▪ Abstract  Granular materials segregate. Small differences in either size or density lead to flow-induced segregation, a complex phenomenon without parallel in fluids. Modeling of mixing and segregation processes requires the confluence of several tools, including continuum and discrete descriptions (particle dynamics, Monte Carlo simulations, cellular automata computations) and, often, considerable geometrical insight. None of these viewpoints, however, is wholly satisfactory by itself. Moreover, continuum and discrete descriptions of granular flows are regime dependent, and this fact may require adopting different subviewpoints. This review organizes a body of knowledge that forms—albeit imperfectly—the beginnings of an expandable continuum framework for the description of mixing and segregation of granular materials. We focus primarily on noncohesive particles, possibly differing in size, density, shape, etc. We present segregation mechanisms and models for size and density segregation and introduce chaotic advection, which appears in noncircular tumblers. Chaotic advection interacts in nontrivial ways with segregation in granular materials and leads to unique equilibrium structures that serve as a prototype for systems displaying organization in the midst of disorder.

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The focus of this work is analysis of mixing in a rotating cylinder—a prototype system for mixing of granular materials—with the objective of understanding and highlighting the role of flow on the dynamics of the process. The analysis is restricted to low speeds of rotation, when the free surface of the granular solids is nearly flat, and when particles are identical so that segregation is unimportant. The flow is divided into two regions: a rapid flow region of the cascading layer at the free surface, and a fixed bed of particles rotating at the angular speed of the cylinder. A continuum model, in which averages are taken across the layer, is used to analyze the flow in the layer. Good agreement is obtained between the predictions of the flow model for the layer thickness profile and experimental results obtained by digital image analysis. The dynamics of the mixing process are studied by advecting tracer particles by the flow and allowing for particle diffusion in the cascading layer. The mixing model predictions for distribution of tracer particles and mixing rates are compared qualitatively and quantitatively to experimental data. Optimal operating conditions, at which mixing rates are maximum, are determined.

We study the rheology of granular mixtures in a steady, fully developed, gravity-driven flow on an inclined plane, by means of discrete element method (DEM) simulations. Results are presented for a single component system and binary mixtures with particles of different size and density. Inclination angles, composition, size ratios and density ratios are varied to obtain different segregated configurations at equilibrium. Steady state profiles of the mean velocity, volume fractions, shear stress, shear rate, inertial number and apparent viscosity across the depth of the flowing layer are reported for the different cases. The viscosity varies with height and is found to depend on the local bulk density and composition, which, in turn, depend on the size ratio, the mass ratio and the degree of segregation. For a single component system, a viscoplastic rheological model [P. Jop et al., Nature 441, 727 (2006)] describes the data quite well. We propose a modification of the model for the case of mixtures. The mixture model predicts the viscosity for both well-mixed and segregated granular mixtures differing in size, density or both, using the same model parameters as obtained for the single component system. The predictions of a model for the volume fraction of the mixtures also agree well with simulation results.

Simultaneous mixing and segregation of granular materials is of considerable practical importance; the interplay among both processes is, however, poorly understood from a fundamental viewpoint. The focus of this work is radial segregation-core formation-due to density in a rotating cylinder. The flow regime considered is the cascading or continuous flow regime where a thin layer of solids flows along a nearly flat free surface, while the remaining particles rotate as a fixed bed along with the cylinder. The essence of the formation of a central segregated core of the more dense particles lies in the flow, mixing, and segregation in the cascading layer. The work involves experiments and analysis. A constitutive model for the segregation flux in cascading layers is proposed and validated by particle dynamics and Monte Carlo simulations for steady flow down an inclined plane. The model contains a single parameter, the dimensionless segregation velocity ͑␤͒, which is treated as a fitting parameter here. Experimental results for the equilibrium segregation of steel balls and glass beads are presented for different fractions and different extents of filling. There is a good match between theoretical predictions and all experimental results when the value of dimensionless segregation velocity is taken to be ␤ϭ2. The extent of segregation is found to increase with increase in the dimensionless segregation velocity and dimensionless diffusivity but is independent of the level of filling. Lagrangian simulations based on the theory and experiments demonstrate the competition between segregation and mixing. In the case of slow mixing, the intensity of segregation monotonically decreases to an equilibrium value; for fast mixing, however, there exists an optimal mixing time at which the best mixing is obtained.

