Abstract
An array of spheres descending slowly through a viscous fluid always clumps [J.M. Crowley, J. Fluid Mech. {\bf 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 neighbours in a horizontal one-dimensional lattice and settling at Reynolds number $\sim 10^{-4}$ 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 emergent spring stiffness 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 wavenumber and lattice spacing, separating the regimes of algebraically growing waves and clumping, in quantitative agreement with our experiments. Anisotropic shape thus suppresses the classic linear instability of sedimenting sphere arrays, introduces a new conserved variable, and opens a window to the physics of transient growth of linearly stable modes.
Highlights
The collective settling of particles in viscous fluids is a classic and notoriously difficult problem in the physics of strongly interacting driven systems
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
VI we use the mass-and-spring analogy to define a natural inner product with respect to which we show that AðqÞ is non-normal, with physical consequences that we discuss in detail
Summary
The collective settling of particles in viscous fluids is a classic and notoriously difficult problem in the physics of strongly interacting driven systems. How nonspherical shape alters this central and inescapable feature of the sedimentation of sphere arrays We pursue this question experimentally and theoretically through the simple yet unexplored case of a freely sedimenting linear array of orientable apolar particles. Explicit construction of the dynamical equations of motion for Stokesian sedimenting spheroids, at the level of pair hydrodynamic interactions, determines the values of coefficients in our coarse-grained theory, and accounts for the experimentally observed instability boundary in the q − d plane. We observe transient algebraic growth of perturbations in the linearly stable regime in our experiments, and in the numerical solution of the far-field equations We term this growth “nonmodal” since it occurs even when all modes of the dynamical matrix are neutral or decaying [32,33,34,35].
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