| The sedimentation of solid spheres in a fluid is a classical problem in fluid mechanics. The long-range interparticle hydrodynamic interactions drive considerable fluctuations of the velocity about its mean and these fluctuations play an essential role in the behavior of a sedimenting suspension. Caflisch and Luke calculated that these fluctuations should diverge with system size.; We investigate the velocity fluctuations as a function of cell thickness; we find that the magnitude of the initial fluctuations does increase with cell size. However, for sufficiently thick cells the magnitude also decreases over the entire time of the experiment. This decay makes it impossible to define a steady-state value except for the thin cells that have been predominantly studied to date. We conclude that the decreasing fluctuations are driven by the broadening of the front which separates the suspension from the particle-free fluid above it, inducing a vertical stratification of the particle concentration which suppresses large fluctuations. We develop a simple scaling picture that quantitatively accounts for the time dependent behavior observed for thick cells.; To understand sedimentation it is essential to measure velocity fluctuations under steady state conditions. It is better to switch to a different frame of reference by using a fluidized bed where liquid is pumped upwards through the particles to create a drag force that exactly counteracts gravity. We show that when the bed is in steady state, the local volume fractions, correlation lengths and velocity fluctuations vary as functions of bed height. A simple model quantitatively predicts the volume fraction as a function of height when segregation effects dominate while a physical scaling argument augments this model when mixing effect becomes important. Despite these complications, the local velocity fluctuations scale in a simple way with correlation lengths and volume fractions.; In conclusion, velocity fluctuation in both sedimentation and fluidization is not "universal." The correlation lengths and magnitude of the fluctuations depend strongly on particle polydispersities, container size, particle concentration gradient, etc. However, locally everywhere, a simple scaling relationship holds for the velocity fluctuations, correlation lengths and volume fractions. |
