Bifurcations in Stokes Flow Sedimentation

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.

Figures (11)

• Figure 1: Spatial trajectories of five helical ribbons with identical shape and initial orientation. Panels show (a) top and (b--c) side views, with trajectories distinguished by color. The red trajectory is that of an idealized cocentered particle (of length $L$), while the rest have a center of mass offset of $0.002L$ in 4 randomly chosen directions. All particles are initialized with $\hat{\mathbf y}$ aligned with gravity, corresponding to Euler angles $(\theta,\psi)=(0,\pi/2)$.
• Figure 2: Orientation dynamics of a cocentered particle. (a) gravity vector trajectories in the particle's body coordinate system (b) Euler angles of tilt ($\theta$) and spin ($\psi$). Yellow circles are centers. Green diamonds are saddles. Planes of parity-time reversal symmetry are labeled according to their normal vector in body coordinates, which is the center of mass offset direction under which the symmetry is preserved: (green) $\hat{x}$ plane, (blue) $\hat{y}$ plane, (red) $\hat{z}$ plane. In the phase plane, yellow bars at $\theta=0$ and $\theta=\pi$ correspond to the centers along $\hat{z}$ which are stretched into lines by the mapping.
• Figure 3: Helical ribbons with holes where metal spheres are epoxied to move the center of mass (COM) along principal axes. (a) COM offset in $x$ direction which is the intermediate axis (labeled by eigenvalues of $\mathbb{b}_m$). (b) COM offset in $y$ direction which is the minor axis. (c) COM offset in $z$ direction which is the major axis.
• Figure 4: Increasing $y$ (minor axis) offset of the center of mass. (a) through (e) show experimental phase diagrams as the center of mass moves along the $y$ axis. (f) shows a bifurcation diagram of the tilt ($\theta$) position of the fixed points from numerical simulations. Saddles are green and centers are yellow. Vertical lines in (f) show the center of mass offset of each experimental data set.
• Figure 5: Increasing $z$ offset of the center of mass. (a) through (e) show experimental phase diagrams as the center of mass moves along the $z$ axis. (f) shows a bifurcation diagram of the spin ($\psi$) position of the fixed points from numerical simulations. Saddles are green and centers are yellow. Vertical lines in (f) show the center of mass offset of each experimental data set.