Dynamics of a conducting drop in a time-periodic electric field

Abstract

The nonlinear dynamical behaviour of a conducting drop in a time-periodic electric field is studied. Taylor's (1964) theory on the equilibrium shape is extended to derive a dynamical equation in the form of an ordinary differential equation for a conducting drop in an arbitrary time-dependent, uniform electric field based on a spheroidal approximation for the drop shape and the weak viscosity effect. The dynamics is then investigated via the classical two-timing analysis and the Poincaré map analysis of the resulting dynamical equation. The analysis reveals that in the neighbourhood of a stable steady solution, an time-dependent change of drop shape can be obtained from an O(ε) resonant forcing. It is also shown that the probability of drop breakup via chaotic oscillation can be maximized by choosing an optimal frequency for a fixed forcing amplitude. As a preliminary analysis, the effect of weak viscosity on the oscillation frequency modification in a steady electric field is also studied by using the domain perturbation technique. Differently from other methods based on the theory of viscous dissipation, the viscous pressure correction is directly obtained from a consideration of the perturbed velocity field due to weak viscosity.