Dimitry I. Kopelevich and Hsueh-Chia Chang

The statistics of sedimenting particles in a viscous fluid remains an open problem. The classical theory of Batchelor (1972) assumes a homogeneous spatial distribution to obtain the average particle sedimentation velocity. In his theory, the diverging volume integral of the Stokes force exerted by the particles on the fluid is made to converge in an ad hoc manner that may explain the significant deviation between the theoretical and observed velocities. Using the same homogeneous assumption, Caflisch and Luke (1985) showed that the variance of the velocity distribution diverges with increasing system size, viz. 1/3 power with respect to the particle number. Koch and Shaqfeh (1991) suggested that a screening effect must be introduced to break the homogeneous particle distribution to obtain finite statistics, but their theoretical scaling law failed to agree with experiments.

Recent experimental measurements by Segre, Herbolzheimer and Chaikin (1997) and by Nikolai and Guazzelli (1995) indicate that the variance is definitely finite and for small containers, dependent on the container size. This led to a new theory by Bruneau, Feuillebois, Blawzdziewicz and Anthore (1998) and scaling arguments by Brenner (1999) that showed that wall effects can yield finite velocity variance. However, the measured velocity variance remains finite for large containers when wall effects and container size are not important. Furthermore, experiments of Segre et al. show that there exist small vortices of particle concentration. The size of these vortices is independent of box size for sufficiently large boxes. These observations cannot be explained by the new wall-effect theories.

We report a new theory that explicitly accounts for inhomogeneous distribution due to screening effects, the vortex instability, the finite variance for particle velocity and a new average sedimentation velocity that is in better agreement with experimental data. The sedimenting suspension is described by Stokes equations for the fluid dynamics and a Liouville equation for the dynamics of point particles. This coupled set of nonlinear equations allows us to capture the effect of particle distribution on the fluid velocity and vice versa. It represents a rigorous mean-field approach. We solve these equations in the limit of negligible particle inertia and find that the homogeneous distribution is unstable. This instability results from the screening effect of coherent particle structures which introduces a positive feedback through the fluid velocity that further promotes inhomogeneous particle distribution. After an initial growth, a vortex with specific intensity and dimension (roughly 20 to 30 mean interparticle spacings) is selected. Hence, this vortex instability breaks the uncorrelated particle position of a homogeneous distribution and thus suppresses the divergence of velocity variance. We estimate the size and intensity of the vortices through bifurcation analysis of our equations. They are found to be consistent with those observed by Serge et al.