Particle Sedimentation in a Fluid at Low Reynolds Number: A Generalization of Hindered Settling Described by a Two-Phase Continuum Model

Abstract

Particle-fluid separation by settling is a ubiquitous process in Earth and Planetary Sciences. The settling velocity of particles is controlled by a balance between buoyancy and drag forces. Since the seminal work of George Gabriel Stokes, parameterizations for the reduction of particle velocities caused by viscous dissipation due to mutual interactions have been described by a non-linear mapping between particle volume fraction and separation velocity. We argue that these parameterizations neglect important physical behavior at high particle volume fractions (>80% of the maximum packing) and are only appropriate when considering suspensions where the particle volume fraction does not evolve in space or time. We introduce a more general model that accounts for the energy dissipation caused by changes in local particle volume fraction, which introduces a new term similar to the compaction term at higher particle volume fraction. This term depends on a consolidation/compaction viscosity that measures the resistance to changes in solid volume fraction. We derive closure equations for this compaction viscosity under dilute and concentrated particle volume fraction limits. Numerical simulations show that the extended hindered settling model predicts two significant differences compared to traditional hindered settling. First, while the steepening of particle volume fraction fronts remains, a dynamic instability is also generated at the front. Second, the rate of growth and structure of a cumulate layer growing above a no-flux boundary is affected by the compaction-like term and predicts the trapping of a higher volume fraction of interstitial melt in a correspondingly thicker cumulate layer.