The effect of a finite conductivity on the stability of a falling liquid film subjected to a normal electric field

The dynamics of a liquid film falling down an inclined plane wall has been studied in a wide range of situations. In particular, electric fields have been studied as a tool to destabilize or destroy a film. In this work, we consider the effect of a finite, but large, conductivity in the liquid, compared to the case of a perfectly conducting film. The finite conductivity introduces new dissipative terms, as the Joule effect, but also new forms of transforming electrical energy into mechanical energy, through surface stresses. Assuming long wave sinusoidal deformations, a modified dispersion relation is found, that incorporates the new stabilizing and destabilizing factors. The result can be understood in terms of the energy processes involved.