Meheboob ALAM
Jawaharlal Nehru Center for Advanced Scientific, Bangalore, India
An exact order-parameter description of granular plane Couette flow from nonlinear stability analysis: Landau equation
Meheboob Alam and Priyanka Shukla
When a dense granular material is sheared in shear-cell experiments, shearing remains confined to a narrow localized zone (``shearband'') near the moving boundary. Such shear-banding has also been realized in the molecular dynamics simulations of granular plane Couette flow for a range of densities (even without gravity) in the rapid flow regime. (This shear-banding phenomenon seems to be generic in many complex fluids, like suspensions, polymeric fluids, etc.) Here our goal is to explore whether we could describe the shear-banding phenomenon in plane Couette flow via an {\it order-parameter} equation. Starting from the continuum equations of rapid granular flows, we derived the Landau equation for the plane Couette flow using {\it nonlinear} stability theory. When the shear flow is perturbed by `finite-amplitude' disturbances, it is shown that the square of the disturbance-amplitude satisfies the Landau equation: A^{-1}\frac{dA}{dt}&=& a^{(0)}+ Aa^{(1)}+A^{2} a^{(2)} + \ldots This `order-parameter' equation describes the onset and the subsequent dynamics of shearband formation near the critical point. To find the actual behavior of flow due to finite-amplitude disturbances, we need to calculate the Landau coefficients $ a^{(n)}$ for $n>0$ which can be expressed in terms of a suitable inner-product of the nonlinear terms and the eigenfunctions of the linearized problem. It is shown that the coefficients $a^{(n)}$ for odd '$n$' are zero; the coefficients $a^{(n)}$ for even values of `$n$' are non-zero which are of interest for the study of shearband formation. In the context of granular flows, this is the first derivation of Landau equation from the standard set of continuum equations. It may be noted that the present order-parameter equation is `exact' in the sense that there are no fitting parameters (unlike in some recently postulated empirical order-parameter equation for granular flows).

Riccardo ARTONI
DIPIC, Universita di Padova, Italy
A fluctuating energy model for dense granular flows
Riccardo Artoni, Andrea Santomaso and Paolo Canu
We address the slow, dense flow of granular materials as a continuum with the incompressible Navier-Stokes equations plus the fluctuating energy balance for granular temperature. The pseudo-fluid is given an apparent viscosity, for which we choose, for the sake of simplicity, an Arrhenius-like dependence from granular temperature, as in ordinary liquids. We derive the fluctuating energy balance following Savage (J. Fluid Mech (1998) 377, 1-26); this balance includes a 'mobility enhancing' term due to shear stress which comes out directly from the derivation and a jamming, dissipative term which we assume, as a constitutive hypothesis, dependent on the isotropic part of the stress tensor and on shear rate. We propose a 'chemical' interpretation of the phenomenology described by the model in terms of reaction rates: the absolute rate of the process is governed by shear rate as a kinetic constant, while the 'activity' of the process is given by the distance from Mohr-Coulomb yield stress. Results for some 2-D standard geometries of flow are reported, as suggested in the paper by GDR Mi.Di. (Eur. Phys. J. E 14 (4), 341-365; vertical chute, rotating drum, silo). These results agree semi-quantitatively with experimental and DEM observations. In particular, our model seems to be able to reproduct the formation of stagnant zones of a characteristic form (e.g. wedge-shaped static zones in a silo with flat bottom) without prescribing them a-priori as it was often done in the past (for necessity) with erosion techniques. This is made possible by means of the progressive cooling term related to pressure's rearranging action. Despite its simple formulation and numerical implementation, the model and approach seem very promising to predict several original features of the dense flowing of granular materials, giving insight into a possible continuum view of jamming dynamics. The model is now being tuned quantitatively to literature and our own experimental data; in the near future we are going to work both on the 3-D implementation and on giving a stronger physical meaning to the phenomenology called into play.

