LINEAR GROWTH OF INSTABILITIES ON A LIQUID-METAL UNDER NORMAL ELECTRIC-FIELD

It is well known that an electric field that is applied normally to the free surface of a conducting fluid has a destabilizing effect. Here we study the linear growth of electro-capillary instabilities in the very general case where the viscosity of the fluid and its thickness are of any value. We derive the asymptotic behaviour in various regimes and give analytical dispersion equations in the case of thin or thick, inviscid or viscous films and compare with previous results. Then we present a diagram of the corresponding simplifications of the dispersion relation and as we derive them directly from the general equation, we are able to derive their conditions of validity explicitly. We also show the similarities and the differences with the case of a liquid falling from a solid flat plane (Rayleigh-Taylor instabilities) without electric field and with ferrofluid instabilities.