scholarly journals Asymmetric flux models for particle-size segregation in granular avalanches

2014 ◽  
Vol 757 ◽  
pp. 297-329 ◽  
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.

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

2008 ◽  
Vol 19 (1) ◽  
pp. 61-86 ◽  
Author(s):  
M. SHEARER ◽  
J. M. N. T. GRAY ◽  
A. R. THORNTON
Keyword(s):  
Particle Size ◽  
Free Surface ◽  
Shear Flow ◽  
Free Surface Flows ◽  
Size Segregation ◽  
Small Particles ◽  
Surface Flows ◽  
Scalar Conservation Law ◽  
Large Particles ◽  
Scalar Conservation

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.

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

2016 ◽  
Vol 794 ◽  
pp. 460-505 ◽  
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.

scholarly journals Effects of dynamic fragmentation on the impact force exerted on rigid barrier: centrifuge modelling

2019 ◽  
Vol 56 (9) ◽  
pp. 1215-1224 ◽  
Author(s):  
C.W.W. Ng ◽  
C.E. Choi ◽  
D.K.H. Cheung ◽  
Y. Cui
Keyword(s):  
Volume Fraction ◽  
Impact Force ◽  
Inertial Particles ◽  
Size Segregation ◽  
Centrifuge Modelling ◽  
Rigid Barrier ◽  
Flow Energy ◽  
Dynamic Fragmentation ◽  
Large Particles ◽  
The Impact

Bi-dispersity is a prerequisite for grain-size segregation, which transports the largest particles to the flow front. These large and inertial particles can fragment upon impacting a barrier. The amount of fragmentation during impact strongly influences the force exerted on a rigid barrier. Centrifuge modelling was adopted to replicate the stresses for studying the effects of bi-dispersity in a granular assembly and dynamic fragmentation on the impact force exerted on a model rigid barrier. To study the effects of bi-dispersity, the ratio between the diameters of small and large particles (δs/δl), characterizing the particle-size distribution (PSD), was varied as 0.08, 0.26, and 0.56. The volume fraction of the large particles was kept constant. A δs/δl tending towards unity characterizes inertial flow that exerts sharp impulses, and a diminishing δs/δl characterizes the progressive attenuation of these sharp impulses by the small particles. Flows dominated by grain-contact stresses (δs/δl < 0.26), as characterized by the Savage number, are effective at attenuating dispersive stresses of the large particles, which are responsible for reducing dynamic fragmentation. By contrast, flows dominated by grain-inertial stresses (δs/δl > 0.26) exhibit up to 66% more impulses and 4.3 times more fragmentation. Dynamic fragmentation of bi-disperse flows impacting a rigid barrier can dissipate about 30% of the total flow energy.

Effect of Particle Size on Fracture Toughness of Spherical-Silica Particle Filled Epoxy Composites

2005 ◽  
Vol 297-300 ◽  
pp. 207-212 ◽  
Author(s):  
Soon Chul Kwon ◽  
Tadaharu Adachi ◽  
Wakako Araki ◽  
Akihiko Yamaji
Keyword(s):  
Fracture Toughness ◽  
Particle Size ◽  
Epoxy Resin ◽  
Small Particle ◽  
Volume Fraction ◽  
Silica Particles ◽  
Bending Test ◽  
High Fracture Toughness ◽  
Particle Content ◽  
Spherical Silica

We investigated the particle size effects on the fracture toughness of epoxy resin composites reinforced with spherical-silica particles. The silica particles had different mean particle diameters of between 1.56 and 0.24µm and were filled with bisphenol A-type epoxy resin under different mixture ratios of small and large particles and a constant volume fraction for all particles of 0.30. As the content with the added smaller particle increased, the viscosity of each composite before curing remarkably increased. We conducted the single edge notched bending test (SENB) to measure the mode I fracture toughness of each composite. The fracture surface with the small particle content exhibited more rough areas than the surface with larger particles. The fracture toughness increased below the small particle content of 0.8 and saturated above it. Therefore, near the small particle content of 0.8, the composite had a relatively low viscosity and a high fracture toughness.

scholarly journals Time-dependent solutions for particle-size segregation in shallow granular avalanches

2006 ◽  
Vol 462 (2067) ◽  
pp. 947-972 ◽  
Author(s):  
J.M.N.T Gray ◽  
M Shearer ◽  
A.R Thornton
Keyword(s):  
Particle Size ◽  
Logarithmic Singularity ◽  
Free Surface Flows ◽  
Time Dependent ◽  
Homogeneous Material ◽  
Recent Theory ◽  
Size Segregation ◽  
Rock Falls ◽  
Wide Range ◽  
Graded Layers

