Stable solutions of a scalar conservation law for particle-size segregation in dense granular avalanches

Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters.

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2004.1420 ◽

2005 ◽

Vol 461 (2057) ◽

pp. 1447-1473 ◽

Cited By ~ 155

Author(s):

J.M.N.T Gray ◽

A.R Thornton

Keyword(s):

Free Surface ◽

Plug Flow ◽

Mixture Theory ◽

Free Surface Flows ◽

Size Segregation ◽

Void Space ◽

Small Particles ◽

Surface Flows ◽

Large Particles ◽

Steady State Solutions

Abstract Granular materials composed of a mixture of grain sizes are notoriously prone to segregation during shaking or transport. In this paper, a binary mixture theory is used to formulate a model for kinetic sieving of large and small particles in thin, rapidly flowing avalanches, which occur in many industrial and geophysical free-surface flows. The model is based on a simple percolation idea, in which the small particles preferentially fall into underlying void space and lever large particles upwards. Exact steady-state solutions have been constructed for general steady uniform velocity fields, as well as time-dependent solutions for plug-flow, that exploit the decoupling of material columns in the avalanche. All the solutions indicate the development of concentration shocks, which are frequently observed in experiments. A shock-capturing numerical algorithm is formulated to solve general problems and is used to investigate segregation in flows with weak shear.

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

Journal of Fluid Mechanics ◽

10.1017/s0022112005007676 ◽

2006 ◽

Vol 550 (-1) ◽

pp. 1 ◽

Cited By ~ 56

Author(s):

A. R. THORNTON ◽

J. M. N. T. GRAY ◽

A. J. HOGG

Keyword(s):

Particle Size ◽

Free Surface ◽

Mixture Theory ◽

Free Surface Flows ◽

Size Segregation ◽

Surface Flows ◽

Three Phase ◽

Phase Mixture

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Asymmetric flux models for particle-size segregation in granular avalanches

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.503 ◽

2014 ◽

Vol 757 ◽

pp. 297-329 ◽

Cited By ~ 30

Author(s):

P. Gajjar ◽

J. M. N. T. Gray

Keyword(s):

Particle Size ◽

Small Particle ◽

Volume Fraction ◽

Free Surface Flows ◽

Two Dimensions ◽

Combined Processes ◽

Maximum Point ◽

Size Segregation ◽

Flux Function ◽

Large Particles

AbstractParticle-size segregation commonly occurs in both wet and dry granular free-surface flows through the combined processes of kinetic sieving and squeeze expulsion. As the granular material is sheared downslope, the particle matrix dilates slightly and small grains tend to percolate down through the gaps, because they are more likely than the large grains to fit into the available space. Larger particles are then levered upwards in order to maintain an almost uniform solids volume fraction through the depth. Recent experimental observations suggest that a single small particle can percolate downwards through a matrix of large particles faster than a large particle can be levered upwards through a matrix of fines. In this paper, this effect is modelled by using a flux function that is asymmetric about its maximum point, differing from the symmetric quadratic form used in recent models of particle-size segregation. For illustration, a cubic flux function is examined in this paper, which can be either a convex or a non-convex function of the small-particle concentration. The method of characteristics is used to derive exact steady-state solutions for non-diffuse segregation in two dimensions, with an inflow concentration that is (i) homogeneous and (ii) normally graded, with small particles above the large. As well as generating shocks and expansion fans, the new asymmetric flux function generates semi-shocks, which have characteristics intersecting with the shock just from one side. In the absence of diffusive remixing, these can significantly enhance the distance over which complete segregation occurs.

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

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.170 ◽

2016 ◽

Vol 794 ◽

pp. 460-505 ◽

Cited By ~ 19

Author(s):

P. Gajjar ◽

K. van der Vaart ◽

A. R. Thornton ◽

C. G. Johnson ◽

C. Ancey ◽

...

Keyword(s):

Particle Motion ◽

Wave Structure ◽

Free Surface Flows ◽

Pyroclastic Flows ◽

Breaking Wave ◽

Size Segregation ◽

Small Particles ◽

Flux Function ◽

Large Particles ◽

Over The Top

Debris and pyroclastic flows often have bouldery flow fronts, which act as a natural dam resisting further advance. Counter intuitively, these resistive fronts can lead to enhanced run-out, because they can be shouldered aside to form static levees that self-channelise the flow. At the heart of this behaviour is the inherent process of size segregation, with different sized particles readily separating into distinct vertical layers through a combination of kinetic sieving and squeeze expulsion. The result is an upward coarsening of the size distribution with the largest grains collecting at the top of the flow, where the flow velocity is greatest, allowing them to be preferentially transported to the front. Here, the large grains may be overrun, resegregated towards the surface and recirculated before being shouldered aside into lateral levees. A key element of this recirculation mechanism is the formation of a breaking size-segregation wave, which allows large particles that have been overrun to rise up into the faster moving parts of the flow as small particles are sheared over the top. Observations from experiments and discrete particle simulations in a moving-bed flume indicate that, whilst most large particles recirculate quickly at the front, a few recirculate very slowly through regions of many small particles at the rear. This behaviour is modelled in this paper using asymmetric segregation flux functions. Exact non-diffuse solutions are derived for the steady wave structure using the method of characteristics with a cubic segregation flux. Three different structures emerge, dependent on the degree of asymmetry and the non-convexity of the segregation flux function. In particular, a novel ‘lens-tail’ solution is found for segregation fluxes that have a large amount of non-convexity, with an additional expansion fan and compression wave forming a ‘tail’ upstream of the ‘lens’ region. Analysis of exact solutions for the particle motion shows that the large particle motion through the ‘lens-tail’ is fundamentally different to the classical ‘lens’ solutions. A few large particles starting near the bottom of the breaking wave pass through the ‘tail’, where they travel in a region of many small particles with a very small vertical velocity, and take significantly longer to recirculate.

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