Analytical Solution for Spheroidal Drop under Axisymmetric Linearized Boundary Conditions

Abstract

A liquid spheroidal drop freely suspended in another fluid is considered under arbitrary axisymmetric boundary conditions, which are linearized with respect to the velocity field and can result, in particular, from an axisymmetric external flow and an electric field being applied either separately or in combination. All nonlinear effects including inertia and surface charge convection are assumed to be negligible, whereas the drop and the ambient fluid are assumed to be leaky dielectrics and to have different viscosities. Central to the analysis are the reformulated stress boundary conditions and representation of the velocity field inside and outside the drop in terms of non-Stokes stream functions. In the prolate spheroidal coordinates, the stream functions are expanded into infinite series of spheroidal harmonics, and the reformulated velocity and stress boundary conditions yield a first-order difference equation for the series coefficients, which admits an exact nonrecursive solution. “Steady” spheroidal drops then correspond to minimums of a kinematic condition error that admits a simple efficient approximation. Under the simultaneous presence of aligned linear flow and a uniform electric field with corresponding capillary numbers $Ca$ and $Ca_E$, a spherical drop is stationary when ${Ca}=k\,{Ca}_E$ and becomes prolate/oblate when ${Ca}\gtrless k\,{Ca}_E$, where $k$ is proportional to the Taylor discriminating function and depends on ratios of viscosities, dielectric constants, and electric conductivities of the two phases. A spheroidal drop is “steady” when ${Ca}=k_1\,{\rm Ca}_E+k_2$, where $k_1$ and $k_2$ depend on the spheroid's axes ratio $d$ and approach $k$ and 0, respectively, as $d\to1$. The results show that when the Taylor deformation parameter $D$ is in the range $[-0.5,0.4]$, this relationship can be used for finding any of the three $Ca$, $Ca_E$, and $D$ when the other two are given.