A model for electrokinetic flow with deformable interfaces

A model is introduced to describe the deformation of a fluid drop in the two-phase flow of immiscible, ionic fluid electrolytes in the presence of an impulsively applied DC electric field. The starting point is the Poisson-Nernst-Planck equations in the zero Reynolds number or Stokes flow regime for symmetric binary electrolytes. These are analyzed by introducing matched asymptotic expansions in the limit when the thickness of the induced charge Debye layers that form adjacent to the interface is much less than the initial size of the undeformed drop. This leads to a hybrid or multiscale model that consists of boundary integral equations for the electrostatic potential and the interface fluid velocity together with relations that contain the coupling between the electrostatic and fluid fields within the thin Debye layers. The drop interface is assumed to be sharp, i.e. of zero thickness relative to the scale of the Debye layers, and the induced charge resides in the Debye layers with no surface charge on the sharp drop interface. A motivation for the study is to understand the influence of bulk electrokinetic effects on interfaces that are deformable and can evolve in time, in the simple context of absence of surface charge, while accurately resolving both Debye layer structure and interface shape and dynamics. Dimensionless groups that appear in the model include the ratio of material properties in the interior and exterior electrolyte phases for the electrical permittivity, available molar ion concentration, the fluid viscosity, and the ion diffusivities. The typical speed at which ions diffuse across the Debye layers is the velocity scale, which appears in a dimensionless ratio with the interfacial capillary velocity. The main control parameter is the applied voltage, which appears in dimensionless ratio with the thermal voltage and in a second, independent ratio of the scaled electrostatic stress to viscous stress. The numerical implementation of the model and results of the numerical simulations will be presented, together with comparison to a small-deformation analysis and discussion of the dependence of the results on system parameters.