## 3 Governing Equations  
### Mathematical Model  
In order to simulate nanofluid hydrothermal treatment in existence of electric field, we should combined equations of electric fields with those of hydrothermal. The formulas of electric field are:  
\[\
abla.\left(\overset{\rightarrow}{E}\epsilon\right)=q \tag{2}\]  
\[\left(-\
abla\varphi\right)=\overset{\rightarrow}{E} \tag{3}\]  
\[\
abla.\overset{\rightarrow}{J}+\frac{\partial q}{\partial t}=0 \tag{4}\]  
There exist two model for charge distribution: (1) conductivity model [28] and (2) mobility model [29]. In first model, electro-convection rely on temperature gradient. But in second model, electro-convection  
Figure 2: Electric density distribution injected by the bottom electrode. is independent of temperature gradient in the liquid. In the case of free charge origination, second model is more acceptable according to experimental results. Electric current density can be defined as [30]:  
\[\overset{\rightarrow}{\mathrm{J}}=\sigma\overset{\rightarrow}{E}-D\
abla q+q \overset{\rightarrow}{V} \tag{5}\]  
Where \(\sigma\overset{\rightarrow}{E}\) is ionic mobility, \(D\
abla q\) is diffusion [31], \(q\overset{\rightarrow}{V}\) is convection. According to Equations (4) and (5), the equation for electric charge density can be obtained as follow:  
\[ \begin{split}&\frac{\partial\overset{\rightarrow}{q}}{ \partial t}+\overset{\rightarrow}{v}\frac{\partial\overset{\rightarrow}{q} }{\partial y}+\overset{\rightarrow}{u}\frac{\partial\overset{\rightarrow} {q}}{\partial x}\\ &+\frac{1}{\mathrm{Re}\,\mathrm{Pr}_{E}}\Bigg{[}q\Bigg{(}\frac{ \partial\overset{\rightarrow}{E_{g}}}{\partial y}+\frac{\partial\overset{ \rightarrow}{E_{s}}}{\partial x}\Bigg{)}+\overset{\rightarrow}{E_{s}}\frac{ \partial\overset{\rightarrow}{q}}{\partial y}+\overset{\rightarrow}{E_{s}} \frac{\partial\overset{\rightarrow}{q}}{\partial x}\Bigg{]}\\ &=\frac{\rho_{uf}}{\mu_{uf}}/\rho_{f}}\frac{1}{\mathrm{Re}\,D_{e}} \Bigg{(}\frac{\partial^{2}\overset{\rightarrow}{q}}{\partial y^{2}}+\frac{ \partial^{2}\overset{\rightarrow}{q}}{\partial x^{2}}\Bigg{)}\end{split}  \tag{6}\]  
According to [30] the diffusion term can be taken negligible. Also \(D\
abla q\) in Equation (5) can be taken negligible and \(\sigma=bq\)[31]. So Equation (5) can be considered as:  
\[\overset{\rightarrow}{\mathrm{J}}=q\overset{\rightarrow}{V}+qb\overset{ \rightarrow}{E} \tag{7}\]  
In presence of electric field Coulomb forces should be added to momentum equation and Joule heating effect should be added in energy equation. So we have:  
\[\begin{split}& \begin{bmatrix}\
abla.\overset{\rightarrow}{V}=0 \\ \rho_{uf}\Bigg{(}\frac{\partial\overset{\rightarrow}{V}}{\partial t}+ \Bigg{(}\overset{\rightarrow}{V}.\
abla\Bigg{)}\overset{\rightarrow}{V} \Bigg{)}=-\
abla p+\mu_{uf}\
abla^{2}\overset{\rightarrow}{V}+q\overset{ \rightarrow}{E}\\ \Big{(}\rho C_{p}\Big{)}_{uf}\Bigg{(}\frac{\partial T}{\partial t}+ \Bigg{(}\overset{\rightarrow}{V}.\
abla\Bigg{)}T\Bigg{)}=k_{uf}\
abla^{2}T+ \overset{\rightarrow}{\mathrm{J}}.\overset{\rightarrow}{E}\\ \
abla.\overset{\rightarrow}{\mathrm{J}}+\frac{\partial q}{\partial t}=0\\ \
abla.\varepsilon\overset{\rightarrow}{E}=q\\ \overset{\rightarrow}{E}=-\
abla\varphi\end{bmatrix} \\ \rho_{uf},\big{(}\rho C_{p}\big{)}_{uf},\alpha_{uf},\mu_{uf}\) and \(k_{nf}\) are defined as [6]:  
\[\rho_{uf}=\rho_{s}\phi+\rho_{f}(\mathrm{I}-\phi) \tag{9}\]  
\[\big{(}\rho C_{p}\big{)}_{uf}=\big{(}\rho C_{p}\big{)}_{s}\phi+\big{(}\rho C_{p }\big{)}_{f}\left(\mathrm{I}-\phi\right) \tag{10}\]  
\[\alpha_{uf}=k_{uf}\big{/}\big{(}\rho C_{p}\big{)}_{uf} \tag{11}\]  
\[\mu_{uf}=\frac{\mu_{uf}}{\left(\mathrm{I}-\phi\right)^{2^{3}}} \tag{12}\]\[k_{n_{f}}=k_{f}\frac{-2\phi(k_{f}-k_{s})+2k_{f}+k_{s}}{+\phi(k_{f}-k_{s})_{s}+2k_{ f}+k} \tag{13}\]  
The thermo physical properties of the working fluid are given in Table1. Non-dimensional parameters are introduced as follows:  
\[