Rapid shallow granular free-surface flows develop in a wide range of industrial and geophysical flows, ranging from rotating kilns and blenders to rock-falls, snow slab-avalanches and debris-flows. Within these flows, grains of different sizes often separate out into inversely graded layers, with the large particles on top of the fines, by a process called kinetic sieving. In this paper, a recent theory is used to construct exact time-dependent two-dimensional solutions for the development of the particle-size distribution in inclined chute flows. The first problem assumes the flow is initially homogeneously mixed and is fed at the inflow with homogeneous material of the same concentration. Concentration shocks develop during the flow and the particles eventually separate out into inversely graded layers sufficiently far downstream. Sections with a monotonically decreasing shock height, between these layers, steepen and break in finite time. The second problem assumes that the material is normally graded, with the small particles on top of the coarse ones. In this case, shock waves, concentration expansions, non-centred expanding shock regions and breaking shocks develop. As the parameters are varied, nonlinearity leads to fundamental topological changes in the solution, and, in simple-shear, a logarithmic singularity prevents a steady-state solution from being attained.

scholarly journals Particle-size and -density segregation in granular free-surface flows

2015 ◽  
Vol 779 ◽  
pp. 622-668 ◽  
Author(s):  
J. M. N. T. Gray ◽  
C. Ancey
Keyword(s):  
Particle Size ◽  
Steady State ◽  
Parameter Space ◽  
Peclet Number ◽  
Particle Density ◽  
Critical Concentration ◽  
Free Surface Flows ◽  
Size Segregation ◽  
Péclet Number ◽  
Concentration Changes

When a mixture of particles, which differ in both their size and their density, avalanches downslope, the grains can either segregate into layers or remain mixed, dependent on the balance between particle-size and particle-density segregation. In this paper, binary mixture theory is used to generalize models for particle-size segregation to include density differences between the grains. This adds considerable complexity to the theory, since the bulk velocity is compressible and does not uncouple from the evolving concentration fields. For prescribed lateral velocities, a parabolic equation for the segregation is derived which automatically accounts for bulk compressibility. It is similar to theories for particle-size segregation, but has modified segregation and diffusion rates. For zero diffusion, the theory reduces to a quasilinear first-order hyperbolic equation that admits solutions with discontinuous shocks, expansion fans and one-sided semi-shocks. The distance for complete segregation is investigated for different inflow concentrations, particle-size segregation rates and particle-density ratios. There is a significant region of parameter space where the grains do not separate completely, but remain partially mixed at the critical concentration at which size and density segregation are in exact balance. Within this region, a particle may rise or fall dependent on the overall composition. Outside this region of parameter space, either size segregation or density segregation dominates and particles rise or fall dependent on which physical mechanism has the upper hand. Two-dimensional steady-state solutions that include particle diffusion are computed numerically using a standard Galerkin solver. These simulations show that it is possible to define a Péclet number for segregation that accounts for both size and density differences between the grains. When this Péclet number exceeds 10 the simple hyperbolic solutions provide a very useful approximation for the segregation distance and the height of rapid concentration changes in the full diffusive solution. Exact one-dimensional solutions with diffusion are derived for the steady-state far-field concentration.

Investigations of the effects of particle properties on the wear resistance of the particle reinforced composites using a novel wear model

2016 ◽  
Vol 05 (02) ◽  
pp. 1650013 ◽  
Author(s):  
T. Ram Prabhu
Keyword(s):  
Particle Size ◽  
Wear Resistance ◽  
Volume Fraction ◽  
Two Dimensions ◽  
Reinforced Composites ◽  
Wear Model ◽  
Spring Force ◽  
Particle Reinforced Composites ◽  
The Matrix ◽  
Applied Forces

A wear model is developed based on the discrete lattice spring–mass approach to study the effects of particle volume fraction, size, and stiffness on the wear resistance of particle reinforced composites. To study these effects, we have considered three volume fractions (10%, 20% and 30%), two sizes ([Formula: see text] and [Formula: see text] sites), and two different stiffness of particles embedded in the matrix in a regular pattern. In this model, we have discretized the composite system ([Formula: see text] sites) into the lumped masses connected with interaction spring elements in two dimensions. The interaction elements are assumed as linear elastic and ideal plastic under applied forces. Each mass is connected to its first and second nearest neighbors by springs. The matrix and particles sites are differentiated by choosing the different stiffness values. The counter surface is simulated as a rigid body that moves on the composite material at a constant sliding speed along the horizontal direction. The governing equations are formed by equating the spring force between the pair of sites given by Hooke’s law plus external contact forces and the force due to the motion of the site given by the equation of motion. The equations are solved for the plastic strain accumulated in the springs using an explicit time stepping procedure based on a finite difference form of the above equations. If the total strain accumulated in the spring elements connected to a lump mass site exceeds the failure strain, the springs are considered to be broken, and the mass site is removed or worn away from the lattice and accounts as a wear loss. The model predicts that (i) increasing volume fraction, reducing particle size and increasing particle stiffness enhance the wear resistance of the particle reinforced composites, (ii) the particle stiffness is the most significant factor affecting the wear resistance of the composites, and (iii) the wear resistance reduced above the critical volume fraction ([Formula: see text]), and [Formula: see text] increases with increasing particle size. Finally, we have qualitatively compared the model results with our previously published experimental results to prove the effectiveness of the model to analysis the complex wear systems.

