Granular flow down an inclined plane: Bagnold scaling and rheology

Objective decomposition of the stress tensor in granular flows

A model for the stress tensor in granular flows [Volfson, Tsimring, and Aranson, Phys. Rev. Lett. 90, 254301 (2003)] is correctly generalized to an objective form that is independent of the coordinate system. The objective representation correctly models the isotropic and anisotropic parts of the stress tensor, whereas the original model for stress tensor components is dependent on the coordinate system. This general objective form of the model also relaxes the assumption in the original model that the principal axes of the granular stress tensor be coaxial with that of the "fluid" stress tensor. This generalization expands the applicability of the model to a wider class of granular flows. The objective representation is also useful in analyzing other models based on additive decomposition of the stress tensor in granular flows.

Exactly solvable model for driven dissipative systems

We introduce a solvable stochastic model inspired by granular gases for driven dissipative systems. We characterize far from equilibrium steady states of such systems through the non-Boltzmann energy distribution and compare different measures of effective temperatures. As an example we demonstrate that fluctuation-dissipation relations hold, however, with an effective temperature differing from the effective temperature defined from the average energy.

Velocity correlations in dense granular flows

Velocity fluctuations of grains flowing down a rough inclined plane are experimentally studied. The grains at the free surface exhibit fluctuating motions, which are correlated over a few grain diameters. The characteristic correlation length is shown to depend on the inclination of the plane and not on the thickness of the flowing layer. This result strongly supports the idea that dense granular flows are controlled by a characteristic length larger than the particle diameter.

Effects of surface friction on a two-dimensional granular system: cooling bound system

Experiments performed by Phys. Rev. E 62, 2380 (2000)] on two-particle collisions and dynamics emphasized the importance of the role played by substrate friction, in particular kinetic friction, on the particle dynamics after collisions on a substrate. We present a numerical model which accounts for collisional and surface frictional dissipation and their influence on particle dynamics for a quasi-two-dimensional cooling initially dilute granular material. This model makes the simplifying assumption that the collision dynamics is determined solely by the incoming velocity and angular velocities of the colliding particles. We apply this model to a numerical simulation of a monolayer of monodisperse particles moving on a substrate, enclosed between inelastic walls. We find that surface friction-in particular, kinetic friction-plays a dominant role in determining the dynamics of quasi-two-dimensional multiparticle systems where the particles are in continuous contact with a substrate. Results from simulations performed for different system sizes indicate that surface friction and the inelastic walls lead to clustering of the particles in and near the vicinity of the walls. We find that the rate of decrease of average total kinetic energy is the highest when the majority of the particles have just collided and are experiencing kinetic frictional forces and torques. We also find from our calculations that, on average, particle-wall collisions lead to more dissipation than particle-particle collisions for a single particle for fixed restitutional parameters.

Anisotropy-driven dynamics in vibrated granular rods

The dynamics of a set of rods bouncing on a vertically vibrated plate is investigated using experiments, simulations, and theoretical analysis. The experiments and simulations are performed within an annulus to impose periodic boundary conditions. Rods tilted with respect to the vertical are observed to spontaneously develop a horizontal velocity depending on the acceleration of the plate. For high plate acceleration, the rods are observed to always move in the direction of tilt. However, the rods are also observed to move opposite to direction of tilt for a small range of plate acceleration and rod tilt. A phase diagram of the observed motion is presented as a function of plate acceleration and the tilt of the rods which is varied by changing the number of rods inside the annulus. Next we introduce a molecular dynamics method to simulate the dynamics of the rods using the dimensions and dissipation parameters from the experiments. We reproduce the observed horizontal rod speeds as a function of rod tilt and plate acceleration in the simulations. By decreasing the friction between the rods and the base plate to zero in the simulation, we identify the friction during the collision as the crucial ingredient for occurrence of the horizontal motion. Guided by the data from the experiments and the simulations, we construct a mechanical model for the dynamics of the rods in the limit of thin rods. The starting point of the analysis is the collision of a single rod with an oscillating plate. Three friction regimes are identified: slide, slip-stick, and slip reversal. A formula is derived for the observed horizontal velocity as a function of tilt angle. Good agreement for the horizontal velocity as a function of rod tilt and plate acceleration is found between experiments, simulations and theory.