Emilien AZEMA
LMGC, Montpellier, France
Influence of particle shape in granular media
Emilien Azema, Farhang Radjai, Robert Peyroux and Gilles Saussine
We perform a detailed analysis of the contact force network in a dense confined packing of pentagonal particles simulated by means of the contact dynamics method. The effect of particle shape is evidenced by comparing the data from pentagon packing and from a packing with identical characteristics except for the circular shape of the particles. A counterintuitive finding of this work is that, under steady shearing, the pentagon packing develops a lower structural anisotropy than the disk packing. We show that this weakness is compensated by a higher force anisotropy, leading to enhanced shear strength of the pentagon packing. We revisit strong and weak force networks in the pentagon packing, but our simulation data provide also evidence for a large class of very weak forces carried mainly by vertex-to-edge contacts. The strong force chains are mostly composed of edge-to-edge contacts with a marked zig-zag aspect and a decreasing exponential probability distribution as in a disk packing.

CEA, Saclay, France
Steady states in granular surface flows: Investigation through nonsmooth contact dynamics simulations
Daniel Bonamy, Mathieu Renouf, Berengere Dubrulle, Francois Daviaud and Pierre-Henry Chavanis
Steady surface flows in rotating drum are investigated through non-smooth contact dynamics simulations. Profiles of volume fraction, velocity, rms velocity, strain rate, and stress tensor ${\bf \Sigma}$ are measured at the midpoint along the length of the surface-flowing layer, where the flow is generally considered as steady and homogeneous. Analysis of these data and their interrelations suggest that the inertial number $I$ (defined as the ratio between the typical time of deformation and the typical time of confinement) is the relevant dimensionless parameter to describe the transition from the quasistatic part of the packing to the flowing part. Constitutive laws relating the components of the stress tensor to the inertial number are determined. The effective friction $\mu=\Sigma_{xz}/\Sigma_{zz}$ id found to increase logarithmically with $I$, independently of the rotating speed, as observed in other geometries. On the other hand, the ratio $k= \Sigma_{xx}/\Sigma_{zz}$ is found (i) to be significantly different from $k=1$ and (ii) to depend non univocally on $I$, in contrast to what was observed in the other geometries. We then considered an alternative description inspired from the modelling of turbulent fluids. Assuming that in steady states, forcing and dissipation equilibrates locally, one can obtained a full characterization of the surface flows through two general functions that relate granular temperature, vorticity, and stream function. We checked successfully these predictions on the steady surface flows observed in the simulated rotating drum. Some possible ways to derive these two functions using statistical physics will be finally discussed briefly.

LPTMC, Paris, France
Simulation of a 3d vibrated granular system
A. Burdeau and P. Viot
The existence of dissipation in granular systems leads to large non-Gaussian features in the velocity distributions. These deviations can be interpreted as the signature of correlations between particle velocities. Recently, Baxter and Olafsen studied a system where the velocity distributions of granular particles were found to exhibit Gaussian behavior. The experiments were done on a bidisperse system, with heavy particles in a dense first layer on a vibrating base, and lighter particles in a second layer. They observed that the horizontal velocity distribution of the upper-layer particles displayed a quasi-Gaussian behavior over a wide range of excitation frequencies. In order to understand this striking phenomenon, we have performed a Molecular dynamics simulation of viscoelastic spheres in physical conditions which closely mimic the experimental conditions. Our simulations are in very good agreement with experimental results over a wide range of parameters. In addition, we have investigated several microscopic quantities, and in particular the existence of correlations between angular and translational velocities of granular particles.

