## 5 Results and Discussion  
Electric filed effect on hydrothermal behavior of nanofluid in an enclosure with moving lower and sinusoidal upper walls is presented. Table 2 illustrates the properties of Ethylene glycol and Fe\({}_{3}\)O\({}_{4}\). Calculations are prepared for various values of supplied voltage ( \(\Delta\varphi\) = 0, 4, 6 and 10 \(kV\)), volume fraction of nanoparticles ( \(\phi\) = 0% and 4%) and Reynolds number numbers ( \(Re\) = 3000,4500  and 6000). In all calculations, the Prandtl number (\(Pr\)) and Eckert number (\(Ec\)) are set to 149.54, 0.0 and 10\({}^{-6}\).  
The influence of Reynolds number and supplied voltage on streamlines and isotherm are shown in Figures 4-6. At \(Re\) = 3000, one main eddy and two very small eddied at upper corners of the enclosure exist in streamline.  
As electric filed applied the main eddy turn in to two smaller ones. Also isotherm becomes denser near the hot wall due to existence these eddies. As Reynolds number increases up to 6000, the upper right eddy becomes stronger and isotherms become denser near the bottom wall.  
Since convective heat transfer is helped more successfully at a greater flow rate, Nusselt number is improved at great Reynolds numbers. So isotherm becomes more distorted at greater values of Reynolds number. As electric filed increases, the main cell convert to two eddies which are rotates opposite direction.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\rho(kg\) / \(m^{3})\) & \(C_{p}(j\) / \(kgk)\) & \(k(W\) / \(m.k)\) & \(\mu(Pa.s)\) \\ \hline Ethylene glycol & 1110 & 2400 & 0.26 & 0.0162 \\ Fe\({}_{3}\)O\({}_{4}\) & _5200_ & _670_ & \(6\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Thermo physical properties of Ethylene glycol and nanoparticles.  
Figure 4: Effect of supplied voltage on streamlines and isotherm when \(Re\) = 3000, \(\phi\) = 0.04.  
Figure 5: Effect of supplied voltage on streamlines and isotherm when \(Re=4500,\phi=0.04\).  
Figure 6: Effect of supplied voltage on streamlines and isotherm when \(Re=6000,\phi=0.04\).  
Figure 7: Effects of Reynolds number and supplied voltage on local Nusselt number for (**a**) \(Re=3000\) ; (**b**) \(Re=4500\) ; (**c**) \(Re=6000\) when \(\phi=0.04\).  
Figures 7 and 8 depict the influences of \(\Delta\varphi\) and \(Re\) on \(Nu_{{}_{loc}}\) and \(Nu_{{}_{ave}}\) along the lid wall. As Reynolds number increases, Nusselt number increases due to decrease in thermal boundary layer thickness. Increasing supplied voltage make the isotherms more distorted. Local Nusselt number profiles have extremums at higher values of supplied voltage because of existence of thermal plumes.  
Nusselt number is an increasing function of supplied voltage. In absence of electric filed, Nusselt number for \(Re=6000\) is \(1.117\) times higher than that of \(Re=3000\) while in presence of electric filed \(\left(\Delta\varphi=10\right),\) Nusselt number for \(Re=6000\) is \(2.298385\) times lower than that of \(Re=3000.\) Also it can be concluded that Nusselt number at \(\Delta\varphi=10\) for \(Re=3000,\)\(4500\) and \(6000\) are times higher than those of obtained at \(\Delta\varphi=0\). This observation confirms that impact of electric filed is more marked for lower Reynolds number.