On dense granular flows

The behaviour of dense assemblies of dry grains submitted to continuous shear deformation has been the subject of many experiments and discrete particle simulations. This paper is a collective work carried out among the French research group Groupement de Recherche Milieux Divisés (GDR MiDi). It proceeds from the collection of results on steady uniform granular flows obtained by different groups in six different geometries both in experiments and numerical works. The goal is to achieve a coherent presentation of the relevant quantities to be measured i.e. flowing thresholds, kinematic profiles, effective friction, etc. First, a quantitative comparison between data coming from different experiments in the same geometry identifies the robust features in each case. Second, a transverse analysis of the data across the different configurations, allows us to identify the relevant dimensionless parameters, the different flow regimes and to propose simple interpretations. The present work, more than a simple juxtaposition of results, demonstrates the richness of granular flows and underlines the open problem of defining a single rheology.

On dense granular flows

The behaviour of dense assemblies of dry grains submitted to continuous shear deformation has been the subject of many experiments and discrete particle simulations. This paper is a collective work carried out among the French research group Groupement de Recherche Milieux Divisés (GDR MiDi). It proceeds from the collection of results on steady uniform granular flows obtained by different groups in six different geometries both in experiments and numerical works. The goal is to achieve a coherent presentation of the relevant quantities to be measured i.e. flowing thresholds, kinematic profiles, effective friction, etc. First, a quantitative comparison between data coming from different experiments in the same geometry identifies the robust features in each case. Second, a transverse analysis of the data across the different configurations, allows us to identify the relevant dimensionless parameters, the different flow regimes and to propose simple interpretations. The present work, more than a simple juxtaposition of results, demonstrates the richness of granular flows and underlines the open problem of defining a single rheology.

Dynamics of drag and force distributions for projectile impact in a granular medium

Our experiments and molecular dynamics simulations on a projectile penetrating a two-dimensional granular medium reveal that the mean deceleration of the projectile is constant and proportional to the impact velocity. Thus, the time taken for a projectile to decelerate to a stop is independent of its impact velocity. The simulations show that the probability distribution function of forces on grains is time independent during a projectile's deceleration in the medium. At all times the force distribution function decreases exponentially for large forces.

Granular packings with moving side walls

The effects of movement of the side walls of a confined granular packing are studied by discrete element, molecular dynamics simulations. The dynamical evolution of the stress is studied as a function of wall movement both in the direction of gravity as well as opposite to it. For all wall velocities explored, the stress in the final state of the system after wall movement is fundamentally different from the original state obtained by pouring particles into the container and letting them settle under the influence of gravity. The original packing possesses a hydrostaticlike region at the top of the container which crosses over to a depth-independent stress. As the walls are moved in the direction opposite to gravity, the saturation stress first reaches a minimum value independent of the wall velocity, then increases to a steady-state value dependent on the wall velocity. After wall movement ceases and the packing reaches equilibrium, the stress profile fits the classic Janssen form for high wall velocities, while some deviations remain for low wall velocities. The wall movement greatly increases the number of particle-wall and particle-particle forces at the Coulomb criterion. Varying the wall velocity has only small effects on the particle structure of the final packing so long as the walls travel a similar distance.

Stick-slip dynamics of a granular layer under shear

Stick-slip regime of shear granular flows is studied theoretically and numerically. Numerical experiments are carried out for a thin Couette cell using soft-particle molecular dynamics code in two dimensions. We apply order parameter theory of partially fluidized granular flows and find a good agreement with simulations and experiments by Nasuno et al.

Superstable granular heap in a thin channel

We observed experimentally a new regime for granular flows in an inclined channel with a flow-rate-controlled system. For high flow rates, the flow occurs atop a static granular heap whose angle is considerably higher than those usually exhibited by granular heaps. The properties of such superstable heaps (SSH) are drastically affected by a change in the channel width W. This indicates that the unusual stability of these heaps can be accounted for by the flowing layer and its friction on the sidewalls. A simple depth-averaged model, assuming Coulomb friction, shows that the SSH angle scales as h/W (W being the channel width), and that grain size plays no part.

