Sedimentation in a dispersion with vertical inhomogeneities

Abstract

Statistical studies of hydrodynamic interactions between many particles in a dilute dispersion raise a problem of divergent integrals. This problem arises in particular when calculating the average velocity of sedimentation of solid spheres in a viscous fluid. The solution to this problem was given by Batchelor (1972) for monodisperse suspensions of spheres, on the basis of an assumption of homogeneity. This assumption is removed here. The problem of divergent integrals is reconsidered. The solution treats as successive steps: the average flow due to random statistically independent point forces; the average flow due to random statistically independent solid spheres, without hydrodynamic interactions; the average sedimentation velocity of random, pairwise-dependent solid spheres with hydrodynamic interactions, in a dilute suspension. Considering the case of identical spheres, and assuming homogeneity in any horizontal plane, an expression is obtained for the average sedimentation velocity of a sphere in an otherwise inhomogeneous dispersion. The formula is written in terms of integrals involving probability distributions. It reduces, when the suspension is homogeneous, to a formula obtained by Batchelor.The probability distributions are not calculated in this paper. In order to evaluate numerically the average velocity of sedimentation, a simple expression for the pair distribution function is assumed, and two different concentration profiles are considered, viz. a sinusoidal variation and a step function. In the case of sinusoidal concentration wave, it is found that the contribution of the inhomogeneity is, for small wavelengths, comparable in magnitude to that calculated for a homogeneous dispersion by Batchelor (1972), i.e. –6.55c.The difference in velocity between the crest and trough of the wave is an increasing function of the wavelength. For a step function in concentration, particles at the top of the cloud start to fall faster, this effect being limited to a top layer about 10 radii thick.For future study of the long-term behaviour of a sedimenting cloud, the evolution of the pair distribution function should be added to the present theory.