We consider the segregation of spheres of equal size and different density flowing over an inclined plane, theoretically and computationally by means of distinct element method (DEM) simulations. In the first part of the work, we study the settling of a single higher-density particle in the flow of otherwise identical particles. We show that the motion of the high-density tracer particle can be understood in terms of the buoyancy and drag forces acting on it. The buoyancy force is given by Archimedes principle, with an effective volume associated with the particle, which depends upon the local packing fraction, $\phi $. The buoyancy arises primarily from normal forces acting on the particle, and tangential forces have a negligible contribution. The drag force on a sphere of diameter $d$ sinking with a velocity $v$ in a granular medium of apparent viscosity $\eta $ is given by a modified Stokes law, ${F}_{d} = c\pi \eta dv$. The coefficient ($c$) is found to decrease with packing fraction. In the second part of the work, we consider the case of binary granular mixtures of particles of the same size but differing in density. A continuum model for segregation is presented, based on the single-particle results. The number fraction profile for the heavy particles at equilibrium is obtained in terms of the effective temperature, defined by a fluctuation–dissipation relation. The model predicts the equilibrium number fraction profiles at different inclination angles and for different mass ratios of the particles, which match the DEM results very well. Finally, a complete model for the theoretical prediction of the flow and number fraction profiles for a mixture of particles of different density is presented, which combines the segregation model with a model for the rheology of mixtures. The model predictions agree quite well with the simulation results.

Granular surface flows are important in industrial practice and natural systems, but the understanding of such flows is at present incomplete. We present a combined theoretical and experimental study of quasi-two-dimensional heap formation by pouring particles continuously at a point. Two cases are considered: open systems and closed systems. Experimental results show that the shear rate in the flowing layer is nearly independent of the mass flow rate, and the angle of static friction at the bed–layer interface increases with flow rate. Predictions of the model for the flowing layer thickness and interface angles are in good agreement with experiments.

An experimental study of the flow of different materials ͑steel balls, glass beads, and sand͒ in quasi-twodimensional rotating cylinders is carried out using flow visualization. The flow in the rotating cylinder comprises of a thin-flowing surface layer with the remaining particles rotating as a fixed bed. Experimental results indicate that the scaled layer thickness increases with increasing Froude number (Frϭ 2 R/g, where is the angular speed, R is the cylinder radius, and g the acceleration due to gravity͒ and with increase in size ratio (sϭd/R, where d is the particle diameter͒. The free surface profile, is nearly flat at low Fr and becomes increasingly S shaped with increasing Fr. The layer thickness profiles, which are symmetric at low Fr become skewed at high values of Fr and small s. The dynamic angles of repose for all the materials studied show a near-linear increase with rotational speed (). Scaling analysis of the experimental data shows that the shape of the scaled surface profiles and the scaled layer thickness profiles are nearly identical when Froude number and size ratio are held constant, for each material. The surface profiles and layer thickness profiles are also found to be nearly independent of the material used. The dynamic angle of repose (␤), however, does not scale with Fr and s and depends on the particle properties. The experimental results are compared to continuum models for flow in the layer. The models of Elperin and Vikhansky ͓Europhys. Lett. 42, 619 ͑1998͔͒ and Makse ͓Phys. Rev. Lett. 83, 3186 ͑1999͔͒ show good agreement at low Fr while that of Khakhar et al. ͓Phys. Fluids, 9, 31 ͑1997͔͒ gives good predictions over the entire range of parameters considered. An analysis of the data indicate that the velocity gradient (␥ ) is nearly constant along the layer at low Fr, and the value calculated at the layer midpoint varies as ␥ 0 ϰ͓g sin(␤ 0 Ϫ␤ s )/d cos ␤ s ͔ 1/2 for all the experimental data, where ␤ s is the static angle of repose and ␤ 0 is the interface angle at the layer midpoint. An extension of ''heap'' models ͑BCRE, BRdG͒ is used to predict the interface angle profiles, which are in reasonable agreement with experimental measurements.