# Numerical simulation of deformation of a droplet in a stationary electric field using DG

## Abstract

Numerical simulation of deformation of a droplet in a stationary electric field is performed in the present research. The droplet is suspended in another immiscible fluid with the same density and viscosity but a different dielectric property (permittivity). By applying the electric field, the fluids are polarized that gives rise to mechanical forces and deformation. A two-way coupling occurs because of the forces exerted from the electric field on the droplet and the deformation of the droplet which changes the geometry for the electric field calculations. The droplet continues to deform until a force balance between the electric force, pressure and the surface tension is achieved and the droplet becomes a spheroid. An electromechanical approach is adopted to solve the above mentioned problem, which includes solving the governing equations of both the electric and fluid fields, computing the coupling forces and capturing the movement of the interface of the droplet and the surrounding fluid. A one-fluid approach is followed, which enables us to solve one set of the governing equations for both the droplet and the surrounding fluid. The interface is represented as the zero iso-value of a level set function and an advection equation is solved to find the movement of the interface. A diffuse interface model is used to regularize the jump in the fluid and electric properties. The governing equations of the electric and fluid fields and the level set advection equation are discretized using the Discontinuous Galerkin Finite Element method (DG) in the BoSSS code for solving conservation laws. The electric field is computed from the electric potential by considering the electrostatic equations. To find the electric potential, a Laplace equation is solved which has a jump in the permittivity at the interface. The Laplace equation is discretized using the interior penalty method (IP) which we modified for the case of high jumps in the permittivity. Assuming that the fluids are linear dielectric materials, the electric force is the dielectrophoretic force which is computed from the Kortweg-Helmholtz formula. This force is added as a body force to the incompressible Navier-Stokes equations, which are the governing equations for the fluid flow. Considering that there is no jump in the fluid properties, a single phase solver of the Navier-Stokes equations including the surface tension at the interface is developed. The surface tension force is added as a body force to the Navier-Stokes equations using the continuum surface force model (CSF). This model is known for producing a spurious velocity field. To decrease the spurious velocities, the surface tension term is calculated by using high degree polynomials for a precise calculation of the normal vector and curvature. To solve the incompressible Navier-Stokes equations using the DG method, a projection scheme with a consistent Neumann pressure boundary condition is employed and the same polynomial order for the velocity and pressure (equal-order method) is applied. Using the above-mentioned pressure boundary condition leads to an optimal convergence rate of k + 1 in the L2-norm for the pressure, which is not reported from other DG solvers. However, using the DG method, we have observed that discontinuities in the solutions at the cell boundaries can affect the solution accuracy and even cause a numerical instability. These accuracy and stability issues occur when the derivatives of the solution are computed. Therefore a flux-based method for calculation of the derivatives of the flow variables was adopted. As the results showed considerably improved accuracy and stability characteristics, we used the proposed method also in solving the above mentioned coupled problem.

Item Type: Ph.D. Thesis
Erschienen: 2014
Creators: Emamy, Nehzat
Title: Numerical simulation of deformation of a droplet in a stationary electric field using DG
Language: English
Abstract:

Numerical simulation of deformation of a droplet in a stationary electric field is performed in the present research. The droplet is suspended in another immiscible fluid with the same density and viscosity but a different dielectric property (permittivity). By applying the electric field, the fluids are polarized that gives rise to mechanical forces and deformation. A two-way coupling occurs because of the forces exerted from the electric field on the droplet and the deformation of the droplet which changes the geometry for the electric field calculations. The droplet continues to deform until a force balance between the electric force, pressure and the surface tension is achieved and the droplet becomes a spheroid. An electromechanical approach is adopted to solve the above mentioned problem, which includes solving the governing equations of both the electric and fluid fields, computing the coupling forces and capturing the movement of the interface of the droplet and the surrounding fluid. A one-fluid approach is followed, which enables us to solve one set of the governing equations for both the droplet and the surrounding fluid. The interface is represented as the zero iso-value of a level set function and an advection equation is solved to find the movement of the interface. A diffuse interface model is used to regularize the jump in the fluid and electric properties. The governing equations of the electric and fluid fields and the level set advection equation are discretized using the Discontinuous Galerkin Finite Element method (DG) in the BoSSS code for solving conservation laws. The electric field is computed from the electric potential by considering the electrostatic equations. To find the electric potential, a Laplace equation is solved which has a jump in the permittivity at the interface. The Laplace equation is discretized using the interior penalty method (IP) which we modified for the case of high jumps in the permittivity. Assuming that the fluids are linear dielectric materials, the electric force is the dielectrophoretic force which is computed from the Kortweg-Helmholtz formula. This force is added as a body force to the incompressible Navier-Stokes equations, which are the governing equations for the fluid flow. Considering that there is no jump in the fluid properties, a single phase solver of the Navier-Stokes equations including the surface tension at the interface is developed. The surface tension force is added as a body force to the Navier-Stokes equations using the continuum surface force model (CSF). This model is known for producing a spurious velocity field. To decrease the spurious velocities, the surface tension term is calculated by using high degree polynomials for a precise calculation of the normal vector and curvature. To solve the incompressible Navier-Stokes equations using the DG method, a projection scheme with a consistent Neumann pressure boundary condition is employed and the same polynomial order for the velocity and pressure (equal-order method) is applied. Using the above-mentioned pressure boundary condition leads to an optimal convergence rate of k + 1 in the L2-norm for the pressure, which is not reported from other DG solvers. However, using the DG method, we have observed that discontinuities in the solutions at the cell boundaries can affect the solution accuracy and even cause a numerical instability. These accuracy and stability issues occur when the derivatives of the solution are computed. Therefore a flux-based method for calculation of the derivatives of the flow variables was adopted. As the results showed considerably improved accuracy and stability characteristics, we used the proposed method also in solving the above mentioned coupled problem.

Uncontrolled Keywords: Electro-fluid-dynamics (EFD), Droplet deformation, Two-way coupling, Electrostatics, Perfect dielectric, Dielectrophoretic force, Incompressible Navier-Stokes equations, projection scheme, Multiphase flow, level set method, surface tension, Curvature computation, Spurious currents, High-order, discontinuous Galerkin Finite Element method (DG), Symmetric interior penalty method (SIPG), Diffusion-dependent penalty parameter, BoSSS
Divisions: 16 Department of Mechanical Engineering > Fluid Dynamics (fdy)
16 Department of Mechanical Engineering
Zentrale Einrichtungen
Exzellenzinitiative
Date Deposited: 15 Sep 2013 19:55