The weakly inertial settling of particles in a viscous fluid

In this paper we investigate the influence of fluid inertia on the settling of a finite assemblage of solid spherical particles in a constant gravity field at small Reynolds number, Re. We show that the first effect of fluid inertia on particle velocities scales as Re, for times much larger than the viscous-relaxation time. In this case the Eulerian acceleration terms associated with the unsteadiness of the stresslet in the far velocity field and the entire local fluid inertia (acceleration and advective terms) contribute at O(Re). As a particular example, Oseen velocities are calculated of two spheres falling along the line of their centres. The inertia-induced relative motion between the particles is in excellent agreement with previous experimental results.