EHD instability of insulating liquids due to charge injection from the free surface

We examine the hydrodynamic instability of a layer of insulating liquid with one free surface subjected to a unipolar injection at this free surface. In the stationary state it is shown that there is no surface charge at the air/liquid interface. The usual convective instability is predicted to occur for small liquid thickness d, the critical voltage being higher than when the free surface is equipotential. For high enough values of d, another instability mechanism is expected with a marked surface deformation of large wavelength; qualitative arguments and an order of magnitude analysis are given