M. Golubitsky, M. Krupa and C. Lim
Time-reversibility and particle sedimentation
SIAM J. Appl. Math.
51 (1) (1991) 49-72.
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.