# Magnetohydrodynamic Nanoliquid Thin Film Sprayed on a Stretching Cylinder with Heat Transfer
## 6 Conclusions  
This article explores the analytical solution of magnetohydrodynamic thin film CuO-H\({}_{2}\)O nanofluid sprayed on a stretching cylinder accompanying transfer of heat. The solution of the problem has been achieved by using analytical technique HAM (Homotopy Analysis Method) for the velocity profile and temperature distribution. Pressure distribution and rate of spray are also investigated. From the given figures, it is clear that the types of nanofluids and various parameters have an effective contribution in the flow, transfer of heat, pressure distribution and rate of spray. The solution has been displayed in the diagrams and the influences of all the parameters included in the problem on the CuO-H\({}_{2}\)O nanofluid have been described graphically for checking their effects on velocity profile, temperature distribution and pressure distribution accompanying spray distribution. It is predicted that the model of the problem can be used in technical procedures, cooling/heating techniques and in the modeling of coating processes by employing the constitutive equations of the problem. The findings of the research are the following:  
1. The velocity depreciates for the thin film nanofluid parameter \(\beta_{1}\), Reynolds number \(Re\) and magnetic field parameter \(M\). 2. Temperature diminishes for the thin film parameter \(\beta_{1}\), Prandtl number \(Pr\) and it enhances for the Reynolds number \(Re\) and it elevates for the magnetic field parameter \(M\). 3. Volume fraction parameter \(\phi\) has no sensitive effect on velocity and temperature. 4. Pressure depreciates for the Reynolds number \(Re\) and magnetic field parameter \(M\) while it elevates for the thin film parameter \(\beta_{1}\), volume fraction parameter \(\phi\). 5. Film size \(\beta_{1}\) enhances with the spray rate, but nonlinearly. 6. Comparisons of HAM solution with the numerical method solution for velocity and temperature are performed and the results are found to be in good agreement. 7. The residual errors show the authentication of the present work. 8. Skin friction coefficient (\(C_{f}\)) as a function of Reynolds number \(Re\) decreases with the increase of nanoparticle volume fraction \(\phi\). 9. Nusselt number \(Nu\) as a function of Reynolds number \(Re\) increases with the increase of nanoparticle volume fraction \(\phi\). 10. The present study shows an excellent agreement with the published experimental work. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **[PERSON] [45]** & **[PERSON] [45]** & **Present Study (When \(h=-0.40\), \(\beta_{1}=0.10\), \(M=0.10\), \(Pr=6.80\))** \\ \hline \(Re\) & CuO & Al\({}_{2}\)O\({}_{3}\) & CuO \\
0.1 & 1.846686 & 1.70097 & 0.227864 \\
1 & 4.324278 & 4.113995 & 2.25099 \\
2 & 5.996721 & 5.711652 & 4.40516 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the effects of different kinds of nanoparticles on Nusselt number (\(Nu\)) as a function of Reynolds number \(Re\) when \(\phi=0.04\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Sheikhoeslami [45]** & **Sheikhoeslami [45]** & **Present Study (When \(h=-0.20\), \(\beta_{1}=0.50\), \(M=0.10\), \(Pr=6.80\))** \\ \hline \(Re\) & CuO & Al\({}_{2}\)O\({}_{3}\) & CuO \\
0.1 & 0.679741 & 0.703897 & 0.577056 \\
1 & 1.194617 & 1.224377 & 0.215528 \\
2 & 1.579317 & 1.615502 & 0.384453 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the effects of different kinds of nanoparticles on skin friction coefficient (\(C_{f}\)) as a function of Reynolds number \(Re\) when \(\phi=0.04\). **Acknowledgments:** The authors greatly acknowledge with thanks the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia for technical and financial support. The authors are cordially thankful to the honorable reviewers for their constructive comments to improve the quality of the paper. **Author Contributions:** [PERSON] and [PERSON] modeled and solved the problem. [PERSON] wrote the paper also. [PERSON] and [PERSON] made the corrections.