Start‐up Brinkman electrophoresis of a dielectric sphere for Happel and Kuwabara models

Abstract

The starting electrophoresis in a charged porous continuum response of a homogeneous suspension of spherical particles to the sudden application of an external electric field is analyzed semianalytically through the use of a unit cell model. The electric double layer in the porous Brinkman medium surrounding each sphere is assumed to be thin but finite, and the effect of dynamic electroosmosis within it is included. The field equation for the fluid outside the double layers is solved based on the unit cell model under the start‐up Brinkman model. Expressions of the time‐dependent electrophoretic and settling velocities of the particle in the Laplace transform as functions of the permeability, the relative mass density, the electrokinetic particle radius, and the volume fraction of the particles are performed for two different boundary conditions at the fictitious surface of the cell. Numerical results indicate that the timescale for the development of electrophoresis and sedimentation is small for a high permeability, an assemblage of spheres through a dielectric porous medium with a higher particle volume fraction and a smaller particle to porous fluid density ratio; the electrophoretic mobility is a monotonically increasing function of the electrokinetic radius of the sphere at any instant. The start‐up of electrophoretic mobility decreases with an increase in the particle volume fraction at a small value of the particle‐to‐medium density ratio, but it may increase as the particle volume fraction increases at a large value of this density. The particle interaction effect in a swarm on the time‐dependent electrophoresis is much smaller than that on the starting sedimentation of the particles. The detailed comparison of the results of the cell model with different boundary conditions at the virtual surface of the cell is obtained and illustrated graphically.