Figure 3.1 illustrates the 3D model considered in this study. A small liquid droplet is suspended in another immiscible fluid and is exposed to a uniform electric field parallel to the Y-axis. This field is generated by applying different electric potentials to the parallel plates in the X-Z planes.
The center of the droplet is placed in the middle of the rectangular cube. The liquids are assumed to be incompressible Newtonian with the same density, ρ, so that the drop is under neutrally buoyant condition. The interface separating the two fluids is assumed to have a constant interfacial tension coefficient. The size of the computational domain is 10
times the droplet radius in the direction of the electric field and 8 times the droplet radius in the directions normal to the electric field. At the initial stage, the shape of the droplet is assumed to be spherical, the center of droplet is located at the centre of the parallel-plate capacitor and both fluids are motionless.
Figure 3.1 Model of a suspended droplet in an electric field
In order to investigate the dynamics of droplet deformation in an electric field it is necessary to solve the Navier-Stokes equations, describing the fluid motion, as well as track the interfaces between both fluids. The laminar two-phase flow system studied here is coupled with the applied electric field and the electric charge conservation law. Additional body forces are added to the Navier-Stokes equations for considering the surface tension (Fst) and electric stress (Fes).
(3.4)
where u denotes fluid velocity, is the identity matrix, ρ is the fluid density, µ is the dynamic viscosity and P is the pressure. In the current simulation, no-slip boundary
conditions are applied for the electrodes and pressure outlet conditions are applied for other boundaries.
To describe the evolution of the droplet shape, the Level-Set method [18], suitable for free boundary problems, is applied. In the Level-Set method, the interface is considered to have a finite thickness of the same order as the mesh size instead of zero thickness. The physical property changes smoothly from the value on one side of the interface to the value on the other side in the interfacial transitional zone. The method describes the evolution of the interface between the two fluids tracing an iso-potential curve of the level set function . In general, in droplet and in ambient fluid . The interface is represented by the 0.5 contour of the level set function ( ). The movement of the interface is governed by a differential equation for this function. To keep the level set function a distance function, a reinitialization process is needed. Ideally, the interface should not change its position during this reinitialization procedure, but in many applications the zero level set can become distorted by parasitic numerical inaccuracies, if the gradients in the neighborhood of the interface are either very large or very small. For this reason, an improved reinitialization method is used.
Level-Set methods automatically deal with topological changes and it is in general easy to obtain high order of accuracy. The time evolution of the interface is modeled via transport of the level set function due to the underlying physical velocity field. The function is governed by the equation
(3.5)
where is the parameter controlling the interface thickness and is the reinitialization parameter. A suitable value for is the maximum velocity magnitude occurring in the model. The density and viscosity, which are different for oil and water, are automatically calculated from the level set variable , as well as the surface tension force.
The surface tension force is given by
where is the interface normal and is the Dirac-delta function that is nonzero only at the fluid interface. The interface normal is calculated from
(3.7)
The use of a Dirac-delta function will ideally create a sharp interface in the mathematical formulation. However, to implement this in the numerical simulation, the Dirac-delta function should be approximated by
(3.8)
The electric force causes the deformation and it can be calculated from the electric field distribution, which depends on the position and shape of the droplet. In the absence of any time-varying magnetic field, the curl of the electric field is zero and the electric field can be expressed in terms of the electric potential V.
(3.9)
The charge conservation in each medium can be expressed as follows
(3.10)
where is the electric conductivity of the medium.
Assuming that the electric relaxation time is less than the time scale of the fluid motion, in a two-fluid system the electrical conductivity is constant within each fluid and Eq. (3.10) for electric potential (V) can be reduced to Laplace equation in each medium
(3.11)
It is assumed that there is no space charge in the fluids except the surface charge on the interface, created by the difference between permittivities and conductivities of both
fluids. At the interface between the two fluid media, the electric potential and normal component of electric current density are continuous.
(3.12)
where represents a jump across the interface. The above boundary conditions at the interface between two fluids can be embedded in the governing equation Eq. (3.10) for electric potential with variable electric conductivity in the different fluid regions of the system.
After solving Eq. (3.10), the electric potential can be obtained and then the electric field strength can be calculated using Eq. (3.9). The current density ( and the electric displacement (D) can also be found from
(3.13)
where is the permittivity of vacuum and is the relative permittivity of medium (ratio of the absolute permittivity and that of vacuum). Assuming that the fluids are incompressible, the electric stress can be calculated by taking the divergence of the Maxwell stress tensor, which couples electrostatic and hydrodynamic phenomena. Neglecting the effect of magnetic field, the Maxwell stress tensor can be defined as follows:
(3.14)
The momentum equation is modified by inserting the electric force, , which can be
determined by calculating the divergence of the Maxwell stress tensor ( :
(3.15)
The conductivities and relative permittivities for each fluid are constant, but different. The volume fraction changes from zero in one fluid to one in the other one. In order to
have all the physical properties in the interface, the two phase relative permittivity ( and conductivity ( can be defined based on the volume fraction of the phases:
(3.16)
where and are the volume fractions of the droplet and the continuous phase,
respectively. Using Eq. (3.16), the physical properties change smoothly from the value on one side to the value on the other side. The governing equations for two-phase flow and electric field have been solved with the commercial software COMSOL based on the Finite Element Method (FEM) [19].