Abstract
An array of spheres descending slowly through a viscous fluid always clumps [J. M. Crowley, J. Fluid Mech. 45, 151 (1971)]. 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 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.
- Received 12 February 2020
- Revised 3 August 2020
- Accepted 25 August 2020
DOI:https://doi.org/10.1103/PhysRevX.10.041016
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
A landmark in the study of particles sedimenting in a viscous fluid dates to the nineteenth century, when George Stokes derived an expression for the settling velocity of a sphere. The collective sedimentation of many particles, however, is a complex problem because each particle affects the motion of other particles via the long-range flow it sets up. This problem has led to many debates and insights into the many-body physics of systems with long-range interactions. In most natural settings, the sedimenting particles are nonspherical, adding further richness to this problem because both their orientation and relative positions are relevant to their interactions. Our theory and experiments on arrays of coinlike objects falling in a viscous fluid highlight a fundamental difference between the settling dynamics of spheres and orientable objects.
A linear array of spheres is known to be unstable—particles clump and draw in lagging neighbors as they fall, collecting in initially denser regions. Our studies on small disks released in a regular array show a competing effect: The orientation of disks can make them glide away from each other, which counteracts their tendency to clump. Our experiments reveal this process as a graceful wave motion for oriented shapes. Our theoretical analysis shows that this wave formally resembles elastic waves in a system of masses connected by springs. The waves are ultimately disrupted by an even more interesting mechanism: Perturbations do not grow exponentially as in a typical unstable system, but more delicately, as a power of time, making this instability mathematically distinct from that of spheres.
Nonspherical sedimenting particles are abundant in marine snow and elsewhere. An understanding of their collective sedimentation—crucial, for example, to global carbon sequestration—must take into account our central results.
