Surface flows of granular mixtures: II. Segregation with grains of different size

We present the generalization of a theoretical model for segregation of granular mixtures due to surface flows, published in J. Phys. I France 6, 1295 (1996). Our generalized model is valid for grains differing by their size and/or their surface properties; in the present paper, we describe the case of two species with the same surface properties but two different sizes. The rolling stream is assumed to be homogeneous. Exchanges between the grains at rest and the rolling stream are modelized via binary collisions. The model predicts that during the filling of a two-dimensional silo, continuous segregation appears inside the static phase: small (respectively large) grains tend to stop uphill (respectively downhill), although both species remain present everywhere. This fits the observations when the size difference between the species is small. When the size difference is large, a different regime is observed. We argue that in this case, segregation occurs directly inside the rolling stream. Received: 25 February 1998 / Received in final form and Accepted: 6 July 1998     

Kinematic segregation of granular mixtures in sandpiles

We study the segregation of granular mixtures in two-dimensional silos using a recently proposed set of coupled equations for surface flows of grains. We study the thick flow regime, where the grains are segregated in the rolling phase. We incorporate this dynamical segregation process, called kinematic sieving, free-surface segregation or percolation, into the theoretical formalism and calculate the profiles of the rolling species and the concentration of grains in the bulk in the steady state. Our solution shows the segregation of the mixture with the large grains being found at the bottom of the pile in qualitative agreement with experiments. Received: 6 July 1998 / Revised and Accepted: 13 August 1998     

Analysis of radial segregation of granular mixtures in a rotating drum

This paper considers the segregation of a granular mixture in a rotating drum. Extending a recent kinematic model for grain transport on sandpile surfaces to the case of rotating drums, an analysis is presented for radial segregation in the rolling regime, where a thin layer is avalanching down while the rest of the material follows rigid body rotation. We argue that segregation is driven not just by differences in the angle of repose of the species, as has been assumed in earlier investigations, but also by differences in the size and surface properties of the grains. The cases of grains differing only in size (slightly or widely) and only in surface properties are considered, and the predictions are in qualitative agreement with observations. The model yields results inconsistent with the assumptions for more general cases, and we speculate on how this may be corrected. Received 4 June 1999 and Received in final form 28 September 1999     

Cluster formation, pressure and density measurements in a granular medium fluidized by vibrations

We report experimental results on the behavior of an ensemble of inelastically colliding particles, excited by a vibrated piston in a vertical cylinder. When the particle number is increased, we observe a transition from a regime where the particles have erratic motions (“granular gas”) to a collective behavior where all the particles bounce like a nearly solid body. In the gas-like regime, we measure the density of particles as a function of the altitude and the pressure as a function of the number N of particles. The atmosphere is found to be exponential far enough from the piston, and the “granular temperature”, T, dependence on the piston velocity, V, is of the form , where is a decreasing function of N. This may explain previous conflicting numerical results. Received 1 February 1999     

On the validity of Hertz contact law for granular material acoustics

We discuss the acoustical behavior of a 1D model of granular medium, which is a chain of identical spherical beads. In this geometry, we are able to test quantitatively alternative models to the Hertz theory of contact between elastic solids. We compare the predictions of the different models to experimental results that concern linear sound wave propagation in the chain submitted to a static force, and nonlinear solitary wave propagation in an unconstrained chain. We use elastic, elastic-plastic and brittle materials, the beads roughness extends on one order of magnitude, and we also use oxidized metallic beads. We demonstrate experimentally that at low static forces, for all types of beads, the linear acoustic waves propagate in the system as predicted by Hertz's theory. At larger forces, after onset of permanent plastic deformation at the contacts, the brass beads exhibit non Hertzian behavior, and hysteresis. Except in the case of brass beads, the nonlinear waves follow the predictions of Hertz theory. Revised: 28 May 1998 / Accepted: 27 July 1998     

