Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory

TLDR: In this paper, Batchelor et al. derived formulae for the mean velocity of the particles of each species correct to order ϕ, that is, with allowance for the effect of pair interactions.

Abstract: Small rigid spherical partials are settling under gravity through Newtonian fluid, and the volume fraction of the particles (ϕ) is small although sufficiently large for the effects of interactions between pairs of particles to be significant. Two neighbouring particles interact both hydrodynamically (with low-Reynolds-number flow about each particle) and through the exertion of a mutual force of molecular or electrical origin which is mainly repulsive; and they also diffuse relatively to each other by Brownian motion. The dispersion contains several species of particle which differ in radius and density.The purpose of the paper is to derive formulae for the mean velocity of the particles of each species correct to order ϕ, that is, with allowance for the effect of pair interactions. The method devised for the calculation of the mean velocity in a monodisperse system (Batchelor 1972) is first generalized to give the mean additional velocity of a particle of species i due to the presence of a particle of species j in terms of the pair mobility functions and the probability distribution pii(r) for the relative position of an i and a j particle. The second step is to determine pij(r) from a differential equation of Fokker-Planck type representing the effects of relative motion of the two particles due to gravity, the interparticle force, and Brownian diffusion. The solution of this equation is investigated for a range of special conditions, including large values of the Peclet number (negligible effect of Brownian motion); small values of the Ptclet number; and extreme values of the ratio of the radii of the two spheres. There are found to be three different limits for pij(r) corresponding to different ways of approaching the state of equal sphere radii, equal sphere densities, and zero Brownian relative diffusivity.Consideration of the effect of relative diffusion on the pair-distribution function shows the existence of an effective interactive force between the two particles and consequently a contribution to the mean velocity of the particles of each species. The direct contributions to the mean velocity of particles of one species due to Brownian diffusion and to the interparticle force are non-zero whenever the pair-distribution function is non-isotropic, that is, at all except large values of the Peclet number.The forms taken by the expression for the mean velocity of the particles of one species in the various cases listed above are examined. Numerical values will be presented in Part 2.