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.

We quantify the transient deformation of a droplet immersed in a weakly conductive (leaky dielectric) fluid upon exposure to a uniform DC electric field. Capillary forces are assumed to be sufficiently large that the drop only slightly deviates from its equilibrium spherical shape. In particular, we account for transient (or linear) fluid inertia via the unsteady Stokes equations, and also account for a finite electrical relaxation time over which the drop interface charges. The temporal droplet deformation is governed by two dimensionless groups: (i) the ratio of capillary to momentum diffusion time scales: an Ohnesorge number Oh and (ii) the ratio of charge relaxation to momentum diffusion time scales, which we denote by Sa. If charge and momentum relaxation occur quickly compared to interface deformation, Sa ≪ 1 and Oh ≫ 1 for the droplet and medium, a monotonic deformation is acquired. In contrast, Sa > 1 and Oh < 1 for either phase can lead to a non-monotonic development in the deformation. Numerical values for the deformation are calculated by inverting an analytical expression obtained in the Laplace domain, and are corroborated by asymptotic expansions at early and late times. The droplet and medium behave as perfect dielectrics at early times, which always favors an initial prolate (parallel to the applied field) deformation. As a consequence, for a final oblate (normal to the applied field) deformation, there is a shape transition from prolate to oblate at intermediate times. This transition is caused by the accumulation of sufficient charge at the interface to generate electrical and viscous shear stresses. Notably, after the transition, there may be an “overshoot” in the deformation, i.e., the magnitude exceeds its steady-state value, which is proceeded by an algebraic tail describing the arrival towards the final, steady deformation. Our work demonstrates that transient inertia or a non-zero electrical relaxation time can yield non-monotonic electrohydrodynamic drop deformation.

Suspension stability on time scales of drop deformation leads to rich and unexplored breakup phenomena.

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