Lattice Boltzmann modeling of two-phase electrohydrodynamic flows under unipolar charge injection

In this work, a two-dimensional droplet confined between two parallel electrodes under the combined effects of a nonuniform electric field and unipolar charge injection is numerically investigated using the lattice Boltzmann method (LBM). Under the non-Ohmic regime, the interfacial tension and electric forces at the droplet surface cooperate with the volumetric Coulomb force, leading to complex deformation and motion of the droplet while at the same time inducing a bulk electroconvective flow. After we validate the model by comparing with analytical solutions at the hydrostatic state, we perform a quantitative analysis on the droplet deformation factor D and bulk flow stability criteria Tc under different parameters, including the electric capillary number Ca, the electric Rayleigh number T, the permittivity ratio r, and the mobility ratio Kr. It is found that the bulk flow significantly modifies the magnitude of D, which in turn decreases Tc of the electroconvective flow. For a droplet repelled by the anode, r>1, an interesting linear relationship can be observed in the D-Ca curves. However, for a droplet attracted to the anode, r<1, the system is potentially unstable. After first evolving into a quasisteady state, the droplet successively experiences steady flow, periodic flow, second steady flow, and oscillatory flow with increasing T. Moreover, discontinuities can be observed in the D-T curves due to the transitions of bulk flow.