Francois CHEVOIR
LMSGC, Institut Navier, Champs-sur-Marne, France
Flow and jamming of granular mixtures through obstacles
F. Chevoir, F. Gaulard and N. Roussel
Due to the formation of arches, granular materials may jam when flowing through obstacles, as in the case of hoppers. As a way to quantify this process, we study experimentally the flow of binary granular mixtures through sieves, as a function of two parameters : the proportion of large grains and the size ratio between large grains and holes of the sieve. We distinguish three regimes : steady flows, jamming, and progressive clogging. In the case of steady flows, we measure the dependencies of the flow rate on the two parameters and observe a generalization of the law known for mono-disperse grains flowing through a single aperture. Moreover we measure how the critical size of the holes leading to jamming depends on the proportion of large grains. In the case of progressive clogging, we measure the slowing down of the flow rate and identify two mechanisms associated to the trapping of the large grains in the holes of the sieves and then to the formation of a filtration cake.

PMMH, ESPCI, Paris, France
Penetration of an intruder into a 2 dimensional disordered granular medium
Pierre Cixous, Evelyne Kolb, Jean-Claude Charmet, Denis Vallet and Eric Clement
We report on 2D experiments aiming to understand the mobility inside a dense and disordered granular system made of bidisperse cylinders. We investigate different packing fractions close to jamming by fixing the number of grains and changing the surface of the horizontal rectangular cell. A cylindrical intruder is moving at a constant velocity relatively to the granular medium. We simultaneously record the force experienced by the intruder during its penetration and monitor by use of a CCD camera the positions of the grains, which allows determining the displacement fields produced by the intruder motion. We will present our first results on the coupling of the level of forces and their fluctuations with the rearrangements of grains as a function of the packing fraction of the granular medium and the speed of the intruder.

GMCM, Universite de Rennes, France
An experimental study of the creeping flow of a granular material
Jerome Crasous, Sebastien Kiesgen de Richter and Jean-Francois Metayer
We studied experimentally the creeping flow of a granular material submitted to gravity between two parallel plates. The velocity profile is measured with particle tracking for the higher velocity, and with a dynamic light scattering technique for the lower velocity. With the combination of those two techniques, we are able to measure a mean velocity from 1m/s to 1nm/s. The creeping part of the velocity profile is found to decay logarithmically with a characteristic length xi=1.1d, and the logarithmic decay is found to hold on at least 15*xi.

Laboratoire FAST, Orsay, France
Experimental evidence of a neutral angle in granular avalanche dynamics
R. Fischer, P. Gondret, B. Perrin and M. Rabaud
A fine analysis of the statistics of dry granular avalanches reveals that the angle of repose at which ends an avalanche is correlated experimentally to the angle of maximal stability at which the avalanche starts. This correlation gives evidence of a well defined intermediate ``neutral" angle that characterizes the corresponding granular pile. In addition, we characterize the time dynamics of the avalanche and show that the avalanche time duration is correlated to the avalanche amplitude, being smaller for higher avalanche amplitude. The time relaxation of the pile slope during any avalanche is governed by the deviation of the initial starting angle from the neutral angle and follows the same master curve when plotted in a normalized way. A simple modelisation recovers most of the results.

Department of Mechanical and Aerospace Engineering University of California, San Diego
From Granular Networks to Graded Continua
J. D. Goddard
This paper is concerned with the derivation of "structured" or multipolar continuum models for the quasi-static mechanics of granular media. As one motivation, the higher spatial gradients in such models provide a multi-scale description with the capability of regularizing the descriptions of shear bands arising from material instability. A second motivation is the systematic development of continuum models from discrete network models, of the type that occur in numerous fields of physics and engineering. The present work represents a continuing effort to develop an energy-based procedure for the definition of continuum stress and kinematics [1]. Based on a variant of the so-called "generalized additive models" (GAMs) of statistics [2], it addresses the fundamental statistical mechanics and thermodynamics may have limited applicability. After a necessary modification of the energy-based metric proposed in [1], some "toy" models of flow in narrow slits are treated to illustrate the extraction of strain gradients from extremely noisy data. As discussed further, the challenging and fundamental problem still remains of providing a sound physical foundation, e.g. an energy principle, for otherwise purely statistical methods. References 1, J.D. Goddard. From granular matter to generalized continuum. In P. Mariano, G. Capriz, and P. Giovine, editors, Mathematical models of granular matter, Lecture Notes in Mathematics, to appear. Springer, Berlin, 2007 (http://maeresearch.ucsd.edu/goddard/archives/). 2, S.N. Wood. Stable and efficient multiple smoothing parameter estimation for generalized additive models. J. Am. Statistical Assoc., 99(467):673-86, 2004.

