obtained: \(F=F_{0}(\eta)\), \(H=H_{0}(\eta)\), and \(G=G_{0}(\eta)\). Furthermore, in order to test the numerical procedure, the following linear eigenvalue problem has to be solved:  
\[ \begin{array}{l}\frac{1}{(1-\phi)^{2S}}F_{0}^{\prime\prime\prime}+\left(1- \phi+\phi\frac{\rho_{5}}{\rho_{f}}\right)\left[\frac{n+1}{2}\left(f_{0}F_{0}^ {\prime\prime}+h_{0}F_{0}^{\prime\prime}\ \ +f_{0}^{\prime\prime}\ F_{0}+f_{0}^{\prime\prime}\ H_{0}\right)\right.\\ \left.-n(2f_{0}^{\prime}\ F_{0}^{\prime}+f_{0}^{\prime}\ H_{0}^{\prime}+h_{0}^{ \prime}F_{0}^{\prime})\left.+\gamma F_{0}^{\prime}\right]+\left(1-\phi+\phi \frac{\left(\rho\beta\right)_{4}}{\left(\rho\beta\right)_{f}}\right)\lambda G _{0}\\ \left.-\left[1+\frac{3\left(\sigma_{s}/\sigma_{f}-1\right)\phi}{\sigma_{s}/ \sigma_{f}+2-\left(\sigma_{s}/\sigma_{f}-1\right)\phi}\right]MF_{0}^{\prime}=0,\end{array}  \tag{31}\]  
\[