  We develop a method for modeling and simulating a class of two-phase flows
consisting of two immiscible incompressible dielectric fluids and their
interactions with imposed external electric fields in two and three dimensions.
We first present a thermodynamically-consistent and reduction-consistent phase
field model for two-phase dielectric fluids. The model honors the conservation
laws and thermodynamic principles, and has the property that, if only one fluid
component is present in the system, the two-phase formulation will exactly
reduce to that of the corresponding single-phase system. In particular, this
model accommodates an equilibrium solution that is compatible with the
zero-velocity requirement based on physics. This property provides a simpler
method for simulating the equilibrium state of two-phase dielectric systems. We
further present an efficient numerical algorithm, together with a
spectral-element (for two dimensions) or a hybrid
Fourier-spectral/spectral-element (for three dimensions) discretization in
space, for simulating this class of problems. This algorithm computes different
dynamic variables successively in an un-coupled fashion, and involves only
coefficient matrices that are time-independent in the resultant linear
algebraic systems upon discretization, even when the physical properties (e.g.
permittivity, density, viscosity) of the two dielectric fluids are different.
This property is crucial and enables us to employ fast Fourier transforms for
three-dimensional problems. Ample numerical simulations of two-phase dielectric
flows under imposed voltage are presented to demonstrate the performance of the
method herein and to compare the simulation results with theoretical models and
experimental data.