PMMH, ESPCI, France, and Universidad Simon Bolivar, Venezuela
Silo Collapse During Discharge: An Experimental Study
Gustavo Gutierrez, Philippe Boltenhagen and Eric Clement
Thin walled high cylindrical structures for the storage of grains are widely used in a large variety of industrial applications. Cylindrical silos are subjected to vertical stresses due to the friction between the stored grains and the walls. Shell buckling during grain discharge is a common source of collapse of silos and this important practical problem is still poorly understood. The purpose of this work is to develop basic insight into the pre-buckling behavior and the buckling transition toward plastic collapse by studying experimentally the different patterns of deformation observed in thin paper cylindrical shells partially filled with spherical glass beads during gravity driven discharge of the grains. We seek to characterize the way the under pressure developed at the internal surface of the silo, caused by the flow of grains during discharge, affects the buckling instability and induces the collapse. We change the height of the column and observed a variety of patterns formed on the surface of the cylindrical shell, close to the collapse threshold. We change rigidity of the shell, the flow rate and the grain sizes. The threshold height is weakly dependent on the sizes of the grains and the diameter of the holes. The time between the opening of the discharge hole and the collapse of the silo, just above the threshold height, was observed to depend on the grain size. We investigate the diamond shaped array of patterns that appear and propagate in different ways, depending on the characteristics of the grains and the shell used. We characterize a pattern growth that rises around the cylinder in a spiral path and develops into a plastic collapsing fold that grows around the collapsing silo.

Lappeenranta University of Technology, Finland
Rough cylindrical object within the granular stream of hard disks
Payman Jalali and Pertti Sarkomaa
We have investigated stresses on rough cylindrical objects confronted with a granular stream of inelastic hard disks. The roughness of cylindrical objects is created via covering their surfaces with hard disks of given size and material. We have employed event-driven simulations using restitution coefficients dependent of the impact velocity in a collision. We report the effect of material property (restitution coefficient) on development of granular shock wave around the object with corresponding stresses exerted on it. The role of the roughness of the object in resulting flow is also studied. Moreover, simulations are performed in two conditions with gravity and without it.

CEA Saclay, France
The jamming transition and beyond: Density dependence of the relevant length and time scales in a horizontally vibrated granular monolayer
F. Lechenault, O. Dauchot, J.-P. Bouchaut and G. Birolli
A dense amorphous monolayer of hard disks is horizontally driven by a glass plate oscillating underneath while confined in a fixed rectangular cell. As the packing fraction is decreased, the system exhibits at a certain packing fraction $\phi_c$ a transition between a totally jammed state in which static and dynamic pressures are the same, to a ``supercooled'' regime in which the kinetic contribution becomes dominant and the static pressure drops to zero. We characterize the diffusion properties of such packing across the transition. Furthermore, we compute the so- called dynamical susceptibility $\chi_4(\tau,\bf{k})$. First we show that the former exhibits two time scales, one of which being strongly peaked in the neighborhood of $\phi_c$. Then, after having identified the coarse-graining length scale for which the latter is well-defined, we find that the cooperative length scale it points out increases as the packing fraction is increased toward the transition and then drops abruptly as $\phi_c$ is crossed, suggesting a critical behaviour across the jamming transition.

