Sedimentation Dynamics of Helical Ribbons

We consider the dynamics of helical ribbons sedimenting at low Reynolds number. Experimentally, we use images from three cameras to reconstruct 3D dynamics with high spatiotemporal resolution. Theoretically, we build ribbons out of hydrodynamically interacting spheres and compute resistance tensor components using the moment expansion of Durlovsky et al. We observe that the typical trajectories of these particles are periodic orbits in orientation space. A simplified picture is that these oscillations occur because the center of projected area shifts relative to the center of mass as the particle spins about its symmetry axis. Mathematically, the translation-rotation mobility tensor has three eigenvalues and when all three are distinct, there are stable and unstable orientations with associated periodic oscillations. We find there are special chiral particles with two degenerate eigenvalues where the stable and unstable points disappear and sedimentation is steady. For our chiral ribbon shape, there is a sequence of these axisymmetric helicoids with steady motion at discrete values of ribbon length.