Nonlinear electrohydrodynamics of slightly deformed oblate drops

The transient deformation of a weakly conducting ('leaky dielectric') drop under a uniform DC electric field is computed via an axisymmetric boundary integral method, which accounts for surface charge convection and a finite relaxation time scale over which the drop interface charges. We focus on drops that attain an ultimate oblate (major axis normal to the applied field) steady-state configuration. The computations predict that as the time scale for interfacial charging increases, a shape transition from prolate deformation (major axis parallel to the applied field) to oblate deformation occurs at intermediate times due to the slow buildup of charge at the surface of the drop. Convection of surface charge towards the equator of the drop is shown to weaken the steady-state oblate deformation. Additionally, convection results in sharp shock-like variations in surface charge density near the equator of the drop. Our numerical results are then compared with an experimental system consisting of a millimetre-sized silicone oil drop suspended in castor oil. Agreement in the transient deformation is observed between our numerical results and experimental measurements for moderate electric field strengths. This suggests that both charge relaxation and charge convection are required, in general, to quantify the time-dependent deformation of leaky dielectric drops. Importantly, accurate prediction of the observed modest deformation requires a nonlinear model. Discrepancies between our numerical calculations and experimental results arise as the field strength is increased. We believe that this is due to the observed onset of rotation and three-dimensional flow at such high electric fields in the experiments, which an axisymmetric boundary integral formulation naturally cannot capture.