Settling and fluidization of non Brownian hard spheres in a viscous liquid

A mean field approach is used to estimate the energy dissipation during the homogeneous sedimentation or the particulate fluidization of non Brownian hard spheres in a concentrated suspension of infinite extent. Depending on inertial screening and the range of the hydrodynamic interactions, the effective buoyancy force is determined either from the average suspension density in a Stokes flow or from the fluid density in the turbulent flow regime. An energy balance then yields a settling or fluidization law depending on the particle Reynolds number in reasonable agreement with the Richardson and Zaki correlation and recent experimental results for particle settling or fluidization. We further estimate the energy dissipation in the turbulent boundary layers around the particles to precise the Reynolds number dependence of the hindered settling function in the intermediate flow regime. Received 22 February 1999 and Received in final form 14 June 1999     

Propagation of shock waves in a dusty gas with exponentially varying density

The variation of flow-variables with distance, in the flow-field behind a shock wave propagating in a dusty gas with exponentially varying density, are obtained at different times. The equilibrium flow conditions are assumed to be maintained, and the results are compared with those obtained for a perfect gas. It is found that the presence of small solid particles in the medium has significant effects on the variation of density and pressure. Received 20 October 1999 and Received in final form 9 March 2000     

Boundary effects on the maximal angle of stability of a granular packing

We have studied the maximal angle of stability of a granular packing confined between two walls. The effect of the walls is to increase the angle dramatically. The decay of the angle with the distance between the walls is exponential with a characteristic length which is a function of the beads diameter. The effect of the roughness of the walls has been also studied. Received 1 April 1999     

Magnetic permeability of a diphasic flow, made of liquid gallium and iron beads

Magnetohydrodynamics studies in laboratory experiments have long been restricted to low magnetic Reynolds number flows, mainly as a result of the very high magnetic diffusivity λ = 1/μσ of common conducting fluids (μ is the fluid's magnetic permeability and σ its electrical conductivity). The best conductivities are found in liquid metals which have a unit magnetic permeability, relative to vacuum. We show experimentally that a suspension of solid particles with a high magnetic permeability in a liquid metal yields an effective medium that has a high electrical conductivity and an enhanced magnetic permeability. The dispersion of the beads results from the turbulent fluid motion. The range of accessible magnetic Reynolds number can be increased by a factor of as much as 4 in our experimental setup. Received 6 March 2000 and Received in final form 13 July 2000     

Convection in horizontally shaken granular material

In horizontally shaken granular material different types of pattern formation have been reported. We want to deal with the convection instability which has been observed in experiments and which recently has been investigated numerically. Using two dimensional molecular dynamics we show that the convection pattern depends crucially on the inelastic properties of the material. The concept of restitution coefficient provides arguments for the change of the behaviour with varying inelasticity. Received 3 March 1999     

A model for ripple instabilities in granular media

We extend the model of surface granular flow proposed in [#!bcre!#] to account for the effect of an external `wind', which acts as to dislodge particles from the static bed, such that a stationary state of flowing grains is reached. We discuss in detail how this mechanism can be described in a phenomenological way, and show that a flat bed is linearly unstable against ripple formation in a certain region of parameter space. We focus in particular on the (realistic) case where the migration velocity of the instability is much smaller than the grains' velocity. In this limit, the full dispersion relation can be established. We relate the critical wave vector to the mean hopping length and to the ratio of the flight time to the `stick' time. We provide an intuitive interpretation of the instability. Received: 30 January 1998 / Revised: 12 May 1998 / Accepted: 8 June 1998     

Influence of roughness and dilatancy for dense granular flow along an inclined wall

In a previous work [#!ref1!#], the flow velocity of a steady two-dimensional granular flow along an inclined wall was investigated. The scaling law for the velocity field was found in good agreement with recent experimental results. The purpose of the present paper is to reformulate in more systematic manner and in a somewhat more general context the equations of mass and momentum conservation for dense granular flow, and also to present some new results with particular emphasis on roughness influence and dynamic dilatancy. Theoretical results are found in good agreement with experiments. Received 19 July 1999 and Received in final form 14 October 1999     

Dynamics of density waves in a two dimensional funnel on an inclined plane

We report here experiments on two-dimensional funnel flow of diameter glass beads on an inclined plane. We have investigated the properties of the flow according to the outlet size D of the funnel and the gravity. We have identified three different regimes. For small funnel outlet sizes, there is no significant change in flow density: the flow is rather steady and homogeneous. For intermediate outlet sizes (), the flow is intermittent, consisting of spatially ordered density waves propagating upwards. At bigger outlet sizes, density waves do not exhibit any ordering and the flow dynamics becomes chaotic. In addition, we find that the flow dynamics is independent of the funnel opening angle except close to the channel flow configuration. Finally, it is stressed that the interactions between the beads and the inclined plane play a crucial role in the mechanism of formation of density waves. Received: 9 July 1998 / Received in final form and Accepted: 18 September 1998     

