Convergence of multiple scattering series for two-body hydrodynamic effects on shear viscosity of suspensions of spheres

An efficient computational scheme has been developed that includes many-body hydrodynamics for viscosity in a suspension of spheres following the approach of Jones and Muthukumar [J. Chem. Phys. 89, 6406 (1988)] for friction coefficients. The method combines the hydrodynamic scattering formalism of Freed and Muthukumar [J. Chem. Phys. 76, 6186 (1982)] with angular momentum diagrammatic techniques from quantum mechanics. The concentration dependence of the viscosity is expressed as a series of finite integrals labeled by the equivalent of angular momentum quantum numbers. This series is smoothly convergent in contrast to the case of the alternate formalism of a series of terms of inverse powers of the sphere separation. A comparison between different methods and the slowness of convergence of the series are addressed in detail. At the two-body level, the virial expansion for the shear viscosity η of a suspension of hard spheres of volume fraction Φ is determined to be η/η0=1+2.5Φ+4.8292Φ2+O(Φ3), where η0 is the viscosity of the fluid and the correction for the Brownian motion [G. K. Batchelor, J. Fluid Mech. 83, 97 (1977)] has been ignored. Our value for the two-body coefficient differs in 3.4% from the extrapolated value 5.0019 obtained by Cichocki and Felderhof [J. Chem. Phys. 89, 1049 (1988)].