The sedimentation rate of disordered suspensions

Abstract

An explicit expression for the sedimentation velocity at low particle Reynolds number in a concentrated suspension is derived and evaluated for two different approximations to the hydrodynamic interactions: a strict pairwise additive approximation and a far-field, or Rotne–Prager, approximation. It is shown that the simple Rotne–Prager approximation gives a very accurate prediction for the sedimentation velocity of random suspensions from the dilute limit all the way up to close packing. The pairwise additive approximation, however, fails completely, predicting an aphysical negative sedimentation velocity above a volume fraction φ ≈ 0.23. The explanation for these different behaviors is shown to be linked to the "effective medium" behavior of the suspensions. It is shown analytically and by Stokesian dynamics simulation that a suspension of neutrally buoyant particles may be modeled as a homogeneous fluid with an effective viscosity, but a sedimenting suspension cannot. As a result, the Rotne–Prager approximation actually captures the correct features of the many-body interactions in sedimentation. An analytical expression for the sedimentation rate, which is in good agreement with experiment, is obtained using the Percus–Yevick hard-sphere distribution function.