Falling plumes of point particles in viscous fluid

The growth of radial bulges on the conduit of a falling viscous plume of particles, reported by Pignatel et al. for a finite starting plume [F. Pignatel, M. Nicolas, É. Guazzelli, and D. Saintillan, “Falling jets of particles in viscous fluids,” Phys. Fluids 21, 123303 (2009) https://doi.org/10.1063/1.3276235], is investigated both numerically and analytically. As a model for the plume conduit, an infinite vertical cylinder of identical non-Brownian point particles falling under gravity in Stokes flow is considered. Numerically, this is implemented with periodic boundary conditions of a large, but finite, period. The quasi-periodic numerical simulations exhibit qualitatively similar behaviour to that previously observed for the finite plume, demonstrating that neither the plume head nor the plume source play a role in the growth of the radial bulges. This growth is instead shown to be due to fluctuations in the average number density of particles along the plume about its mean value n, which leads to an initial growth rate proportional to n^−1/2. The typical length scale of the bulges, which is of the order of 10 plume radii, results from the particle plume responding most strongly to density fluctuations in the axial direction on this scale. Large radial bulges undergo a nonlinear wave-breaking mechanism, which entrains ambient fluid and reduces the magnitude of perturbations on the plume surface. This contributes towards an outwards diffusion of the plume in which the increase in radius, at sufficiently large times, is proportional to t^2/3.