Moving Finite Size Particles in a Flow: A Physical Example for Pitchfork Bifurcations of Tori

TL;DR

This paper explores how finite-sized particles in a fluid flow exhibit complex long-term behaviors, including bifurcations of tori, depending on particle size and system parameters, providing a physical example of pitchfork bifurcations.

Contribution

It demonstrates the occurrence of pitchfork bifurcations of tori in the dynamics of finite-size particles in fluid flows, linking bifurcation theory with physical particle behavior.

Findings

Particles settle on different attractors based on size.

Identification of bifurcations of tori as parameters vary.

Physical example of pitchfork bifurcations in dynamical systems.

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 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. This way we provide a physical example for sub- and supercritical pitchfork bifurcations of tori.