Average pressure and velocity fields in non-uniform suspensions of spheres in Stokes flow

A widely used method for the approximate numerical simulation of the bulk behavior of particle suspensions consists in filling the entire space with copies of a fundamental cell in which N particles are arranged according to some probability distribution. Until now this method has only been used for suspensions that are spatially uniform in the mean. The case of spatially non-uniform systems, on the other hand, has not been considered. Here the average velocity and pressure fields for such a non-uniform suspension of identical rigid spheres in Stokes flow are calculated, and analytic solutions expressed in terms of multipole coefficients are presented. The results match and extend others obtained by the authors in parallel work using a completely different approach. In particular, the definition of a quantity to be identified with the mixture pressure is fully supported by the present results. An explicit result for the structure of the viscous stress in the suspension is also found. It is shown that, for spatially non-uniform systems, the stress contains a non-symmetric contribution analogous to a baroclinic source of vorticity. As a byproduct of the analysis, certain integrals of two periodic functions introduced by Hasimoto are calculated. These integrals would arise in similar problems, e.g. the electric field produced by electric multipoles in a periodic cubic structure.