Waves, Algebraic Growth, and Clumping in Sedimenting Disk Arrays

Abstract

An array of spheres descending slowly through a viscous fluid always clumps [J. M. Crowley, J. Fluid Mech. 45, 151 (1971)JFLSA70022-112010.1017/S0022112071003045]. We show that anisotropic particle shape qualitatively transforms this iconic instability of collective sedimentation. In experiment and theory on disks, aligned facing their neighbors in a horizontal one-dimensional lattice and settling at Reynolds number ∼10^{-4} in a quasi-two-dimensional slab geometry, we find that for large enough lattice spacing the coupling of disk orientation and translation rescues the array from the clumping instability. Despite the absence of inertia, the resulting dynamics displays the wavelike excitations of a mass-and-spring array, with a conserved “momentum” in the form of the collective tilt of the disks and an effective spring stiffness emerging from the viscous hydrodynamic interaction. However, the non-normal character of the dynamical matrix leads to algebraic growth of perturbations even in the linearly stable regime. Stability analysis demarcates a phase boundary in the plane of wave number and lattice spacing, separating the regimes of algebraically growing waves and clumping, in quantitative agreement with our experiments. Through the use of particle shape to suppress a classic sedimentation instability, our work uncovers an unexpected conservation law and hidden Hamiltonian dynamics which in turn open a window to the physics of transient growth of linearly stable modes.