The effective shear viscosity for two-phase flow

Abstract

A formula is derived for the effective shear viscosity of two-phase flow for arbitrary volume fractions of the two phases. Using a background medium viscosity one may make a distinction between the local velocity field and the total velocity field. The local velocity field in a fluid element is due to the shear pressure in other fluid elements plus the velocity field due to external forces. An explicit expression for this local field is given. Analogous to the polarizability in a dielectric one may define a pressurizability of a fluid element giving the shear pressure in the fluid element in terms of the local field. Neglecting correlations between different fluid elements a formula is found for the effective viscosity. An appropriate choice of the background viscosity then leads to a formula for the effective viscosity analogous to Bruggeman's formula for the dielectric constant of a two-phase material. This gives reason to expect that the approximation will be good for arbitrary values of the volume fractions. If the viscosity of one of the phases is taken infinite one may compare with results for suspensions of hard spheres. The effective viscosity is found to diverge at a volume fraction below close packing. Experimental results suggest such a divergence, and agree reasonably well with the formula given. For small volume fractions our result reduces to Einstein's formula.