Hydrodynamic diffusion in dilute sedimenting suspensions at moderate Reynolds numbers

Abstract

The hydrodynamic diffusion of particles or bubbles sedimenting in a liquid is considered for moderate particle Reynolds numbers, Re∼O(1), and small particle volume fractions, φ≪1. Here, Re≡ρUta/μ, Ut is the mean terminal velocity of the particles, a is their mean radius, and ρ and μ are the density and viscosity of the fluid. The interactions that control the diffusion occur when one particle is in the wake of another particle with a separation that is large compared with the particle radius a. The conditionally averaged velocity disturbance in the wake of a particle is screened at an O(aφ−1) distance behind the particle. Beyond this distance, the wake spreads more quickly because of a deficit of particles in the center and an excess at the periphery of the wake of a test particle. This structure is produced by lift forces, which tend to resist the advection of particles into the wake. Screening is required in order to obtain finite values of the variances of the vertical velocities of the particles and fluid. The particle diffusion is anomalous even at wavelengths larger than the screening length; the mean-square displacements of the particles in the vertical and horizontal directions grow in proportion to t3/2 and t ln2 t, respectively. The calculations are carried out for a suspension with a small degree of polydispersity.