scholarly journals Segregation, recirculation and deposition of coarse particles near two-dimensional avalanche fronts

2009 ◽  
Vol 629 ◽  
pp. 387-423 ◽  
Author(s):  
J. M. N. T. GRAY ◽  
C. ANCEY
Keyword(s):  
Free Surface ◽  
Size Distribution ◽  
Two Dimensions ◽  
Bulk Flow ◽  
Coarse Grained ◽  
Low Flow ◽  
Two Dimensional ◽  
Parallel Plates ◽  
Size Segregation ◽  
Large Particles

Stratification patterns are formed when a bidisperse mixture of large rough grains and smaller more mobile particles is poured between parallel plates to form a heap. At low flow rates discrete avalanches flow down the free surface and are brought to rest by the propagation of shock waves. Experiments performed in this paper show that the larger particles are segregated to the top of the avalanche, where the velocity is greatest, and are transported to the flow front. Here the particles are overrun but may rise to the free surface again by size segregation to create a recirculating coarse-grained front. Once the front is established composite images show that there is a steady regime in which any additional large grains that reach the front are deposited. This flow is therefore analogous to finger formation in geophysical mass flows, where the larger less mobile particles are shouldered aside to spontaneously form static lateral levees rather than being removed by basal deposition in two dimensions. At the heart of all these phenomena is a dynamic feedback between the bulk flow and the evolving particle-size distribution within the avalanche. A fully coupled theory for such segregation–mobility feedback effects is beyond the scope of this paper. However, it is shown how to derive a simplified uncoupled travelling-wave solution for the avalanche motion and reconstruct the bulk two-dimensional flow field using assumed velocity profiles through the avalanche depth. This allows a simple hyperbolic segregation theory to be used to construct exact solutions for the particle concentration and for the recirculation within the bulk flow. Depending on the material composition and the strength of the segregation and deposition, there are three types of solution. The coarse-particle front grows in length if more large particles arrive than can be deposited. If there are fewer large grains and if the segregation is strong enough, a breaking size-segregation wave forms at a unique position behind the front. It consists of two expansion fans, two shocks and a central ‘eye’ of constant concentration that are arranged in a ‘lens-like’ structure. Coarse grains just behind the front are recirculated, while those reaching the head are overrun and deposited. Upstream of the wave, the size distribution resembles a small-particle ‘sandwich’ with a raft of rapidly flowing large particles on top and a coarse deposited layer at the bottom, consistent with the experimental observations made here. If the segregation is weak, the central eye degenerates, and all the large particles are deposited without recirculation.

scholarly journals Segregation-induced fingering instabilities in granular free-surface flows

2012 ◽  
Vol 709 ◽  
pp. 543-580 ◽  
Author(s):  
M. J. Woodhouse ◽  
A. R. Thornton ◽  
C. G. Johnson ◽  
B. P. Kokelaar ◽  
J. M. N. T. Gray
Keyword(s):  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Free Surface Flows ◽  
Numerical Results ◽  
Small Scale ◽  
Size Segregation ◽  
Numerical Resolution ◽  
Bulk Motion ◽  
Large Particles ◽  
Ill Posed

AbstractParticle-size segregation can have a significant feedback on the bulk motion of granular avalanches when the larger grains experience greater resistance to motion than the fine grains. When such segregation-mobility feedback effects occur the flow may form digitate lobate fingers or spontaneously self-channelize to form lateral levees that enhance run-out distance. This is particularly important in geophysical mass flows, such as pyroclastic currents, snow avalanches and debris flows, where run-out distance is of crucial importance in hazards assessment. A model for finger formation in a bidisperse granular avalanche is developed by coupling a depth-averaged description of the preferential transport of large particles towards the front with an established avalanche model. The coupling is achieved through a concentration-dependent friction coefficient, which results in a system of non-strictly hyperbolic equations. We compute numerical solutions to the flow of a bidisperse mixture of small mobile particles and larger more resistive grains down an inclined chute. The numerical results demonstrate that our model is able to describe the formation of a front rich in large particles, the instability of this front and the subsequent evolution of elongated fingers bounded by large-rich lateral levees, as observed in small-scale laboratory experiments. However, our numerical results are grid dependent, with the number of fingers increasing as the numerical resolution is increased. We investigate this pathology by examining the linear stability of a steady uniform flow, which shows that arbitrarily small wavelength perturbations grow exponentially quickly. Furthermore, we find that on a curve in parameter space the growth rate is unbounded above as the wavelength of perturbations is decreased and so the system of equations on this curve is ill-posed. This indicates that the model captures the physical mechanisms that drive the instability, but additional dissipation mechanisms, such as those considered in the realm of flow rheology, are required to set the length scale of the fingers that develop.

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