Sedimenting disk arrays: waves and transient growth

We study experimentally the Stokesian sedimentation (Re ∼ 10^-4) of one-dimensional lattice of disks in a quasi two-dimensional geometry with trajectories of disk centres lying in a plane. We induce initial positional perturbations over a uniformly spaced configuration of disks with their separation vectors and normals aligned, and perpendicular to gravity. For various perturbation wavenumbers q and interparticle separations d, we find two classes of behaviour: (i) wavelike propagation of orientations coupled with number-density fluctuations and (ii) clumping instability resembling that of spheres, decorated with orientations. We construct equations of motion using pairwise addition of forces and torques in an array of spheroids. Linear stability analysis of the dispersion relation predicts non-modal transient growth, which we observe experimentally; and a phase boundary in the q-d plane separating neutrally stable and unstable regimes, consistent with our experimental observations. We predict a critical density of the lattice of spheroids below which all wavenumbers are asymptotically linearly stable, showing that orientable particles need not be subject to the inevitable clumping instability of spheres.