Behaviour of a conducting drop in a viscous fluid subject to an electric field
The slow deformation of a conducting drop surrounded by a viscous insulating fluid subject to a uniform electric field is considered. Two analytic models are presented for inviscid drops. The first makes use of a time-evolving spheroidal shape along with an energy balance to determine the drop behaviour. For fields below a critical value there exist equilibrium shapes. In this case, the evolution of the drop to the equilibrium shape is obtained. Above the critical value no equilibrium shapes exist and the drop has a period of slow elongation before undergoing rapid expansion. The spheroidal model is shown to be accurate up to aspect ratios of about 5. The second model uses slenderbody theory to model the drop behaviour. A similarity solution, exhibiting a finite-time singularity is obtained. Finally, detailed numerical computations, based on a boundary integral formulation, are presented for inviscid and viscous drops. The deformation of the drop right up to breakup is obtained. The type of breakup seen depends on the viscosity ratio of the drop to the surrounding fluid, and on the electric field strength. The different types of breakup seen are small droplets being emitted from the ends of the drop with a charge greater than the Rayleigh limit, the formation of what appear to be conical ends with the subsequent ejection of thin jet-like structures, or the formation of thin jet-like structures without the conical ends. Also, local analytic solutions that allow for a conical end are derived. However, none of the analytic solutions seem to correspond well with the numerical results. Finally, the behaviour of the drop near the critical electric field strength is examined in detail and time scales for the drop evolution are determined.