Nicolas RIVIER
IPCMS, Universite Louis Pasteur, Strasbourg, France
Dynamics of jammed granular matter
N. Rivier
Dry granular matter is modelled as a graph of grains linked by purely repulsive contacts. Its stability (jamming) is insured by odd circuits that prevent the grains from rolling on each other. A topological dynamical matrix is associated with the graph; it has a spectrum of low-energy excitations characteristic of dry, disordered granular matter. In the limit of large stiffness-to load ratio, dry granular matter has two possible dynamical states, dry fluid and jammed, rigid but fragile solid. Hard granular materials differ in three ways from the structural rigidity analysis of elastic networks, from floppy to overconstrained, through isostatic at the rigidity percolation threshold. a) Stresses carried by edges have a sign constraint. It follows that (repulsive) self-stresses cannot be sustained in an overconstrained network: the two grains connected by a redundant edge pull apart, and the network becomes isostatic. On the other hand, a floppy network will rearrange and add edges, if it can, to become isostatic. b) But hinges in a floppy network can only buckle through grains rolling on each other, a motion blocked by odd circuits. In the absence of odd circuits, the material is a dry fluid, a three-dimensional ball-bearing. c) Odd circuits do not occur in isolation, but surround closed so-called R-loops (odd vorticity). In disordered granular materials, the largest R-loop extends across the entire material. The odd circuits surrounding the largest R-loop are responsible for the high density (independent of the size of the material and of the dimension) of low-energy excitations and for the extended corresponding eigenstates, with localized and uncorrelated stresses in (disordered) granular matter at the jamming transition, obtained in simulations and in a scaling analysis [1,2] [1] C.S. OíHern, S.A. Langer, A.J. Liu, S.R. Nagel, Phys. Rev. Letters 88, 075507 (2002) [2] M. Wyart, S.R. Nagel, T.A. Witten, Europhys. Lett. 72, 486 (2005) .

Cinvestav, Monterrey, Mexico
A Torricelli-like granular flow?
Hector Pacheco-Martinez, , Henk Jan van Gerner, and J. Carlos Ruiz-Suarez
Experiments and computational simulations are carried out to study the behavior of a granular column in a silo whose walls are able to vibrate horizontally. If the walls are still, the Janssenís effect is observed. When the walls of the silo vibrate, the bed is brought to a steady-fluidized-state with no convection and it behaves like a hydrostatic system. Archimedean buoyancy is observed on light intruders. We also study the dynamics of the granular outflow through openings at the bottom of the silo in order to search for possible deviations from Beverloo's law.

Universidad Simon Bolivar, Venezuela
Vertical granular transport in a vibrated U-tube
Ivan Sanchez, Ramon Darias, Ricardo Paredes and Gustavo Gutierrez
Energy injected on a granular system using vibrations generates many interesting phenomena like pattern formation, segregation and collective motion of the grains. Narrow tubular containers enhance the role of the walls in the interchange of energy, and are extremely important from the industrial point of view. Many industrial processes involve the transport of grains through pipes and are unavoidably affected by mechanical vibration, therefore the reliability of granular transport processes can be greatly affected by the knowledge of the physics of granular bulk flow induced by vibrations. We study experimentally the collective motion of grains inside a U shaped tube undergoing vertical low frequency oscillations. We observe that the height differences between the granular columns grow with time when the system is shaken at sufficiently low frequencies. One of the columns can grow until the other one is empty. This growth can be exponentially divergent, or saturating, depending on the size of the grains. We develop a very simple quantitative model that captures relevant features of the observed instability. The model is based on the idea of cyclic fluidization that occurs when the granular medium is fluidized during a fraction of the period of each oscillation. The exponentially divergent growth can be quenched by removing the air, whereas the saturating behavior can occur in the absence of air. A good agreement between theory and experiment is obtained.

