Differential multiphase models for polydispersed suspensions and particulate solids

We report some novel applications of the differential scheme to construct the effective viscosity and moduli of suspensions of spheres of diverse sizes. Exact closed form expression for the effective viscosity of droplets of diverse sizes is derived, which includes the generalised Einstein formula of Brinkman and Roscoe. Exact closed form solutions for the effective moduli of a particulate solid consisting of rigid spheres of diverse sizes embedded in an elastic solid shows that the Poisson's ratio of the composite should approach the value 1/5 at high volume fraction. The effect of a limited range of sizes of the included phase and locally-inhomogeneous distribution of inclusions is taken into account by considering a three-component system, with a fictitious inclusion phase made up of the matrix (solvent) material, which limits the space available for the real inclusions. (C) 1997 Elsevier Science B.V.