Transient electrophoresis of a conducting cylindrical colloidal particle suspended in a Brinkman medium

Abstract

The time-dependent electrophoresis of an infinitely cylindrical particle in an electrolyte solution, saturated in a charged porous medium after the sudden application of a transverse or tangential step electric field, is investigated semi-theoretically with an arbitrary double-layer thickness in an arbitrary direction relative to the cylinder. The time-dependent modified Brinkman equation with an electric force term, which governs the fluid flow field, is used to model the porous medium and is solved by using the Laplace transform technique. Explicit formulas, for the time-dependent electrophoretic velocity of the cylindrical particle in Laplace’s transform domain, have been derived for both axially and transversely when the uniform electric fields are imposed. They can also be linearly superimposed for an arbitrarily oriented relative to the electric field. Semi-analytical results for the electrophoretic velocities are presented as functions of the dimensionless elapsed time, the ratio of the particle radius to the Debye length, the particle-to-medium density ratio, and the permeability parameter of the porous medium. The results demonstrate, in general, that the growth of the electrophoretic velocities with the time scale are more slower for high permeability, and the effect of the relaxation time for unsteady electrophoresis is found to be negligible, regardless of the thickness of the double layer, the relative mass density or the permeability of the medium. The normalized transient electrophoretic velocities exhibit a consistent upward trend as the ratio of the particle radius to the Debye screening length increases. Conversely, they display a consistent downward trend as the particle-to-fluid density ratio increases, while all other parameters remain constant. The effect of the relaxation time for the transient electrophoresis is much more important for a cylindrical particle than for a spherical particle due to its smaller specific surface area.