Bifurcations in Stokes Flow Sedimentation

TL;DR

This paper investigates how slight offsets in particle center of mass affect sedimentation dynamics at low Reynolds number, revealing bifurcations between complex and simple behaviors through experiments and simulations.

Contribution

It introduces a unifying framework for understanding sedimentation bifurcations, combining experimental measurements and immersed boundary simulations to identify key dynamical transitions.

Findings

Small center of mass offsets (<1%) cause bifurcations from complex to simple sedimentation.

Simulations reveal an alignment bifurcation surface separating different dynamical regimes.

Symmetry considerations explain the emergence of closed orbits and limit cycles.

Abstract

Particles whose shapes couple translation to rotation display a rich array of behaviors as they sediment at low Reynolds number. We introduce a unifying perspective in which the possible dynamical regimes and bifurcations between them can be understood. We use experimental measurements of helical ribbons, with controlled center of mass offsets, to identify the key bifurcation from complex dynamics to a single attracting state as the magnitude of the offset increases. The sedimentation dynamics are very sensitive to small center of mass offsets, with the bifurcation occurring for offsets less than one percent of the particle length. Using mobility tensors obtained from immersed boundary method simulations, we simulate helical particle sedimentation and identify an alignment bifurcation surface, defined in the three dimensional space of center of mass offsets, that separates simple from complex sedimentation dynamics. Inside this surface we find limit cycles which emerge through Hopf and homoclinic bifurcations. Cocentered particles with coincident centers of force and mobility provide a reference case at the center of the bifurcation surface. We show how the geometric and dynamical symmetries of sedimenting cocentered particles are broken as the center of force offset moves away from the cocentered case. Three parity time-reversal (PT) symmetries exist for all cocentered particles under reflections normal to the eigenvectors of its translation-rotation coupling tensor. When a center of force offset preserves at least one of these PT symmetries, then there are closed orbits for particles inside the alignment bifurcation surface.