Surface flows of granular mixtures

We present the generalization of the minimal model for surface flows of granular mixtures, proposed by Boutreux and de Gennes [J. Phys. I France 6, 1295 (1996)]. The minimal model was valid for grains differing only in their surface properties. The present model also takes into account differences in the size of the grains. We apply the model to study segregation in two-dimensional silos of mixtures of grains differing in size and/or surface properties. When the difference in size is small, the model predicts that a continuous segregation appears in the static phase during the filling of a silo. When the difference in size is wide, we take into account the segregation of the grains in the rolling phase, and the model predicts complete segregation and stratification in agreement with experimental observations. Received 9 September 1998 and Received in final form 4 November 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     

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     

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     

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     

Fragmentation of grains in a two-dimensional packing

A numerical model of fragmentation of a two-dimensional granular medium under pressure is investigated. The fragmentation process is found to be strongly dependent on the type of force used as the criterion for breaking the grains. The fragmentation mode affects the process less dramatically. There is a power-law divergence in the pressure when the medium approaches the full packing limit, . Both log-normal and power-law fragment-size distributions are found. Gravity is demonstrated to be an important factor. Received: 14 December 1997 / Accepted: 17 March 1998     

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     

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     

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     

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     

Compression of a soft sphere packing

Mechanical properties of packings of deformable spheres of polyelectrolyte gel are studied experimentally. These particles are plunged into a brine. They have the property to swell and shrink when the concentration of salt of the solution is varied. An oedometric compression is performed imposing cycles of deformation at constant speed and constant salinity C_s. Under many different conditions, we study the laws of deformation relating the macroscopic compression force F, to the macroscopic strain . We find empirical non linear relations of the type . The values of this exponent m are discussed and compared to the results of measurements on a single sphere compressed on a plane as well as to the results of experiments and simulations on dry model granular assemblies. The swelling and deswelling properties of the spheres are used to perform isotropic compression tests. In this situation we determine the relation between the force at equilibrium and the macroscopic strain . The results are compared with those obtained in the oedometric compression tests. Received 27 January 1998     

II. Recirculation flow induced by a bubble stream rising in a viscous liquid

The recirculation flow induced by the rising motion of a bubble stream in a viscous fluid within an open-top rectangular enclosure is studied. The three-dimensional volume averaged conservation equations are solved by a control-volume method using a hybrid finite differencing scheme to describe the liquid phase hydrodynamics. The momentum exhange between the bubbles and the liquid phase is modeled with a source term equals to the volumetric buoyancy force acting on the gas in the bubble stream. The volumetric buoyancy force accounts for in line interactions between bubbles through the average gas volume fraction in the gas liquid column which depends on the size and the rising velocity of bubbles. The fluid flow within an open-top rectangular enclosure is further investigated by particle image velocimetry for a bubble stream rising in a water-glycerol solution. The measured fluid velocities in a vertical plane are compared with the predictions of the numerical model over a wide range of fluid viscosity (43 mPa s-800 mPa s) and gas flow rates. Finally, the recirculation flows resulting from the interaction of two neighbouring vertical bubble streams are studied. Received: 23 July 1997 / Revised: 19 December 1997 / Accepted: 11 May 1998     

Collision of a 1-D column of beads with a wall

An experimental study of the collision of a column of N beads () with a fixed wall is presented. For a fixed height of fall and a rigid wall, we show that the maximum force felt by the wall is independent of the number of beads N. The duration of impact, the velocity of the deformation wave in the column and an effective restitution coefficient of the column are also measured as a function of N. For a soft wall, we show that the maximum force depends on N. A non-dissipative numerical model, based on a nonlinear interaction law between nearest neighbours, gives results in agreement with the experimental data. Moreover, we show that, after the compression phase, the beads of the top of the column separate one after the other from the column with a velocity greater than the initial one. The beads at the bottom then bounce upwards in block, with a velocity smaller than the initial one. We emphasize that this detachment effect results from the energy redistribution within the whole system during the collision and not from any dissipative effect. Received: 6 February 1998 / Revised and accepted: 26 May 1998     

Lattice Boltzmann simulation of a binary fluid with different phase viscosities and its application to fingering in two dimensions

A thermodynamically consistent lattice Boltzmann scheme for simulating the flow of a binary fluid is extended to allow the fluid components to have different viscosities. The approach is tested for the shear and Poiseuille flow of layered immiscible fluids and for the dispersion relation and the damping of a capillary wave. We then consider the fingering that results when a fluid is displaced by a less viscous fluid in a two-dimensional channel. The finger widths obtained match the results of Reinelt and Saffman [#!Reinelt85!#], but differ somewhat from those of Halpern and Gaver [#!Halpern94!#] for capillary numbers above 2. A limiting finger width close to 1/2 is obtained for high capillary numbers and high viscosity ratios. Received 25 May 1999 and Received in final form 19 November 1999     

Force chain splitting in granular materials: A mechanism for large-scale pseudo-elastic behaviour

We investigate both numerically and analytically the effect of strong disorder on the large-scale properties of the hyperbolic equations for stresses proposed in J.-P. Bouchaud, M.E. Cates, P. Claudin, J. Phys. I 5, 639 (1995), and J.P. Wittmer, P. Claudin, M.E. Cates, J.-P. Bouchaud, Nature 382, 336 (1996); J.P. Wittmer, P. Claudin, M.E. Cates, J. Phys. I 7, 39 (1997). The physical mechanism that we model is the local splitting of the force chains (the characteristics of the hyperbolic equation) by packing defects. In analogy with the theory of light diffusion in a turbid medium, we propose a Boltzmann-like equation to describe these processes. We show that, for isotropic packings, the resulting large-scale effective equations for the stresses have exactly the same structure as those of an elastic body, despite the fact that no displacement field needs to be introduced at all. Correspondingly, the response function evolves from a two-peak structure at short scales to a broad hump at large scales. We find, however, that the Poisson ratio is anomalously large and incompatible with classical elasticity theory that requires the reference state to be thermodynamically stable. Received 13 November 2000 and Received in final form 3 January 2001