Linear stability of an interface between a non-ohmic liquid and air subjected to an electric field and charge injection

The electrohydrodynamic (EHD) instability of an insulating liquid subjected to unipolar injection of ions has been the object of many different studies. It is due to the existence of a potentially unstable distribution of charge in the liquid bulk. Besides that classical instability, it has also been found experimentally that insulating liquids exhibit another kind of EHD instability when subjected to corona discharge from air. This instability, referred to as rose-window instability, is characterized by a pattern of large cells. Both instabilities arise above quite different voltage thresholds. In this paper we write down and numerically solve the linearized equations of motion of the liquid when the air above is considered. Our first aim is to discuss the effect of the air layer on the linear stability threshold for the classical EHD volume instability.