The Oseen problem for a finite collection of spheres settling in a viscous liquid

Abstract

In this paper, we investigate the influence of fluid inertia on the dynamics of the finite collection of rigid spherical particles falling through a viscous liquid in a constant gravity field at small Reynolds numbers. The matched asymptotic expansion combined with the numerical solution of the corresponding many-particle Oseen problem based on the multipole expansion of Lamb’s spherical harmonic solution is exploited to compute the inertial correction to the individual speeds of the spheres. For regular horizontal polygons of spheres of radii a , the expansion rate of the polygon is studied as a function of the number of particles N p and the edge length s / a . For a fixed edge length, the expansion rate is found to be a monotonically increasing function of N p ; for a fixed N p the expansion rate has a minimum at s / a ∼ 5 . The prediction of the leading effect of fluid inertia on the velocity of the particle in the assemblage conforms well to Brenner (1961) [22], Brenner–Cox (1963) [23] and Bretherton (1964) [15] theories. The anticipated Oseen velocities of the two spheres settling in tandem and side-by-side configurations are in excellent agreement with the results of recent experiments (Vanroyen et al. 2005 [26]).