Segregation patterns in three-dimensional granular flows

Flow of size-bidisperse particle mixtures in a spherical tumbler rotating alternately about two perpendicular axes produces segregation patterns that track the location of nonmixing islands predicted by a dynamical systems approach. To better understand the paradoxical accumulation of large particles in regions defined by barriers to transport, we perform discrete element method (DEM) simulations to visualize the three-dimensional structure of the segregation patterns and track individual particles. Our DEM simulations and modeling results indicate that segregation pattern formation in the biaxial spherical tumbler is due to the interaction of size-driven radial segregation with the weak spanwise component of the advective surface flow. Specifically, we find that after large particles segregate to the surface, slow axial drift in the flowing layer, which is inherent to spherical tumblers, is sufficient to drive large particles across nominal transport barriers and into nonmixing islands predicted by an advective flow model in the absence of axial drift. Axial drift alters the periodic dynamics of nonmixing islands, turning them into "sinks" where large particles accumulate even in the presence of collisional diffusion. Overall, our results indicate that weak perturbation of chaotic flow has the potential to alter key dynamical system features (e.g., transport barriers), which ultimately can result in unexpected physical phenomena.

Measuring grain size fractions of bidisperse granular materials using X-ray radiography

The mechanical properties of granular materials such as sand, snow and rice are inherently tied to the size of the constituent particles. When a system is composed of particles of various sizes, it is common for these particles to segregate by size when disturbed. There is therefore a need to measure the particle size distribution within granular media as it evolves over time. However, there are very few experimental techniques available which can measure the particle sizes in situ without disturbing the medium. Here we present a technique to determine the volume fractions of the grain sizes in bidisperse granular materials with the aid of dynamic X-ray radiography. As a result of the penetration of the X-rays into the medium, radiography minimises the effect of walls and boundaries on experimental measurements, which typically dominate optical measurements. The technique proposed here is based on using Fourier transforms of X-ray radiographs to extract local measurements evolving over time that can be related to the particle size distribution. For the case of bidisperse granular media, with two distinct particle sizes, we show that this technique can measure the relative concentration of the two species, which we determine via a heuristic calibration parameter. We validate this technique by comparing discrete element simulations of mixtures of known concentration with experimental measurements derived from X-ray radiography of glass beads. In the future, this technique could be used to measure the grain size distribution in systems of bidisperse dense granular media where the concentration of particles is not known a priori. Additionally, the technique can be used to analyse granular segregation as it evolves dynamically.

Modeling Segregation in Granular Flows

Accurate continuum models of flow and segregation of dense granular flows are now possible. This is the result of extensive comparisons, over the last several years, of computer simulations of increasing accuracy and scale, experiments, and continuum models, in a variety of flows and for a variety of mixtures. Computer simulations-discrete element methods (DEM)-yield remarkably detailed views of granular flow and segregation. Conti-nuum models, however, offer the best possibility for parametric studies of outcomes in what could be a prohibitively large space resulting from the competition between three distinct driving mechanisms: advection, diffusion, and segregation. We present a continuum transport equation-based framework, informed by phenomenological constitutive equations, that accurately predicts segregation in many settings, both industrial and natural. Three-way comparisons among experiments, DEM, and theory are offered wherever possible to validate the approach. In addition to the flows and mixtures described here, many straightforward extensions of the framework appear possible.

Diffusion of size bidisperse spheres in dense granular shear flow

Diffusion is an important particle behavior in granular flow. Although granular diffusion has been studied for decades, the diffusion of size bidisperse particles has not been well understood. In this paper, discrete element method simulations with the Lees-Edwards boundary condition are performed to quantify the relation between the diffusion coefficient (D) and flow parameters for size bidisperse spheres in dense granular flow. The influences of the shear rate (γ[over ̇]), the solids fraction (f), and the diameter ratio (D_{LS}) of particles on diffusion are studied. The effects of the friction coefficient (μ) and the restitution coefficient (e) are also investigated. The results indicate that while small particles diffuse faster than large particles in a binary system the volume weighted average diffusion coefficient is proportional to the shear rate and the square of the volume weighted average particle diameter, d^{2}, and it is inversely proportional to the solids fraction. The quantified relation is given as D=k_{d}γ[over ̇]d^{2}, where k_{d}=0.0186/f, and this relation is not sensitive to the diameter ratio for D_{LS}≤3. The diffusion coefficient is not sensitive to the friction coefficient except for the extreme condition where μ<0.1, and it is also not sensitive to the restitution coefficient between 0.3 and 0.9.

Effect of pressure on segregation in granular shear flows

The effect of confining pressure (overburden) on segregation of granular material is studied in discrete element method (DEM) simulations of horizontal planar shear flow. To mitigate changes to the shear rate due to the changing overburden, a linear with depth variation in the streamwise velocity component is imposed using a simple feedback scheme. Under these conditions, both the rate of segregation and the ultimate degree of segregation in size bidisperse and density bidisperse granular flows decrease with increasing overburden pressure and scale with the overburden pressure normalized by the lithostatic pressure of the particle bed. At overburdens greater than approximately 20 times the lithostatic pressure at the bottom of the bed, the density segregation rate is zero while the size segregation rate is small but nonzero, suggesting that different physical mechanisms drive the two types of segregation. The segregation rate scales close to linearly with the inertial number for both size bidisperse and density bidisperse mixtures under various flow conditions, leading to a proposed pressure dependence term for existing segregation velocity correlations. Surprisingly, particle stiffness has only a minor effect on segregation, although it significantly affects the packing density.

