Axisymmetric deformation and stability of a viscous drop in a steady electric field

We consider a neutrally buoyant and initially uncharged drop in a second liquid subjected to a uniform electric field. Both liquids are taken to be leaky dielectrics. The jump in electrical properties creates an electric stress balanced by hydrodynamic and capillary stresses. Assuming creeping flow conditions and axisymmetry of the problem, the electric and flow fields are solved numerically with boundary integral techniques. The system is characterized by the physical property ratios R (resistivities), Q (permitivities) and lambda (dynamic viscosities). Depending on these parameters, the drop deforms into a prolate or an oblate spheroid. The relative importance of the electric stress and of the drop/medium interfacial tension is measured by the dimensionless electric capillary number, Ca(E). For lambda = 1, we present a survey of the various behaviours obtained for a wide range of R and Q. We delineate regions in the (R, Q)-plane in which the drop either attains a steady shape under any field strength or reaches a fold-point instability past a critical CaE. We identify the latter with linear instability of the steady shape to axisymmetric disturbances. Various break-up modes are identified, as well as more complex behaviours such as bifurcations and transition from unstable to stable solution branches. We also show how the viscosity contrast can stabilize the drop or advance break-up in the different situations encountered for lambda = 1.