Viscosity-induced instability of a one-dimensional lattice of falling spheres

Abstract

When a layer of particles moves through a viscous liquid it experiences forces which tend to disrupt the layer into clusters of particles separated by open channels. A theoretical description of this process is presented and a viscous instability is predicted. The spatial growth of the instability is approximated by eγz, where \[ \gamma = {\textstyle\frac{3}{2}} a/d^2, \] where a is the particle radius and d is the average distance between particles. This result implies that any initial irregularity in a uniform particle distribution will be amplified by viscous forces alone. Significant amplification will occur when the particle has drifted a small multiple of the separation distance, if this separation is not much greater than the particle diameter. Thus, any initially uniform particle layer will form clusters as it drifts through a viscous fluid. The distance in which this clustering occurs will be unaffected by changes in the particle velocity, as long as the Reynolds number remains small. The preferred form of irregularity will consist of small clusters separated by individual particles which trail some distance behind. Experimental verification of these conclusions is presented.

Universal spatiotemporal scaling of distortions in a drifting lattice

“…(4,5) at the linear level by defining the fields φ ± = u x ± νu z where ν = λ 2 /λ 3 . In terms of φ ± , they becomė…”

“…The study of drifting lattices began with the work of Crowley in 1971 [4] who predicted that an array of particles moving through a viscous fluid is unstable to clumping due to hydrodynamic forces alone, a result he verified experimentally by dropping steel balls into turpentine oil. The role of elastic and Brownian forces on this lattice instability was analysed by Lahiri and Ramaswamy in 1997 [3] .…”

Self-organizing microfluidic crystals

“…For instance, a lattice might be linearly unstable, subject to a clumping instability resembling that found for a row of sedimenting spheres. 23,24 Even if a lattice is linearly stable, its basin of attraction might not be signicant if the dynamics of the system are multistable. 25 Addressing these questions requires a quantitative approach.…”

Dynamic Theory of Suspensions with Brownian Effects

“…For the very simple problem of sedimentation through a stationary fluid, Crowley [10], [11], [12] has shown experimentally and analytically that a periodic array of spheres is unstable. Also, Siano [25] has demonstrated experimentally that a spatially uniform distribution of sedimenting particles is unstable for some range of parameters.…”

“…Alternatively one can apply a Kramers-Smoluchovski limit at the N-particle level ( 8) and then a mean field limit afterward resulting in (9.5). 10 A particle undergoes velocity diffusion due to many collisions in each of which it changes its velocity slightly. It thus moves chaotically through velocity space, but smoothly through physical space.…”