## 5 Conclusions  
The study of a stability analysis on mixed convection boundary layer flow filled with nanofluid and thermal radiation over a permeable vertical cylinder lodged in a porous medium has been numerically analyzed and discussed in detail in this paper. It was found that the involving parameters--specifically the mixed convection parameter \(\lambda\), radiation parameter \(Rd\), suction parameter \(S\), nanoparticle volume fraction \(\varphi\) and curvature parameter \(\xi\)--significantly affected the flow field. We can observe that for the three kinds of nanoparticles, Al\({}_{2}\)O\({}_{3}\) and TiO\({}_{2}\) are shown to separate the boundary layer thickness faster--which do not have much difference, and is followed by Cu. We then performed an analysis of stability on the respective model since there exists dual solutions obtained by a selected range of parameters. The main purpose of stability analysis is to analyze the most stable solution from the two solutions obtained from our numerical results. It has been identified that the first solution initiated decaying disturbances, while the second solution initiated growing disturbances. From this observation, we can make a final conclusion that the first solution is stable and physically realizable, while the second solution is not.  
**Acknowledgments:** The authors gratefully acknowledge the financial support received in the form of a FRGS research grant from the Ministry of Higher Education, Malaysia.  
**Author Contributions:** [PERSON], [PERSON], [PERSON] and [PERSON] conceived and designed the experiments; [PERSON] performed the experiments, analyzed the data and wrote the paper; [PERSON] and [PERSON] re-checked the experiments to make sure it is correct.  
**Conflicts of Interest:** The authors declare no conflicts of interest.  
**Nomenclatures**  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Copper**} & \multicolumn{2}{c}{**Aluminium**} & \multicolumn{2}{c}{**Titanium**} \\ \cline{2-7}
**Rd** & **First Solution** & **Second Solution** & **First Solution** & **Second Solution** & **First Solution** & **Second Solution** \\ \hline
0.2 & 0.78526 & \(-\)0.84201 & 0.38032 & \(-\)0.48132 & 0.26269 & \(-\)0.43730 \\
0.4 & 0.78535 & \(-\)0.84211 & 0.38035 & \(-\)0.48214 & 0.26339 & \(-\)0.44001 \\
0.6 & 0.78544 & \(-\)0.84221 & 0.38038 & \(-\)0.48297 & 0.26406 & \(-\)0.44227 \\
0.8 & 0.78553 & \(-\)0.84232 & 0.38041 & \(-\)0.48381 & 0.26471 & \(-\)0.44560 \\ \hline \hline \end{tabular}
\end{table}
Table 4: List of several values of \(Rd\) and the smallest eigenvalue \(\gamma\) when \(\varphi=S=0.1\), \(\gamma=-1.6\) and \(\xi=0.5\) for three types of nanoparticles.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Copper**} & \multicolumn{2}{c}{**Aluminium**} & \multicolumn{2}{c}{**Titanium**} \\ \cline{2-7} \(\lambda\) & **First Solution** & **Second Solution** & **First Solution** & **Second Solution** & **First Solution** & **Second Solution** \\ \hline \(-1.3\) & 0.59663 & \(-\)0.64225 & 0.27145 & \(-\)0.36161 & 0.22169 & \(-\)0.32238 \\ \(-1.4\) & 0.69092 & \(-\)0.74210 & 0.32588 & \(-\)0.42126 & 0.24201 & \(-\)0.37918 \\ \(-1.5\) & 0.78522 & \(-\)0.84195 & 0.38032 & \(-\)0.48091 & 0.26233 & \(-\)0.43597 \\ \(-1.6\) & 0.87951 & \(-\)0.94180 & 0.43474 & \(-\)0.54055 & 0.28265 & \(-\)0.49276 \\ \hline \hline \end{tabular}
\end{table}
Table 3: List of several values of \(\lambda\) and the smallest eigenvalue \(\gamma\) when \(\varphi=Rd=S=0.1\) and \(\xi=0.5\) for three types of nanoparticles.