  In this work, we aim to develop a phase-field based lattice Boltzmann (LB)
method for simulating two-phase electrohydrodynamics (EHD) flows, which allows
for different properties (densities, viscosities, conductivity and
permittivity) of each phase while maintaining thermodynamic consistency. To
this end, we first present a theoretical analysis on the two-phase EHD flows by
using the Onsager's variational principle, which is an extension of Rayleigh's
principle of least energy dissipation and, naturally, guarantees thermodynamic
consistency. It shows that the governing equations of the model include the
hydrodynamic equations, Cahn-Hilliard equation coupled with additional
electrical effect, and the full Poisson-Nernst-Planck electrokinetic equations.
After that, a coupled lattice Boltzmann (LB) scheme is constructed for
simulating two-phase EHD flows. In particular, in order to handle two-phase EHD
flows with a relatively larger electric permittivity ratio, we also introduce a
delicately designed discrete forcing term into the LB equation for
electrostatic field. Moreover, some numerical examples including two-phase EHD
flows in planar layers and charge diffusion of a Gaussian bell are simulated
with the developed LB method. It is shown that our numerical scheme shares a
second-order convergence rate in space in predicting electric potential and
charge density. Finally, we used the current model to simulate the deformation
of a droplet under an electric field and the dynamics of droplet detachment in
reversed electrowetting. Our numerical results align well with the theoretic
solutions, and the available experimental/numerical data, demonstrating that
the proposed method is feasible for simulating two-phase EHD flows.
