Scaling of transient hydrodynamic interactions in hard sphere suspensions

Abstract

Analyses of retarded hydrodynamic interactions between pairs of spheres, computed in Fourier space over the full frequency range, have been performed to investigate scaling of the time-dependent self-diffusion coefficient ${D}_{s}(t,\ensuremath{\varphi}).$ It appears that up to intermediate volume fractions $(\ensuremath{\varphi}l~0.15)$ ${D}_{s}(t,\ensuremath{\varphi})$ shows scaling behavior when both the characteristic time \ensuremath{\tau} is appropriately rescaled and ${D}_{s}(t,\ensuremath{\varphi})$ is normalized by the short-time self-diffusion coefficient of the suspension ${D}_{s}{=D}_{0}(1\ensuremath{-}1.83\ensuremath{\varphi}).$ The rescaled characteristic time is based on matching of the long-time tail of the velocity autocorrelation function with the single-particle result. Scaling is observed for a range of particle to fluid density ratios (for $0l~\ensuremath{\sigma}l~2,$ $\ensuremath{\sigma}={\ensuremath{\rho}}_{s}/\ensuremath{\rho},$ with ${\ensuremath{\rho}}_{s}$ the particle density and \ensuremath{\rho} the fluid mass density). Scaling for higher volume fractions, which is already present when the characteristic time is computed by optimal fitting, might be improved by including three-particle hydrodynamic interactions. The present results support the conclusion that modification of correlation functions in hard sphere suspensions, in order to include effects of two-particle hydrodynamic interactions, is already sufficient to show the existence of scaling of ${D}_{s}(t,\ensuremath{\varphi}).$