Nagoya University, Japan
Dynamics of dissipative system of asymmetric interaction and the N-body problem of the emergence of moving cluster
Yuki Sugiyama, Katsutoshi Masuoka and Takahiro Ishida
Collective motion of self-driven particles is a non-equilibrium dissipative system with asymmetric non-linear interaction. Investigations of such kind of systems can be applied to many other phenomena of dynamical pattern formation of physical, chemical and biological systems. The characteristic properties of such systems are, i) dynamical phase transition (bifurcation) to a non-trivial phase, ii) emergence of macroscopic scale (pattern formation), iii) emergence of macroscopic time-scale (rhythm), iv) Power law behavior of fluctuations, etc. These general properties are fully appeared in minimal Newtonian equation of 1-D system of particles in non-linear asymmetric interaction with dissipative (viscous) term, which is the physical meaning of so called Optimal Velocity Model (, which is first introduced as a traffic flow model.) In such systems, the inseparable relation between the asymmetry and dissipation in non-equilibrium system is clearly understood in comparison of nonlinear-interacting particles of energy conserved system (e.g. Toda Lattice) and oscillation systems. Moreover, N-body problem of the phenomena of emergence of moving clusters are exactly solved as idealized cases for N=2, 3(even in this case a cluster emerges), 4, ... general- N, which provides the essential information such that why the particle-number N (or density) is a control parameter for the instability of a system? why small- N is large enough as many-body system, etc. These properties are sharply contrasted to those of energy-conserved systems.

Laboratoire de Physique, ENS Lyon, France
Leidenfrost granular flows
N. Taberlet, P. Richard and R. Delannay
We present numerical findings on rapid 2D and 3D granular flows on a bumpy base. In the supported regime studied here, a strongly sheared, dilute and agitated layer spontaneously appears at the base of the flow and supports a compact packing of grains moving as a whole. In this regime, the flow behaves like a sliding block on the bumpy base. In particular, for flows on a horizontal base, the average velocity decreases linearly in time and the average kinetic energy decreases linearly with the travelled distance, those features being characteristic of solid-like friction. This allows us to define and measure an effective friction coefficient, which is independent of the mass and velocity of the flow. This coefficient only loosely depends on the value of the micromechanical friction coefficient whereas the influence of the bumpiness of the base is strong. We give evidence that this dilute and agitated layer does not result in significantly less friction. Finally, we show that a steady regime of supported flows can exist on inclines whose angle is carefully chosen.

Department of Physics, University of Oslo, Norway
Granular Rayleigh-Taylor instability: experiments and simulations
Jan Ludvig Vinningland, Oistein Johnsen, Eirik G. Flekkoy, Renaud Toussaint and Knut Jorgen Maaloy
A granular instability is studied experimentally and numerically as a packing of dense granular material above a gap of air falls under gravity in a closed Hele-Shaw cell; a granular analog of the Rayleigh-Taylor instability. The initially flat front defined by the grains subsequently develops into a characteristic pattern of falling granular fingers separated by rising bubbles of air. In contrast to the classical hydrodynamic instability a transient coarsening of the front is observed right from the start by a finger merging process. The coarsening is stabilized by new fingers growing from the center of the rising bubbles. The emerging structures are quantified by means of Fourier analysis and quantitative agreement between experiment and computation is shown. This analysis also reveals scale invariance of the flow structures under overall change of spatial scale.

LMGC, Montpellier, France
Space-filling properties of polydisperse particles
C. Voivret, J.-Y. Delenne, M.S. El Youssoufi and F. Radjai
The space-filling features of granular materials are strongly influenced by the particle size distribution (PSD). Discrete element methods are not computationally efficient enough to allow for the generation of well-represented samples of highly polydisperse particles. We present a PSD model in association with a protocol of geometrical deposition, which enables us to perform a systematic 2D study of the arrangement properties (compactness, connectivity, anisotropy) as a function of the PSD. A remarkable result of this study is the nonlinear variation of compactness with polydispersity. We observes in particular a regime where compactness does not evolve with polydispersity. A transition takes place for a critical polydispersity towards more compact states when the PSD is sufficiently graded to allow for the insertion of small particles in the pores formed by larger ones. This transition is also observed for the contact network which undergoes a transition from order to disorder at the same critical point.The particle connectivity evolves with polydispersity and tends to decrease the orientational order. In the same way the translational order decrease and completely disappear after the transition.