Generation of interfacial instabilities in charged electrified viscous liquid films

This study is concerned with the stability of a two-dimensional incompressible conducting liquid film surrounded by a passive conducting medium, when an electric field is applied in a direction parallel to the initially flat bounding fluid interfaces. Currents generate charges at the bounding interfaces which in turn affect the stress balances there. In the absence of an electric field, the viscous liquid film is stable (instability can be induced by the inclusion of van der Waals forces for ultra thin films). A complete model is presented, at arbitrary Reynolds number, which accounts for conductivity and permittivity contrasts between the fluid and surrounding medium, as well as surface tension. The linear stability of the system is considered for arbitrary Reynolds numbers and it is shown that the stable film can become unstable if, (i) σ_Rε_p >1, or (ii) σ_Rε_p <1 and (σ_R −1)(1−ε_p)<0, where σ_R is the ratio of outer to inner conductivity and ϵ_p is the ratio of inner to outer permittivity. Instability is possible only if the electric field is non-zero and the scalings near bifurcation points that can be used to construct nonlinear theories are calculated. Several asymptotic limits are also considered including zero Reynolds numbers and short or long waves. The instability criteria given above are constructed explicitly in the case of Stokes flow.