The velocity fluctuations present in macroscopically homogeneous suspensions of neutrally buoyant non-Brownian spheres undergoing simple shear flow, and their dependence on the microstructure developed by the suspensions, are investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics simulations. We show that, in the dilute limit, the standard deviation of the velocity fluctuations (the so-called suspension temperature) is proportional to the volume fraction, in both the transverse and the flow directions, and that a theoretical prediction, which considers only the hydrodynamic interactions between isolated pairs of spheres, is in good agreement with the numerical results at low concentrations. We also simulate the velocity fluctuations that would result from a random hard-sphere distribution of spheres in simple shear flow, and thereby investigate the effects of the microstructure on the velocity fluctuations. Analogous results are discussed for the fluctuations in the angular velocity of the suspended spheres. In addition, we present the probability density functions for all the linear and angular velocity components, and for three different concentrations, showing a transition from a Gaussian to an exponential and finally to a stretched exponential functional form as the volume fraction is decreased.
The simulations include a non-hydrodynamic repulsive force between the spheres which, although extremely short range, leads to the development of fore–aft asymmetric distributions for large enough volume fractions, if the range of that force is kept unchanged. On the other hand, we show that, although the pair distribution function recovers its fore–aft symmetry in dilute suspensions, it remains anisotropic and that this anisotropy can be accurately predicted theoretically from the two-sphere solution by assuming the complete absence of any permanent doublets of spheres.
We also present a simple correction to the analysis of laser-Doppler velocimetry measurements, which substantially improves the interpretation of these measurements at low volume fractions even though it involves only the angular velocity of a single sphere in the vorticity direction.
Finally, in an Appendix, we show that, in the dilute limit, the whole velocity autocorrelation function can be predicted using again only two-particle encounters.