# 1 EHDFoam

## 1.1 Introduction

Electrohydrodynamics deal with fluid motion induced by electric fields. In the mid 1960s G.I. Taylor introduced the leaky dieletric model to explain the behaviour of droplets deformed by steady field, and J.R. Melcher used it extensively to develop electrohydrodynamic. Here, we developed a Numerical model of electrohydrodynamics, the model concides the Taylor-Melcher Leaky Dielectric Model with VOF (Volume of Fraction) method. The Numerical solver was created in OpenFOAM-1.5.

## 1.2 Model Equations $\vec{E}=-\nabla U_E$ $\varepsilon\nabla\cdot\vec{E}= - \varepsilon\Delta U_E=\rho_E$ $\frac{\partial \rho_E}{\partial t} + \nabla\cdot\left(\rho_E\vec{V}\right) + \frac{\sigma}{\epsilon}\rho_E =0$ $\vec{F}_b=\rho_E\vec{E} - \frac{1}{2}\left|\vec{E}\right|^2\nabla\varepsilon$ $\frac{\partial\vec{V}}{\partial t} + \nabla\cdot\left(\vec{V}\vec{V}\right)= -\frac{1}{\rho}\nabla p + \frac{1}{\rho}\nabla\cdot\vec{\vec{\tau}} + \frac{1}{\rho}\vec{F}_b + \vec{g}$ $\nabla\cdot\vec{V}=0$ $\vec{\vec{\tau}}=\mu\left(\nabla\vec{V} + \nabla\vec{V}^T\right)$ $\frac{\partial \alpha}{\partial t} + \nabla\cdot\left(\alpha\vec{V}\right) = 0$ $\rho=\rho_1\alpha + \rho_2\left(1-\alpha\right)$ $\mu=\mu_1\alpha + \mu_2\left(1-\alpha\right)$ $\varepsilon=\varepsilon_1\alpha + \varepsilon_2\left(1-\alpha\right)$ $\sigma=\sigma_1\alpha + \sigma_2\left(1-\alpha\right)$