  The deformation of a viscous liquid droplet suspended in another liquid and
subject to an applied electric field is a classic multiphase flow problem best
described by the Melcher-Taylor leaky dielectric model. The main assumption of
the model is that any net charge in the system is concentrated on the interface
between the two liquids as a result of the jump in Ohmic currents from the
bulk. Upon application of the field, the drop can either attain a steady
prolate or oblate shape with toroidal circulating flows both inside and outside
arising from tangential stresses on the interface due to action of the field on
the surface charge distribution. Since the pioneering work of
\cite{taylor1966}, there have been numerous computational and theoretical
studies to predict the deformations measured in experiments. Most existing
theoretical models, however, have either neglected transient charge relaxation
or nonlinear charge convection by the interfacial flow. In this work, we
develop a novel small-deformation theory accurate to second order in electric
capillary number ${O}(Ca_E^2)$ for the complete Melcher-Taylor model that
includes transient charge relaxation, charge convection by the flow, as well as
transient shape deformation. The main result of the paper is the derivation of
coupled evolution equations for the induced electric multipoles and for the
shape functions describing the deformations on the basis of spherical
harmonics. Our results, which are consistent with previous models in the
appropriate limits, show excellent agreement with fully nonlinear numerical
simulations based on an axisymmetric boundary-element formulation and with
existing experimental data in the small-deformation regime.
