A three-phase mixture theory for particle size segregation in shallow granular free-surface flows

Abstract

Particle-size segregation within granular materials is of great technological significance yet it is still very poorly understood. There are several causes for segregation, but this paper focuses on kinetic sieving which is the dominant mechanism in dense gravity-driven shallow free-surface flows, or, granular avalanches. The segregation model is derived from a three-phase mixture theory composed of large particles, small particles and a passive interstitial fluid. Steady-state solutions are constructed for a normally graded inflow in a steady uniform flow field. This problem is of fundamental interest, because it shows how an unstably stratified layer readjusts into a stable configuration. Expansion fans and concentration shocks are generated and sufficiently far downstream inversely-graded segregated layers form, with the larger particles overlying the finer ones. The distance for complete segregation to occur is shown to increase with rising fluid density and tends to infinity as its density approaches that of the grains. If the particles are buoyant then the initial configuration is stable. An exact time-dependent two-dimensional solution is constructed for plug-flow, which exploits the uncoupling of material columns of grains in the absence of shear. This yields insight into the nature of more complex numerical solutions for strong shear, which are computed with a high resolution, shock-capturing numerical scheme.

Asymmetric breaking size-segregation waves in dense granular free-surface flows

“…They also showed that the same rheology used to describe dense steady aerial flows (GDR Midi 2004) also applies to immersed flows, with the interstitial fluid changing the time scale of the particle rearrangements. This is consistent with the experimental results of Vallance & Savage (2000) and the theory of Thornton, Gray & Hogg (2006) who both showed that the role of the interstitial fluid in flows containing different sized constituents is to modify the segregation time scales. These results would suggest that the physical phenomena observed in the experiments above, with a few large particles recirculating very slowly in regions of small particles, are indicative an underlying asymmetry in the particle motion that occurs whether the flow is dry or submerged.…”

“…Bridgwater, Foo & Stephens 1985;Savage & Lun 1988;Bridgwater 1994;Dolgunin & Ukolov 1995;Gray & Thornton 2005;Gray & Chugunov 2006;Thornton et al 2006;May, Shearer & Daniels 2010) all share a similar advection-diffusion structure…”

“…Following Thornton et al (2006) and Thornton & Gray (2008), the particle paths of the large (superscript l) and small grains (superscript s) are given by…”

Particle-size segregation and diffusive remixing in shallow granular avalanches

“…Three specific solutions are investigated when the initial concentration is (i) homogeneous, (ii) reverse graded, and, (iii) normally graded. These show that, for low z 0 , the shocks, in the pure segregation solutions obtained by Gray & Thornton (2005) and Thornton et al (2006), are replaced by rapid smooth transitions. In the homogeneous problem, the boundary conditions are not compatible with the initial conditions, this drives a rapid readjustment to the solution that becomes increasingly strong with larger z 0 .…”

“…The discontinuity is rapidly diffused away, so that significant concentrations of large particles reach the free surface by t = 0.2. This is much quicker than in the no-diffusive remixing solution constructed by Thornton et al (2006), where the large particles reach the free-surface only at t = 0.5. This shows that the remixing process can help the particles to approach a segregated state more quickly than the non-diffusive case.…”