U-shaped disks in Stokes flow: chiral sedimentation of a non-chiral particle

We study the sedimentation of U-shaped circular disks in the Stokes limit of vanishing inertia. We simulate the flow past such disks using a finite-element-based solution of the three-dimensional Stokes equations, accounting for the integrable singularities that develop along their edges. We show that the purely vertical sedimentation of such disks in their upright (upside-down) U orientation is unstable to perturbations about their pitching (rolling) axes. The instability is found to depend only weakly on the size of the container in which the disks sediment, allowing us to analyse their behaviour based on the resistance matrix which governs the evolution of the disk's six rigid-body degrees of freedom in an unbounded fluid. We show that the governing equations can be reduced to two ordinary differential equations which describe the disk's inclination against the direction of gravity. A phase-plane analysis, the results of which are in good agreement with experiments, reveals that the two instabilities generally cause the disk to sediment along complex spiral trajectories while it alternates between pitching- and rolling-dominated motions. The chirality of the trajectories is set by the initial conditions rather than the (non-chiral) shape of the disk. For certain initial orientations, the disk retains its inclination and sediments along a perfectly helical path. The observed behaviour is fundamentally different from that displayed by flat circular disks which sediment without any reorientation. We therefore study the effect of variations in the disk's curvature to show how in the limit of vanishing curvature the behaviour of a flat disk is recovered.