Size segregation is a widely observed consequence of sheared flow in granular media, in which the larger grains rise to the upper surface of the bulk, despite the fact that all grains have the same material density. A two-dimensional molecular dynamics study of the phenomenon is described in this paper. The grains are represented by inelastic disks and the flow occurs down an inclined slope with a rough base. The simulations readily reproduce the segregation effect; the results of a series of measurements of various aspects of this so-called `inverse grading' are reported. [Download reprint]Here is an image of what can be seen during simulations of this type, and here is a short animated sequence (350K).
Molecular dynamics methods are used to study the two-dimensional gravity-driven granular flow through a horizontal aperture. Two distinct approaches to modeling the granular particles are studied: a) Circular particles subject to a strongly repulsive short-range interaction, together with normal and tangential frictional damping forces. b) Rigid nonconvex particles, each consisting of disks arranged as an equilateral triangle, suitably spaced to provide a tangible indentation along each edge; the same repulsive interactions between disks in different grains and normal frictional damping forces are incorporated, but transverse damping is omitted, with the model relying on grain shape to resist sliding motion. In order to allow accurate measurements under steady-state conditions a continuous-feed approach is adopted, in which grains exiting through the hole are returned to the top of the material in the container. For both models the output flow is measured as a function of aperture size and the observed behavior compared with previous theoretical and experimental results. Tests of the degree to which the models reproduce the depth independence of the flow are reported, and the influence of the container width and the nature of the walls are studied. The depth dependence of the pressure, the local stress distribution and the particle flow patterns are also examined. [Download reprint]Here is an animated sequence showing the stress distribution (1200K).
The simulation of the flow of granular materials often leads to behavior that is difficult to quantify and that is therefore best described visually. In this paper we show several examples in which adding visualization to the calculations makes it possible to examine features which are not readily characterized by other means. The problems discussed are granular flow from a silo, the size-grading that occurs in inclined-chute flow, and the surface waves that appear in a thin, vertically vibrated layer.
Using a simplified model of a dissipative discrete-particle fluid we study the formation of surface waves under vertical vibration. The only dissipative forces included in this two-dimensional molecular dynamics simulation act normally to the line of contact during collisions; transverse frictional forces commonly used in granular models are not required. Depending on the parameters, surface waves are observed to form with a half or a quarter the driving frequency. [Download reprint]Here is an image of what can be seen during simulations of this type, and here is a short animated sequence (900K).
Granular flow from a silo: Discrete-particle simulations in three dimensions
D. Hirshfeld and D. C. Rapaport, Eur. Phys. J. E 4, 193 (2001)Molecular (or granular) dynamics methods are used to study the gravity-driven flow of granular material through a horizontal aperture in three dimensions. The grains are spherical and modeled using a short-range repulsive interaction, together with normal and tangential frictional damping forces. The material is contained in a rough-walled cylindrical container with a circular hole in its base, and to permit flow measurements under steady-state conditions a continuous feed approach is employed in which exiting grains are replaced at the upper surface of the material. The dependence of flow velocity and discharge rate on aperture diameter is found to agree with experiment; other quantities such as the kinetic energy and pressure distributions are also examined. [Download reprint]
Stratified horizontal flow in vertically vibrated granular layers
M. Levanon and D. C. Rapaport, Phys. Rev. E 64, 011304 (2001)A layer of granular material on a vertically vibrating sawtooth-shaped base exhibits horizontal flow whose speed and direction depend on the parameters specifying the system in a complex manner. Discrete-particle simulations reveal that the induced flow rate varies with height within the granular layer and oppositely directed flows can occur at different levels. The behavior of the overall flow is readily understood once this novel feature is taken into account. [Download reprint]Here is a Java demonstration of the phenomenon.
Mechanism for granular segregation
D. C. Rapaport, Phys. Rev. E 64, 061304 (2001)A process is described that produces horizontal size segregation in a vertically vibrated layer of granular material. The behavior is a consequence of two distinct phenomena that are unique to excited granular media: vibration which causes the larger particles to rise to the top of the layer, and a vibrating base with a sawtooth surface profile which can produce stratified flows in opposite directions at different heights within the layer. The result of combining these effects is that large and small particles are horizontally driven in opposite directions. The observations reported here are based on computer simulations of granular models in two and three dimensions. [Download reprint]Here is some more on this surprising phenomenon.
Simulational studies of axial granular segregation in a rotating cylinder
D. C. Rapaport, Phys. Rev. E 65, 061306 (2002)Discrete particle simulation methods have been used to study axial segregation in a horizontal rotating cylinder that is partially filled with a mixture of two different kinds of granular particles. Under suitable conditions segregation was found to occur, with the particles separating into a series of bands perpendicular to the axis. In certain cases the band structure exhibited time-dependent behavior, including band formation, merging and motion along the axis, all corresponding to phenomena that arise experimentally. In order to examine how the many parameters specifying the problem affect the segregation process, simulation runs were carried out using a variety of parameter settings, including combinations of friction coefficients not realizable experimentally. Both segregation and desegregation (mixing) were investigated, and cylinders with both explicit end caps and periodic ends were used to help isolate the causes of segregation. [Download reprint]Color versions of some of the pictures in the paper are available (each 30-50K): (a) oblique view, and (b) view from above (different runs). Here is an animated sequence showing segregation (1000K).
Shake, rattle or roll: Things to do with a granular mixture on a computer
D. C. Rapaport, in Computer Simulation Studies in Condensed Matter Physics, XVII, eds D. P. Landau, et al (Springer, 2005), p.7Studies of granular media continue to produce surprising results that are unique to this class of matter. A particularly prominent feature of granular mixtures is the tendency to segregate into their individual components when externally driven in various ways, an effect that has consequences for industrial processing. This article describes the computer simulation of several different modes of segregation in flowing, rotating and vibrating granular media, in some cases reproducing known behavior, in others predicting effects that have yet to be observed experimentally. [Download preprint]
Radial and axial segregation of granular matter in a rotating cylinder: A simulation study
D. C. Rapaport, Phys. Rev. E 75, 031301 (2007)The phenomena of radial and axial segregation in a horizontal rotating cylinder containing a mixture of granular particles of two different species have been modeled using discrete particle simulation. Space-time plots and detailed imagery provide a comprehensive description of what occurs in the system. As is the case experimentally, the nature of the segregation depends on the parameters defining the problem; the radial component of the segregation may be transient or long-lasting, and the axial component may or may not develop. Simulations displaying the different kinds of behavior are described and the particle dynamics associated with the axially segregated state examined. The importance of an appropriate choice of interaction for representing the effective friction force is demonstrated. [Download reprint]
Simulated three-component granular segregation in a rotating drum
D. C. Rapaport, Phys. Rev. E 76, 041302 (2007)Discrete particle simulations are used to model segregation in granular mixtures of three different particle species in a horizontal rotating drum. Axial band formation is observed, with medium-size particles tending to be located between alternating bands of big and small particles. Partial radial segregation also appears; it precedes the axial segregation and is characterized by an inner core region richer in small particles. Axial bands are seen to merge during the long simulation runs, leading to a coarsening of the band pattern; the relocation of particles involved in one such merging event is examined. Overall, the behavior is similar to experiment and represents a generalization of what occurs in the simpler two-component mixture. [Download reprint]
