The viscosity of suspensions — an effective medium approach that incorporates Brownian motion

By an extension of the hydrodynamic equivalent to the Clausius-Mossotti (CM) and Bruggeman (BM) formulas in dielectrics, new theories for the viscosity of a fluid suspension are constructed that include the effects due to Brownian motion hitherto ignored in previous works using the effective medium (EM) approach. This entails knowledge of the averaged dipole polarisability due to the hydrodynamic interaction of (at least) a two-sphere system. When incorporated into the theories, the resultant viscosity of the suspension is clearly temperature dependent. A requirement, desired in this work, is that in the limits of high and low Péclet numbers (Pe = Eija2 / D0), the theories reproduce the well-known (exact) and (semi-exact) results for the low density expansions due to Einstein (1905, 1911) and Batchelor (1977) respectively. We demonstrate this explicitly for the CM case. Our numerical estimates also indicate that this requirement can only be met in the BM case, provided that two-body corrections are included symmetrically in the theory. In the latter, a coupling of the two-particle distribution function and the EM equation results, with the natural consequence that the renormalised Péclet number (Peφ = Eija2 / D(φ)) now becomes the relevant parameter (whose value must be determined self-consistently). The near coincidence of the second order coefficient at low Péclet number with the low density expansion of the BM formula (without Brownian motion) is further discussed, together with a comparison with experiments, computer simulations and more recent studies.