Time-Reversibility and Particle Sedimentation

Abstract

This paper studies an ordinary differential equation (ODE) model, called the Stokeslet model, and describes sedimentation of small clusters of particles in a highly viscous fluid. This model has a trivial solution in which the n particles arrange themselves at the vertices of a regular n-sided polygon. When $n = 3$, Hocking [J. Fluid Mech., 20 (1964), pp. 129–139] and Caflisch et al. [Phys. Fluids, 31 (1988), pp. 3175–3179] prove the existence of periodic motion (in the frame moving with the center of gravity in the cluster) in which the particles form an isosceles triangle. The study of periodic and quasiperiodic solutions of the Stokeslet model is continued, with emphasis on the spatial and time reversal symmetry of the model (time reversibility is due to infinite viscosity and spatial (dihedral) symmetry is due to the assumption of identical particles and the symmetry of the trivial solution). For three particles, the existence of a second family of periodic solutions and a family of quasiperiodic solutions is proved. It is also indicated how the methods generalize to the case of n particles.