Moving finite-size particles in a flow: A physical example of pitchfork bifurcations of tori

Abstract

The motion of small, spherical particles of finite size in fluid flows at low Reynolds numbers is described by the strongly nonlinear Maxey-Riley equations. Due to the Stokes drag, the particle motion is dissipative, giving rise to the possibility of attractors in phase space. We investigate the case of an infinite cellular flow field with time-periodic forcing. The dynamics of this system are studied in a part of the parameter space. We focus particularly on the size of the particles, whose variations are most important in active physical processes, for example, for aggregation and fragmentation of particles. Depending on their size the particles will settle on different attractors in phase space in the long-term limit, corresponding to periodic, quasiperiodic, or chaotic motion. One of the invariant sets that can be observed in a large part of this parameter region is a quasiperiodic motion in the form of a torus. We identify some of the bifurcations that these tori undergo, as particle size and mass ratio relative to the fluid are varied. In this way we provide a physical example for sub- and supercritical pitchfork bifurcations of tori.