Continuum modelling of segregating tridisperse granular chute flow

Segregation and mixing of size multidisperse granular materials remain challenging problems in many industrial applications. In this paper, we apply a continuum-based model that captures the effects of segregation, diffusion and advection for size tridisperse granular flow in quasi-two-dimensional chute flow. The model uses the kinematics of the flow and other physical parameters such as the diffusion coefficient and the percolation length scale, quantities that can be determined directly from experiment, simulation or theory and that are not arbitrarily adjustable. The predictions from the model are consistent with experimentally validated discrete element method (DEM) simulations over a wide range of flow conditions and particle sizes. The degree of segregation depends on the Péclet number, Pe, defined as the ratio of the segregation rate to the diffusion rate, the relative segregation strength κ_ij between particle species i and j, and a characteristic length L, which is determined by the strength of segregation between smallest and largest particles. A parametric study of particle size, κ_ij , Pe and L demonstrates how particle segregation patterns depend on the interplay of advection, segregation and diffusion. Finally, the segregation pattern is also affected by the velocity profile and the degree of basal slip at the chute surface. The model is applicable to different flow geometries, and should be easily adapted to segregation driven by other particle properties such as density and shape.

Sidewall-friction-driven ordering transition in granular channel flows: Implications for granular rheology

We report a transition from a disordered state to an ordered state in the flow of nearly monodisperse granular matter flowing in an inclined channel with planar slide walls and a bumpy base, using discrete element method simulations. For low particle-sidewall friction coefficients, the flowing particles are disordered, however, for high sidewall friction, an ordered state is obtained, characterized by a layering of the particles and hexagonal packing of the particles in each layer. The extent of ordering, quantified by the local bond-orientational order parameter, varies in the cross section of the channel, with the highest ordering near the sidewalls. The flow transition significantly affects the local rheology-the effective friction coefficient is lower, and the packing fraction is higher, in the ordered state compared to the disordered state. A simple model, incorporating the extent of local ordering, is shown to describe the rheology of the system.

Transient response in granular quasi-two-dimensional bounded heap flow

We study the transition between steady flows of noncohesive granular materials in quasi-two-dimensional bounded heaps by suddenly changing the feed rate. In both experiments and simulations, the primary feature of the transition is a wedge of flowing particles that propagates downstream over the rising free surface with a wedge front velocity inversely proportional to the square root of time. An additional longer duration transient process continues after the wedge front reaches the downstream wall. The entire transition is well modeled as a moving boundary problem with a diffusionlike equation derived from local mass balance and a local relation between the flux and the surface slope.

Modelling density segregation in flowing bidisperse granular materials

Preventing segregation in flowing granular mixtures is an ongoing challenge for industrial processes that involve the handling of bulk solids. A recent continuum-based modelling approach accurately predicts spatial concentration fields in a variety of flow geometries for mixtures varying in particle size. This approach captures the interplay between advection, diffusion and segregation using kinematic information obtained from experiments and/or discrete element method (DEM) simulations combined with an empirically determined relation for the segregation velocity. Here, we extend the model to include density-driven segregation, thereby validating the approach for the two important cases of practical interest. DEM simulations of density bidisperse flows of mono-sized particles in a quasi-two-dimensional-bounded heap were performed to determine the dependence of the density-driven segregation velocity on local shear rate and particle concentration. The model yields theoretical predictions of segregation patterns that quantitatively match the DEM simulations over a range of density ratios and flow rates. Matching experiments reproduce the segregation patterns and quantitative segregation profiles obtained in both the simulations and the model, thereby demonstrating that the modelling approach captures the essential physics of density-driven segregation in granular heap flow.

Shear-Rate-Independent Diffusion in Granular Flows

We computationally study the behavior of the diffusion coefficient D in granular flows of monodisperse and bidisperse particles spanning regions of relatively high and low shear rate in open and closed laterally confined heaps. Measurements of D at various flow rates, streamwise positions, and depths collapse onto a single curve when plotted as a function of γd2, where d is the local mean particle diameter and γ is the local shear rate. When γ is large, D is proportional to γd2, as in previous studies. However, for γd2 below a critical value, D is independent of γd2. The acceleration due to gravity g and particle stiffness (or, equivalently, the binary collision time t(c)) together determine the transition in D between regimes. This suggests that while shear rate and particle size determine diffusion at relatively high shear rates in surface-driven flows, diffusion at low shear rates is an elastic phenomenon with time and length scales dependent on gravity (sqrt d/g) and particle stiffness (t(c)sqrt(dg), respectively.