## 3 Stability Analysis
\begin{array}{l}\frac{1}{\mathrm{Pr}}\frac{k_{nf}}{k_{f}}\frac{\partial^{2 }\theta}{\partial\eta^{2}}+\Big{(}1-\phi+\phi\frac{\rho(\rho C)_{nf}}{(\rho C _{\rho})_{f}}\Big{)}\Big{[}\frac{n+1}{2}(f+h)\frac{\partial\theta}{\partial \eta}-(2n-1)\Big{(}\frac{\partial f}{\partial\eta}+\frac{\partial u}{\partial \eta}\Big{)}\theta\\ -(n-1)\tau\Big{(}\frac{\partial f}{\partial\eta}+\frac{\partial u}{\partial \eta}\Big{)}\frac{\partial\theta}{\partial\tau}+(n-1)\tau\Big{(}\frac{\partial f }{\partial\tau}+\frac{\partial u}{\partial\tau}\Big{)}\frac{\partial\theta}{ \partial\eta}-\frac{\partial\theta}{\partial\tau}\Big{]}=0,\end{array}  \tag{24}\]  
subject to the boundary conditions:  
\[ \begin{array}{l}f(0,\tau)=s,\ \ \ h(0,\tau)=0,\ \ \ \theta(0,\tau)=1+K\frac{ \partial\theta}{\partial\eta}(0,\tau),\\ \frac{\partial f}{\partial\eta}(0,\tau)=c+N\frac{\partial^{2}f}{\partial\eta^{2}}( 0,\tau),\ \ \ \frac{\partial u}{\partial\eta}(0,\tau)=c+N\frac{\partial^{2}h}{ \partial\eta^{2}}(0,\tau),\\ \frac{\partial f}{\partial\eta}(\infty,\tau)\to 0,\ \ \ \frac{\partial h}{\partial\eta}( \infty,\tau)\to 0,\ \ \ \theta(\infty,\tau)\to 0.\end{array}  \tag{25}\]  
Furthermore, in order to test the stability of solutions \(f(\eta)=f_{0}(\eta)\), \(h(\eta)=h_{0}(\eta)\), and \(\theta(\eta)=\theta_{0}(\eta)\) in satisfying the steady-state problem (9)-(12), the following is written:  
\[ \begin{array}{l}f(\eta,\tau)=f_{0}(\eta)+e^{-\gamma\tau}F(\eta,\tau),\\ h(\eta,\tau)=h_{0}(\eta)+e^{-\gamma\tau}H(\eta,\tau),\\ \theta(\eta,\tau)=\theta_{0}(\eta)+e^{-\gamma\tau}G(\eta,\tau),\end{array}  \tag{26}\]  
where \(\gamma\) is an unknown eigenvalue parameter. Besides, note that \(F(\eta,\tau)\), \(H(\eta,\tau)\), and \(G(\eta,\tau)\) are smaller relative to \(f_{0}(\eta)\), \(h_{0}(\eta)\), and \(\theta_{0}(\eta)\). Furthermore, by substituting (26) into Equations (22)-(24), along with the boundary conditions (25), the following equations are obtained:  
\[ \begin{array}{l}\frac{1}{(1-\phi)^{25}}\frac{\partial^{3}F}{\partial\eta^{3 }}+\Big{(}1-\phi+\phi\frac{\rho_{s}}{\rho_{f}}\Big{)}\Big{[}\frac{n+1}{2} \Big{(}f_{0}\frac{\partial^{2}F}{\partial\eta^{2}}+h_{0}\frac{\partial^{2}F}{ \partial\eta^{2}}+f_{0}^{\prime\prime}\ F+f_{0}^{\prime\prime}H\Big{)}\\ -n\Big{(}f_{0}^{\prime}\frac{\partial H}{\partial\eta}+2f_{0}^{\prime}\ \frac{\partial F}{ \partial\eta}+h_{0}^{\prime}\frac{\partial F}{\partial\eta}\Big{)}+(n-1)\tau\Big{(} \gamma f_{0}^{\prime}\frac{\partial F}{\partial\eta}+\gamma h_{0}^{\prime}\frac{ \partial F}{\partial\eta}-f_{0}^{\prime}\ \frac{\partial^{2}F}{\partial\eta\sigma T}-h_{0}^{\prime}\frac{\partial^{2}F}{ \partial\eta\sigma T}\Big{)}\\ -(n-1)\tau\Big{(}\gamma f_{0}^{\prime\prime}F+\gamma f_{0}^{\prime\prime}H-f_{0 }^{\prime\prime}\ \frac{\partial F}{\partial\tau}-f_{0}^{\prime\prime}\ \frac{\partial H}{\partial\tau}+\gamma\frac{\partial F}{ \partial\eta}-\frac{\partial^{2}F}{\partial\eta\sigma T}\Big{]}\\ -\Big{[}1+\frac{3\left(\sigma_{s}/\sigma_{f}-1\right)\phi}{\sigma_{s}/\sigma_ {f}+2-\left(\sigma_{s}/\sigma_{f}-1\right)\phi}\Big{]}M\frac{\partial F}{ \partial\eta}+\Big{(}1-\phi+\phi\frac{\rho(\rho\beta_{s})}{(\rho\beta_{f})} \Big{)}\lambda\in\mathcal{C}=0,\end{array}  \tag{27}\]\[