# Magnetohydrodynamic Nanoliquid Thin Film Sprayed on a Stretching Cylinder with Heat Transfer
## 3 Solution of the Problem by HAM
### Zeroth-Order Deformation Problems  
Introducing the nonlinear operators \(\aleph_{f}\) and \(\aleph_{\theta}\) as  
\[\aleph_{f}[f(\zeta,p)]=\zeta\frac{\partial^{2}f(\zeta,p)}{\partial\zeta^{2}}+ \frac{\partial^{2}f(\zeta,p)}{\partial\zeta^{2}}-ReB_{1}(1-\varrho)^{2S}\left[ \left(\frac{\partial f(\zeta,p)}{\partial\zeta}\right)^{2}-f(\zeta,p)\frac{ \partial^{2}f(\zeta,p)}{\partial\zeta^{2}}\right]-M\frac{\partial^{2}f(\zeta, p)}{\partial\zeta}, \tag{43}\]  
\[\aleph_{\theta}[f(\zeta,p),\theta(\zeta,p)]=\zeta\frac{\partial^{2}\theta( \zeta,p)}{\partial\zeta^{2}}+\frac{\partial\theta(\zeta,p)}{\partial\zeta}- ReP\textbf{\emph{P}}\left(\frac{B_{2}}{B_{3}}\right)\left[2\frac{\partial f( \zeta,p)}{\partial\zeta}\theta(\zeta,p)-f(\zeta,p)\frac{\partial\theta(\zeta, p)}{\partial\zeta}\right], \tag{44}\]  
where \(p\) is an embedding parameter such that \(p\)\(\in\)[0, 1]. The equations of zeroth-order deformation are prepared as  
\[(1-p)\textbf{L}_{f}[f(\zeta,p)-f_{0}(\zeta)]=ph\aleph_{f}[f(\zeta,p)], \tag{45}\]  
\[(1-p)\textbf{L}_{\theta}[\theta(\zeta,p)-\theta_{0}(\zeta)]=ph\aleph_{\theta}[ f(\zeta,p),\theta(\zeta,p)], \tag{46}\]  
where \(h\) is denoting the auxiliary non-zero parameter. Equation (45) has the boundary conditions  
\[f(1,p)=1,\ \ \ f(1,p)=1,\ \ \ f^{\prime\prime}(\beta_{1},p)=0. \tag{47}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Thermophysical Quantities** & **Pure Water** & \(\textbf{Al}_{2}\textbf{O}_{3}\) & **CuO** \\ \hline \(\rho(\text{Kg}/\text{m}^{3})\) & 997.1 & 3970 & 6500 \\ \(\text{C}_{p}(\text{J}/\text{kgK})\) & 4179 & 765 & 540 \\ \(K(\text{W}/\text{mk})\) & 0.613 & 25 & 18 \\ \(d_{p}(\text{mm})\) & - & 47 & 29 \\ \(\sigma(\text{$\Omega$m})^{-1}\) & 0.05 & \(10^{-12}\) & \(10^{-10}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Thermophysical quantities of water and nanoparticles [47]. \begin{table}
\begin{tabular}{c c c} \hline
**Coefficients Values** & \(\textbf{Al}_{2}\textbf{O}_{3}\)**-\(\textbf{H}_{2}\textbf{O}\)** & **CuO-\(\textbf{H}_{2}\textbf{O}\)** \\ \hline \(\text{c}_{1}\) & 52.813488759 & \(-\)26.593310846 \\ \(\text{c}_{2}\) & 6.115637295 & \(-\)0.403818333 \\ \(\text{c}_{3}\) & 0.6955745084 & \(-\)33.3516805 \\ \(\text{c}_{4}\) & 4.17455552786 \(\times\)\(10^{-2}\) & 1.915825591 \\ \(\text{c}_{5}\) & 0.176919300241 & 6.42185846658 \(\times\)\(10^{-2}\) \\ \(\text{c}_{6}\) & \(-\)298.19819084 & 48.40336955 \\ \(\text{c}_{7}\) & \(-\)34.532716906 & \(-\)9.787756683 \\ \(\text{c}_{8}\) & \(-\)3.9225289283 & 190.245610009 \\ \(\text{c}_{9}\) & \(-\)0.2354329626 & 10.9285386565 \\ \(\text{c}_{10}\) & \(-\)0.999063481 & \(-\)0.72009983664 \\ \hline \end{tabular}
\end{table} 
Table 2: Coefficients values of the \(\text{Al}_{2}\textbf{O}_{3}\)-\(\textbf{H}_{2}\textbf{O}\) and \(\text{CuO-H}_{2}\textbf{O}\)[47]. The boundary conditions for Equation (46) are  
\[\theta(1,p)=1,\hskip 28.452756 pt\theta^{\prime}(\beta_{1},p)=0. \tag{48}\]  
\[p=0\Rightarrow f(\zeta,0)=f_{0}(\zeta)\hskip 14.226378 pt\textit{and}\hskip 14.226378 ptp=1 \Rightarrow f(\zeta,1)=f(\zeta), \tag{49}\]  
\[p=0\Rightarrow\theta(\zeta,0)=\theta_{0}(\zeta)\hskip 14.226378 pt\textit{and} \hskip 14.226378 ptp=1\Rightarrow\theta(\zeta,1)=\theta(\zeta), \tag{50}\]  
\(f(\zeta,p)\) becomes \(f_{0}(\zeta)\) to \(f(\zeta)\) when \(p\) assumes the values from 0 to 1. Similarly, \(\theta(\zeta,p)\) becomes \(\theta_{0}(\zeta)\) to \(\theta(\zeta)\) when \(p\) assumes the values from 0 to 1. Using Taylor series expansion and Equations (45) and (46), one obtains  
\[f(\zeta,p)=f_{0}(\zeta)+\sum_{m=1}^{\infty}f_{m}(\zeta)p^{m},\hskip 14.226378 ptf _{m}(\zeta)=\frac{1}{m!}\frac{\partial^{m}\!\!f(\zeta,p)}{\partial p^{m}}\mid_ {p=0}, \tag{51}\]  
\[\theta(\zeta,p)=\theta_{0}(\zeta)+\sum_{m=1}^{\infty}\theta_{m}(\zeta)p^{m}, \hskip 14.226378 pt\theta_{m}(\zeta)=\frac{1}{m!}\frac{\partial^{m}\!\!\theta( \zeta,p)}{\partial p^{m}}\mid_{p=0}. \tag{52}\]  
The convergence of the series is sharply relying on \(h\).