## Abstract

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.

Original language | English (US) |
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Article number | 112101 |

Journal | Physics of Fluids |

Volume | 25 |

Issue number | 11 |

DOIs | |

State | Published - Aug 30 2013 |

Externally published | Yes |

### Bibliographical note

Funding Information:We are grateful for fruitful discussions held with Ory Schnitzer. We thank the National Science Foundation through Grant No. CBET-1066853 for support.