## 3 Stability Analysis  
In this respect, we consider the unsteady state of our governing model in order to perform an analysis of stability on the present problem. While Equation (1) remains the same, Equations (2) and (3) replaced by as follows:  
\[\frac{\partial u}{\partial t}+\frac{\mu_{nf}}{\mu_{f}}\frac{\partial u}{\partial r }=\frac{gK\Big{[}\varphi\rho_{s}\beta_{s}+(1-\varphi)\rho_{f}\beta_{f}\Big{]} }{\mu_{f}}\frac{\partial T}{\partial r}, \tag{18}\]  
\[\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T }{\partial r}=\frac{\alpha_{nf}}{r}\frac{\partial}{\partial r}\left(r\frac{ \partial T}{\partial r}\right)-\frac{1}{\left(\rho C_{p}\right)_{nf}}\left( \frac{\partial q_{r}}{\partial r}\right), \tag{19}\]  
where \(t\) is time. The new boundary conditions are now replaced by:  
\[ \begin{array}{rl}t<0:&u=v=0,T=T_{\infty}\text{ for any }x,r,\\ t\geq 0:&v=v_{\eta},T=T_{\infty}+T_{0}\frac{x}{a}\text{ at }r=0,\\ &u\to U_{0}\frac{x}{a},T\to T_{\infty}\text{ as }r\to\infty.\end{array}  \tag{20}\]  
A new dimensionless variable \(\tau\) is now introduced, where \(\tau\) is uniform with the problem of which solutions will be associates with an initial value problem and physically realizable. With the introduction of new dimensionless variable \(\tau\) and Equation (7), we now have:  
\[\psi=\sqrt{2U_{0}a\alpha_{f}}xf(\eta),T=T_{\infty}+\frac{T_{0}x}{a}\theta( \eta),\eta=\frac{r^{2}-a^{2}}{2a\alpha_{f}}\sqrt{\frac{U_{0}\alpha_{f}}{2a}}, \tau=at. \tag{21}\]  
We substitute Equation (21) into Equations (18) and (19), and we get:  
\[\frac{\partial^{2}f}{\partial\eta^{2}}+B\lambda\frac{\partial\theta}{\partial \eta}+\frac{\partial^{2}f}{\partial\eta\sigma}=0, \tag{22}\]  
\[\left[A(1+2\xi\eta)+\frac{8}{3}\xi Rd\right]\frac{\partial^{2}\theta}{\partial \eta^{2}}+\left(2A\xi+\frac{8}{3}\xi Rd\right)\frac{\partial\theta}{\partial \eta}+2f\frac{\partial\theta}{\partial\eta}-2\frac{\partial f}{\partial\eta} \theta-2\frac{\partial\theta}{\partial\tau}=0. \tag{23}\]  
It is worth mentioning that Equations (22) and (23) can be combined together because of the characteristic of porous medium to give a single equation as follows:  
\[ \begin{array}{rl}\left[A(1+2\xi\eta)+\frac{8}{3}\xi Rd\right]\left(\frac{ \partial^{3}f}{\partial\eta^{3}}+\frac{\partial^{3}f}{\partial\eta^{2}\sigma _{f}}\right)+\left(2A\xi+\frac{8}{3}\xi Rd\right)\frac{\partial^{2}f}{\partial \eta^{2}}+\left(2A\xi+\frac{8}{3}\xi Rd-2+2f\right)\frac{\partial^{2}f}{ \partial\eta\partial\tau}-\\ 2\left(\frac{\partial f}{\partial\eta}\right)^{2}+2\frac{\partial f}{\partial \eta}+2f\frac{\partial^{2}f}{\partial\eta^{2}}-2\frac{\partial\theta}{B \lambda\partial\tau}=0.\end{array}  \tag{24}\]  
The boundary conditions in Equation (20) now reduced to:  
\[ \begin{array}{rl}f(0)=S,\frac{\partial f}{\partial\eta}(0)=B\lambda+1,\\ \frac{\partial f}{\partial\eta}(\eta)\to 1\text{ as }\eta\to\infty.\end{array}  \tag{25}\]  
We test the solution stability of \(f(\eta)=f_{0}(\eta)\) and \(\theta(\eta)=\theta_{0}(\eta)\) to satisfy the boundary value problems as in Equation (14) by embraced the analysis proposed by [PERSON] [14] and [PERSON] et al. [15]:  
\[f(\eta,\tau)=f_{0}(\eta)+e^{-\gamma\tau}F(\eta,\tau),\ \theta(\eta,\tau)=\theta_{0}( \eta)+e^{-\gamma\tau}G(\eta,\tau), \tag{26}\]  
where \(\gamma\) is the unknown eigenvalue or also can be described as the rate of growth or decay of disturbances. The eigenvalue solutions give a set of infinite eigenvalues \(\gamma_{1}<\gamma_{2}<\gamma_{3}<\ldots\), and later, the initial decay of disturbances will appear if the smallest eigenvalue is a positive number, and by thenwe can deduce that the solution is stable and significantly realizable. If the smallest eigenvalue shows a negative number, the flow is said to be unstable by the meaning if there is an initial disturbances growth. We substitute Equation (26) into Equation (24) to give a final equation:  
\[