On the Stokes problem for a suspension of spheres at finite concentrations

We consider the problem of the evaluation of the ensemble averaged translational friction coefficient of, and the velocity field produced by, a single nonrotating sphere, moving with uniform velocity through a stationary suspension of similar spheres at finite concentrations, where the average is performed over the spatial distribution of the remaining spheres. The coupled equations for the forces exerted by the spheres on the fluid are iterated to provide a multiple scattering representation of the friction coefficient and velocity field. This expansion can be viewed as an expansion in powers of the concentration, and all terms beyond the trivial leading one are shown to involve divergent integrals because of the long range nature of hydrodynamic disturbances. However, the integrands are shown to form simple geometric series which produce absolutely convergent integrals after the summation. The result displays the phenomenon of hydrodynamic screening (providing a derivation of Darcy’s law) that has been utilized to explain the concentration dependence of the hydrodynamics of polymer solutions. An effective medium approach is presented which incorporates the hydrodynamic screening in lowest order, leading to a theory which is free of divergent integrals.