Effective viscosity of a periodic suspension

Abstract

The effective viscosity of a suspension is defined to be the four-tensor that relates the average deviatoric stress to the average rate of strain. We determine the effective viscosity of an array of spheres centred on the points of a periodic lattice in an incompressible Newtonian fluid. The formulation involves the traction exerted on a single sphere by the fluid, and an integral equation for this traction is derived. For lattices with cubic symmetry the effective viscosity tensor involves just two parameters. They are computed numerically for simple, body-centred and face-centred cubic lattices of spheres with solute concentrations up to 90% of the close-packing concentration. Asymptotic results for high concentrations are obtained for arbitrary lattice geometries, and found to be in good agreement with the numerical results for cubic lattices. The low-concentration asymptotic expansions of Zuzovsky also agree well with the numerical results.