Dynamics and clustering of sedimenting disc lattices

Uniform arrays of particles tend to cluster as they sediment in viscous fluids. Shape anisotropy of the particles enriches these dynamics by modifying the mode-structure and the resulting instabilities of the array. A one-dimensional lattice of sedimenting spheroids in the Stokesian regime displays either an exponential or an algebraic rate of clustering depending on the initial lattice spacing (Chajwa et al. 2020). This is caused by an interplay between the Crowley mechanism which promotes clumping, and a shape-induced drift mechanism which subdues it. We theoretically and experimentally investigate the sedimentation dynamics of one-dimensional lattices of oblate spheroids or discs and show a stark difference in clustering behaviour: the Crowley mechanism results in clumps comprised of several spheroids, whereas the drift mechanism results in pairs of spheroids whose asymptotic behavior is determined by pair-hydrodynamic interactions. We find that a Stokeslet, or point-particle, approximation is insufficient to accurately describe the instability and that the corrections provided by the first-reflection are necessary for obtaining some crucial dynamical features. As opposed to a sharp boundary between exponential growth and neutral eigenvalues under the Stokeslet approximation, the first-reflection correction leads to exponential growth for all initial perturbations, but far more rapid algebraic growth than exponential growth at large lattice spacing $d$. For discs with aspect ratio 0.125, corresponding to the experimental value, the instability growth rate is found to decrease with increasing lattice spacing $d$, approximately as $d^{-4.5}$, which is faster than the $d^{-2}$ for spheres (Crowley, 1971). Sedimenting pairs predominantly come together to form '$\perp$', which our theory accounts for through an analysis that builds on Koch & Shaqfeh (1989).