A Liquid Spheroidal Drop in a Viscous Incompressible Fluid under a Steady Electric Field

Abstract

An analytical approach to the problem of a liquid spheroidal drop in a viscous incompressible fluid under a steady electric field is presented. It is assumed that the drop and ambient fluid are slightly conducting (leaky dielectrics) with no net charge. The velocity field inside and outside the drop is governed by the Stokes equations and is represented in terms of four generalized analytic functions which are found from the velocity and stress boundary conditions. For the case of equal viscosities, the velocity field admits an integral-form solution, whereas for an arbitrary viscosity ratio $\lambda$, it is given by a series of spheroidal harmonics in the prolate spheroidal coordinates. Both solution forms hold for prolate and oblate spheroids. The axes ratio of a spheroid closest to a steady shape (or “steady” spheroid) is a minimizer of the kinematic condition error. However, the entire dependence of the “steady” spheroid's axes ratio on an electric capillary number ${\rm Ca}_E$ can be estimated analytically with the inverse problem: for each spheroid's axes ratio, find ${\rm Ca}_E$ that minimizes the kinematic condition error. The inverse problem, being quadratic with respect to $1/{\rm Ca}_E$, has a simple analytical solution, which identifies critical ${\rm Ca}_E$ for spheroidal drops and includes an “unstable” solution. Remarkably, as functions of ${\rm Ca}_E$, the deformation parameters $D$ for the “steady” spheroids and for “true” steady shapes (obtained based on boundary-integral equations) have the same qualitative behavior for various viscosity, conductivity, and permittivity ratios ($\lambda$, $R$, and $Q$, respectively), and are virtually identical within the range $-0.4\le D\le0.4$ for all investigated $\lambda$, $R$, and $Q$. This shows that the presented analytical approach for spheroidal drops is by far superior to Taylor's small-perturbation theory and Ajayi's second-order correction ($O({\rm Ca}_E)$ and $O({\rm Ca}_E^2)$ theories, respectively) and can replace numerical solutions from boundary-integral equations up to large deformations.