# Magnetohydrodynamic Nanoliquid Thin Film Sprayed on a Stretching Cylinder with Heat Transfer
## 5 Discussion
### Numerical Comparison, Residual Error Sketches and Tables  
In order to obtain the accuracy of the achieved results a proper mechanism is adopted. HAM solution is compared with the numerical method solution which shows an excellent agreement with one another. In Figures 18 and 19 the HAM solution graphs of velocity and temperature overlap with the graphs of numerical method solution. To be more authentic the HAM solution, the residual error _Res_ graphs are drawn in Figures 20 and 21. Similarly to achieve more precision of HAM solution the comparison Table of HAM solution and numerical method solution is prepared in Table 3. The residual error Table 4 also shows the accuracy of results numerically.  
Figure 19: Comparison of the temperature solution of HAM with numerical method solution when \(h=-0.55\), \(\beta_{1}=2.00\), \(M=0.10\), \(Re=0.70\), \(\phi=0.04\), \(Pr=1.10\) for (CuO-H\({}_{2}\)O).  
Figure 20: Residual errors sketch of the velocity solution of HAM when \(\beta_{1}=2.00\), \(M=0.10\), \(Re=0.70\), \(\phi=0.04\), \(Pr=1.10\) and various values of \(h\) for (CuO-H\({}_{2}\)O).  
Figure 18: Comparison of the velocity solution of HAM with numerical method solution when \(h=-0.55\), \(\beta_{1}=2.00\), \(M=0.10\), \(Re=0.70\), \(\phi=0.04\), \(Pr=1.10\) for (CuO-H\({}_{2}\)O).  
Skin Friction Coefficient (\(C_{f}\)), Nusselt Number (Nu) Sketches and Tables Showing Comparison with the Published Experimental Work  
It is experimental proved that the shear stress and rate of heat transfer change by using different kinds of nanofluids. From this, it is concluded that the type of nanofluid is most important in the cooling and heating processes. By selecting CuO as nanoparticle the higher Nusselt number and smaller skin friction coefficient can be achieved. Therefore in the present study CuO-H\({}_{2}\)O is used to investigate the effects of various parameters. The thermo-physical properties of the nanoliquid are taken to be functions of the volume fraction and the thermal conductivity is modeled based on the effective medium theory. The Prandtl number \(Pr\) for the base liquid water is usually around 7. Using  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Velocity**} & \multicolumn{2}{c}{**Temperature**} \\ \hline \(\xi\) & \(f(\zeta)\) & **Residual Errors** & \(\xi\) & \(\theta(\zeta)\) & **Residual Errors** \\ \hline
0.0 & 1. & \(-0.64363\) & 0.0 & 1.0 & 1.53209 \\
0.1 & 1.09667 & \(-0.456915\) & 0.1 & 0.931831 & 1.64776 \\
0.2 & 1.18735 & \(-0.29  
the definition of Prandtl number and the thermo-physical properties of water as listed in Table 1 along with \(\mu_{f}=1\times 10^{-3}\) Pa s at 20 \({}^{\circ}\)C, the Prandtl number of water is evaluated to be \(Pr=6.8173\). This value has been used throughout the computations.  
Effects of CuO-H\({}_{2}\)O nanofluid on skin friction coefficient (\(C_{f}\)) and Nusselt number (\(Nu\)) are demonstrated in Figures 22 and 23. The quantity \(C_{f}\) related to the surface drag is shown as a function of the Reynolds number \(Re\) for different values of \(\phi\). Figure 22 presents the scenario that the skin friction coefficient (\(C_{f}\)) decreases with the increase of nanoparticle volume fraction \(