Seminar on Advances in Modelling the Atomization of Electrohydrodynamic Jets
Date:
Introduction
Electrohydrodynamic (EHD) jets are a highly promising technology in numerous industrial applications, including inkjet printing, spray coating, fuel injection, and others. Therefore, an accurate prediction of droplet size and distribution is essential for the advancement of this technology. However, these predictions are hindered by an insufficient understanding of many aspects of its flow mechanisms, such as whipping behaviour.
Govening Equations
The governing equations needed to solve the two isothermal, incompressible, and immiscible fluids are continuity, momentum, and the interface advection equations. Our simulations employ numerical methods such as the Volume of Fluid (VoF) method and the Finite Volume Method (FVM), along with the implementation of Maxwell’s equations for electrostatics within the governing equations of fluid dynamics based on the Navier-Stokes equations.An important detail is the method for interface tracking, which in our case is the geometric Volume-of-Fluid (VoF) method. With this method, only one scalar, $\alpha$, is defined to represent the liquid and gas phases and, in the end, the sum of the volume fraction is equal to one. An overview of the governing equations is given below.
\[\nabla \cdot \mathbf{u} = 0 ,\] \[\frac{\partial}{\partial t} \left(\rho \mathbf{u} \right)+ \nabla \cdot \left(\rho\mathbf{u} \mathbf{u} \right)= -\nabla p + \mu \nabla^{2} \mathbf{u} + \rho \mathbf{g} +\mathbf{f}_{\gamma}+\mathbf{f}_{e} ,\] \[\frac{\partial \alpha}{\partial t}+\nabla \cdot(\alpha \vec{u})=0 .\]where $\mathbf{u}$ is the velocity vector, composed for a Cartesian coordinate system by the three components, $u_{x}$, $u_{y}$ and $u_{z}$. We are considering a flow with a mass density $\rho$, a dynamic viscosity $\mu$, a local fluid pressure $p$ and subjected to a gravitational acceleration $ {\mathbf{g}}$. In this case into the source term of the momentum, we added the force due to the surface tension $\mathbf{f}{\gamma}$ and one due to the external imposed electric field, as $\mathbf{f}{e}$. In addition to the hidrodynamic governing equations, we also consider the conservation of electric charge density ($\rho_e$). This equation accounts for the distribution and transport of electric charges as,
\[\frac{\partial \rho_{e}}{\partial t}+\nabla \cdot\left(\rho_{e} \mathbf{u} \right) + \nabla \cdot (\sigma \mathbf{E}) = 0\]References
S. Cândido, J. Páscoa, “Dynamics of Three-dimensional Electrohydrodynamic Instabilities on Taylor Cone Jets using a Numerical Approach”, Physics of Fluids, Vol. 35, 2023, doi: 10.1063/5.0151109