Numerical Simulations of Sedimentation

In a sedimenting suspension, spatial and temporal variations in particle concentration drive large fluctuations in the particle velocities, of the same order of magnitude as the mean settling velocity. For particles larger than about 10 microns, this hydrodynamic diffusion dominates the thermal Brownian motion, and in the absence of inertia (i.e. at Reynolds numbers close to zero), the particle velocities are determined solely by the instantaneous particle positions. The video (7.7 MB MPEG-1) shows the settling of a suspension of ~6000 particles from an initial volume fraction of about 13%; hindered settling, where the velocity decreases with increasing particle concentration, keeps the interface sharp.

It can be seen that there are large fluctuations in the particle velocities during settling. In fact, if the particles are randomly distributed, then the velocity fluctuations are proportional to the increasing container size [1]. However, experimental measurements indicate that the velocity fluctuations converge to a finite value as the container dimensions are increased [2,3], but the mechanism by which the velocity fluctuations saturate is not yet clear. We have recently used computer simulations with similar geometries to those used in laboratory experiments [2,3], namely a rigid container bounded in all three directions. We found that the calculated velocity fluctuations then saturate with increasing container dimensions (Fig 1),as observed experimentally, but contrary to earlier simulations with periodic boundary conditions [4].

The simulations were carried out in a tall column, focusing on a small viewing window located about 1/3 of the way up the column, as shown in the cartoon on the right. The bulk suspension region is shown in orange, the sediment layer is shown in red, and the supernatent fluid is shown in blue; the viewing window is shown in green. We have run simulations with two differnent macroscopic boundary conditions. In the first case periodic conditions are used at the top and bottom boundaries, while in the second case we use impermeable walls. The video (6.1 MB MPEG-1) shows particle motion in the viewing window for the two cases. In both simulations we see that there are clusters of particles streaming down through the system in large, coherent wavy columns. This is a consequence of Poisson statistics in the initial distribution, and we see that, with periodic boundary conditions at the top and bottom (first clip), this condition persists for the duration of the simulation. However, in the second clip these coherent structures are destroyed by interactions with the macroscopic inhomogeneities at the sediment-suspension and suspension-supernatent interfaces. In this case the velocity fluctuations decay in time and fluctuations are independent of system size for sufficiently large systems, as observed experimentally. Thus we conclude that, although random(Poisson) distributions of particles lead to velocity fluctuations that grow in proportion to the system size, the large-scale fluctuations drain out at the sediment-suspension and suspension-supernatent interfaces, leading to the observed time-dependent decay of the velocity fluctuations (Fig 2). These large-scale fluctuations in particle density are introduced by the initial conditions, typically shaking or stirring of the suspension, but once these initial fluctuations drain out, they cannot be replaced at a sufficiently high rate to compensate for their convective decay [4].

Measurements of the fluctuations in particle concentration, S(k) = <nkn-k> support this qualitative picture. In Fig. 3 we show the structure factor, S(k), for density fluctuations in the horizontal plane and compare them with those in the initial, well-mixed suspension. The absence of long-wavelength fluctuations in the settling suspension is consistent with experimental observations of a correlation length and with the saturation of the velocity fluctuations with increasing container size [5].

In summary, our simulations show that the laboratory observations are qualitatively affected by the macroscopic boundaries, and are not representative of the inherent the properties of the suspension, as had been supposed, but incorporate important influencesof the macroscopic boundaries. A complete account of the most recent findings can be found in Ref. [6]

References

  1. R. E. Caflish and J. H. C. Luke, Phys. Fluids, 28:759, 1985.
  2. H. Nicolai and E. Guazzelli, Phys. Fluids, 7:3, 1995.
  3. P. N. Segre, E. Herbolzheimer and P. M. Chaikin, Phys. Rev. Lett., 79:2574, 1997.
  4. A. J. C. Ladd, Phys. Rev. Lett., 88:048301, 2002.
  5. N.-Q. Nguyen and A. J. C. Ladd, Phys. Rev. E., 050401(R), 2004.
  6. N.-Q. Nguyen and A. J. C. Ladd, J. Fluid Mech., 525:73-104, 2005.