On the Stokes problem for a suspension of spheres at nonzero concentrations. II. Calculations for effective medium theory

Abstract

We present the detailed calculations for the effective medium approximation to the description of the hydrodynamics of a stationary suspension of randomly distributed, penetrable spheres where the fluid motion is generated by the motion of a single sphere with uniform velocity through the suspension. The formal theory, given in Part I, is reduced to a set of nonlinear integral equations for the concentration dependent effective hydrodynamic interactions in the suspension. The effective medium theory equations are solved approximately by use of two methods. The first is a ’’hydrodynamic’’ approximation which is utilized to evaluate the shear viscosity η, the translational friction coefficient, ζ, and the hydrodynamic screening as expansions in cζ/η. A more general approximation scheme includes the effects of ’’microscopic viscosity’’ and is shown to slightly alter the analytic form of this concentration dependence. Because of this alteration, the method is also applied to suspensions of Brownian spheres, the theory of which is presented elsewhere, to demonstrate that in the mobile case the analytic form is not altered by the inclusion of microscopic viscosity from that generated by the hydrodynamic approximation.