  In this work, the electrohydrodynamic (EHD) instability induced by a unipolar
charge injection is extended from a single-phase dielectric liquid to a
two-phase system that consists of a liquid-air interface. A volume of fluid
(VOF) model based two-phase solver was developed with simplified Maxwell
equations implemented in the open-source platform OpenFOAM\textsuperscript. The
numerically obtained critical value for the linear stability matches well with
the theoretical values. To highlight the effect of the slip boundary at
interface, the deformation of the interface is ignored. A bifurcation diagram
with hysteresis loop linking the linear and finite amplitude criteria, which is
Uf = 0.059, was obtained in this situation. It is concluded that the lack of
viscous effect at interface leads to a significant increase in the flow
intensity, which is the reason for the smaller instability threshold in
two-phase system. The presence of interface also changes the flow structure and
makes the flow vortices shift closer to the interface.
