Nonuniversal Velocity Fluctuations of Sedimenting Particles

Shang-You Tee, P. J. Mucha, Luca Cipelletti, S. Manley, M. P. Brenner, P. N. Segre, D. A. Weitz
2002 Physical Review Letters  
Velocity fluctuations in sedimentation are studied to investigate the origin of a hypothesized universal scale [P. N. Segre, E. Herbolzheimer, and P. M. Chaikin, Phys. Rev. Lett. 79, 2574 (1997)]. Our experiments show that fluctuations decay continuously in time for sufficiently thick cells, never reaching steady state. Simulations and scaling arguments suggest that the decay arises from increasing vertical stratification of particle concentration due to spreading of the sediment front. The
more » ... ment front. The results suggest that the velocity fluctuations in sedimentation depend sensitively on cell geometry. The slow sedimentation of a dilute suspension of particles through a viscous fluid is an important and fundamental problem in fluid mechanics, impacting processes ranging from formation of geological deposits to removal of contaminants in ground water to centrifugation of proteins. Despite its apparent simplicity, the complexities of hydrodynamic interactions between the particles have provoked a long-standing controversy and sedimentation is still intensely debated to this day [1, 2] . A single spherical falling particle falls at the well-known Stokes velocity v s 2=9a 2 g=, where a is the particle radius, is the density difference, g is the gravitational constant, and is the dynamic viscosity. A uniform concentration of particles sediments more slowly than a single particle because of the confinement of the suspension within the finite cell: fluid must rise as particles sink, and this backflow slows the sedimentation. Remarkably, the average particle velocity is universal, independent of the cell size and shape [3] . However, the long-range interparticle hydrodynamic interactions drive considerable fluctuations of the velocity about its mean. Simple arguments and simulations [4] suggest that the magnitude of the velocity fluctuations should diverge with system size. By contrast, experiments [5] conclude that the fluctuations are independent of the system size; indeed recent experiments [1,6] even argue that they are universal, organized into swirls of characteristic size 15a ÿ1=3 , for solid volume fraction , with characteristic velocity fluctuations scaling like v 1=3 . The simplicity of these experimental results is appealing; however, the origin of this scaling remains a mystery [7-9]. The behavior of velocity fluctuations must be resolved if sedimentation is to be understood in even the most rudimentary way. An important key to understanding this problem is the dependence of the characteristic velocity fluctuations on cell thicknesses [8] . Virtually all experimental evidence is restricted to a narrow range of cell thickness [1,5,6]; thus, a careful investigation over an extended range of thicknesses is essential, both to critically test the universality and to help determine its underlying origin. In this Letter, we present experimental evidence suggesting that the magnitude of the initial fluctuations increases with cell size. However, for sufficiently thick cells the magnitude also decreases over the entire time of the experiment, so that a steady state does not exist. To help understand these surprising results, we perform numerical simulations of up to 10 6 particles, which capture the experimentally observed behavior. This suggests that the decrease of the fluctuations is driven by the broadening of the front [10] separating the suspension from the particle-free fluid above it; the resulting stratification of the particle concentration suppresses large fluctuations. A simple scaling picture quantitatively predicts the magnitude and scale of the velocity fluctuations observed experimentally. The decay in the fluctuations makes it impossible to define a steady-state value except for the thin cells that have been predominantly studied to date [1]. This qualitatively changes the essential experimental observations which define this phenomenon. We measure velocity fluctuations using particle image velocimetry (PIV). Monodisperse glass particles, of radius a 26:5 1:8 m and volume fraction 0:001-0:01, are index matched with a mixture of glycerol and water. The resulting viscosity is measured with a rheometer to range from 10-20 cP. These particles have high Peclet number Pe 7 10 6 and low Reynolds number Re 7 10 ÿ4 , so both Brownian diffusion and inertial effects are negligible. The cell depth, d, is varied between 30a d 500a; the width, w, is always greater than d, while the height, h, is always considerably greater. To suppress temperature fluctuations across the sample, the cell is immersed in a stirred water bath at a temperature of T 22:0 0:1 C. A CCD camera images a region of 1:3 1:8 cm 2 , centered far from the cell bottom and the sediment front. The depth of focus of the lens 0:5 cm so the signal samples particles across the entire cell cross section. Initial particle distributions are prepared by vigorously shaking the cell or by stirring with a rotating blade. We determine the vertical velocity fluctuations, v hv ÿ v sed 2 i 1=2 , where v is the local vertical VOLUME 89, NUMBER 5 P H Y S I C A L
doi:10.1103/physrevlett.89.054501 pmid:12144444 fatcat:hfrauh5lqzfl7cvi723mikwzyi