Sedimentation of spheres at small Reynolds number

The effect of fluid inertia on the settling of spheres in a viscous incompressible fluid is studied in the limit of small Reynolds number. The kinetic energy of flow depends on the positions of the spheres, and gives rise to forces on the spheres. In the dilute limit it suffices to study the corresponding pair interaction. The interaction is calculated from the Stokes flow for two spheres settling between plane walls in the point particle limit. The dissipative interaction between a pair of spheres is calculated from the Proudman-Pearson [I. Proudman and J. R. A. Pearson, J. Fluid Mech. 2, 237 (1957)] solution of the Navier-Stokes equations for flow about a sphere in unbounded geometry. The combination of kinetic and dissipative interaction gives rise to a repulsive force of range of the order of the sphere diameter divided by the Reynolds number.