On granular surface flow equations

Conservation equations are written for surface flows (either fluid or granular). The particularity of granular surface flows is then pointed out, namely that the depth of the flowing layer is not a priori fixed, leading to open equations. It is shown how some hypothesis on the flowing layer allows to close the system of equations. A possible hypothesis, similar to that made for a fluid layer, but inspired from granular flow experiments, is presented. The force acting on the flowing layer is discussed. Averaging over the flowing depth, as in shallow water theory, then allows to transform these conservation laws into equations for the evolution of the profile of a granular pile. Apart from their interest for building models, these conservation laws can be used to measure experimentally the effective forces acting on a flowing layer. Received 25 July 1998 and Received in final form 14 January 1999     

Behavior of one inelastic ball bouncing repeatedly off the ground

An experimental study of the behavior of one bead bouncing repeatedly off a static flat horizontal surface is presented. We observe that the number of bounces made by the bead is finite. When the duration between two successive bounces becomes of the order of the impact duration, the bead no longer bounces but oscillates on the elastically deformed surface before coming to rest. This transition is explained with a modified Hertz interaction law in which gravity is taken into account during the interaction. For each bounce, measurement of both the duration of collision and the restitution coefficient have been done. The effective restitution coefficient is essentially constant and close to 1 during almost all bounces before decreasing to zero when the impact velocity vanishes. This is due to an interplay between gravity and viscoelastic dissipation. Received: 2 December 1997 / Accepted: 5 March 1998     

Fingering instability in a water-sand mixture

The temporal evolution of a water-sand interface driven by gravity is experimentally investigated. By means of a Fourier analysis of the evolving interface the growth rates are determined for the different modes appearing in the developing front. To model the observed behavior we apply the idea of the Rayleigh-Taylor instability for two stratified fluids. Carrying out a linear stability analysis we calculate the growth rates from the corresponding dispersion relations for finite and infinite cell sizes. Based on the theoretical results the viscosity of the suspension is estimated to be approximately 100 times higher than that of pure water, in agreement with other experimental findings. Received: 12 December 1997 / Revised: 23 March 1998 / Accepted: 28 April 1998     

Molecular dynamics of comminution in ball mills

We investigate autogenous fragmentation of dry granular material in rotating cylinders using two-dimensional molecular dynamics. By evaluation of spatial force distributions achieved numerically for various rotation velocities we argue that comminution occurs mainly due to the existence of force chains. A statistical analysis of these force chains explains the spatial distribution of comminution efficiency in ball mills as measured experimentally by Rothkegel [1] and Rolf [2]. For animated sequences of our simulations see http://summa.physik.hu-berlin.de/kies/mill/bm.html Received 19 January 2000     

Rolling as a “continuing collision”

We show that two basic mechanical processes, the collision of particles and rolling motion of a sphere on a plane, are intimately related. According to our recent findings, the restitution coefficient for colliding spherical particles , which characterizes the energy loss upon collision, is directly related to the rolling friction coefficient for a viscous sphere on a hard plane. We quantify both coefficients in terms of material constants which allows to determine either of them provided the other is known. This relation between the coefficients may give rise to a novel experimental technique to determine alternatively the coefficient of restitution or the coefficient of rolling friction. Received 5 May 1999     

Force fluctuations in a vertically pushed granular column

We present series of experiments on the resistance force encountered by a bottom piston pushing a vertical granular column confined in a two-dimensional cell. We show that, due to the presence of friction at the boundaries and between the grains, the signal shows many complex features. At slow driving velocities, we observe a transition to a stick-slip dynamic instability. Depending on the granular material used, the elementary stick-slip events may either be well characterized or largely distributed. We present a statistical study on the waiting time between events and the distribution of energy release as a function of the spring stiffness and the driving velocity. Received 5 August 1998 and Received in final form 22 October 1998