Two-dimensional inclined chute flows: transverse motion and segregation

We present an experimental study of two-dimensional dense inclined chute flows consisting of both monodisperse and bidisperse disks. We analyzed the trajectories of the particles within the flow in a steady regime. (i) In monodisperse flows, particles are arranged in layers that are in motion relative to one another, and it is found that the particles have a nonzero probability of being transferred to adjacent layers. We measured the mean time spent by a particle in a given layer. This residence time is found to decrease with increasing layer height. The particle transfer between layers can be interpreted as transverse motion of a diffusive nature. The diffusion coefficient associated with each layer increases linearly with the layer height. (ii) In polydisperse flows consisting of a small percentage (less than 1%) of small disks among large ones, the small particles have a net downward motion on which a fluctuating behavior is superimposed. At short times, the small particle motion can be described as a biased Brownian motion. The ratio of the characteristic time of diffusion to that of convection is found to increase with the layer height, indicating that the segregation process is more efficient in the upper layers of the flow. At longer times, the transverse motion of the small particles seems to differ greatly from a classical biased Brownian motion.

Internal granular dynamics, shear-induced crystallization, and compaction steps

Internal imaging using index matching, and sensitive volume measurement, are used to investigate the spatial order and dynamics of a deep disordered layer of spheres sheared under a fixed load. Shearing triggers a crystallization transition accompanied by a step compaction event. The delay preceding the transition depends strongly on the layer thickness and can require a translation of about 10(5) particle diameters. The mean velocity varies with depth by more than five decades, and its profile is qualitatively altered by the transition.

Partially fluidized shear granular flows: continuum theory and molecular dynamics simulations

The continuum theory of partially fluidized shear granular flows is tested and calibrated using two-dimensional soft particle molecular dynamics simulations. The theory is based on the relaxational dynamics of the order parameter that describes the transition between static and flowing regimes of granular material. We define the order parameter as a fraction of static contacts among all contacts between particles. We also propose and verify by direct simulations the constitutive relation based on the splitting of the shear stress tensor into a"fluid part" proportional to the strain rate tensor, and a remaining "solid part." The ratio of these two parts is a function of the order parameter. The rheology of the fluid component agrees well with the kinetic theory of granular fluids even in the dense regime. Based on the hysteretic bifurcation diagram for a thin shear granular layer obtained in simulations, we construct the "free energy" for the order parameter. The theory calibrated using numerical experiments with the thin granular layer is applied to the surface-driven stationary two-dimensional granular flows in a thick granular layer under gravity.

Order parameter description of stationary partially fluidized shear granular flows

We carry out a detailed comparison of soft-particle molecular dynamics simulations with the theory of partially fluidized shear granular flows. We verify by direct simulations a constitutive relation based on the separation of the shear stress tensor into a fluid part proportional to the strain rate tensor, and a remaining solid part. The ratio of these two components is determined by the order parameter. Based on results of the simulations we construct the "free energy" function for the order parameter. We also present the simulations of the stationary deep 2D granular flows driven by an upper wall and compare it with the continuum theory.

Model for dense granular flows down bumpy inclines

We consider dense flows of spherical grains down an inclined plane on which spherical bumps have been affixed. We propose a theory that models stresses as the superposition of a rate-dependent contribution arising from collisional interactions and a rate-independent part related to enduring frictional contacts among the grains. We show that dense flows consist of three regions. The first is a thin basal layer where grains progressively gain fluctuation energy with increasing distance from the bottom boundary. The second is a core region where the solid volume fraction is constant and the production and dissipation of fluctuation energy are nearly balanced. The last is a thin collisional surface layer where the volume fraction abruptly vanishes as the free surface is approached. We also distinguish basal flows with the smallest possible height, in which the core and surface layers have disappeared. We derive simple closures of the governing equations for the three regions with insight from the numerical simulations of Silbert et al. [Phys. Rev. E64, 051302 (2001)] and the physical experiments of Pouliquen [Phys. Fluids 11, 542 (1999)]. The theory captures the range of inclination angles at which steady, fully developed flows are observed, the corresponding shape of the mean and fluctuation velocity profiles, the dependence of the flow rate on inclination, flow height, interparticle friction, and normal restitution coefficient, and the dependence of the height of basal flows on inclination.

Confined granular packings: structure, stress, and forces

The structure and stresses of static granular packs in cylindrical containers are studied by using large-scale discrete element molecular dynamics simulations in three dimensions. We generate packings by both pouring and sedimentation and examine how the final state depends on the method of construction. The vertical stress becomes depth independent for deep piles and we compare these stress depth profiles to the classical Janssen theory. The majority of the tangential forces for particle-wall contacts are found to be close to the Coulomb failure criterion, in agreement with the theory of Janssen, while particle-particle contacts in the bulk are far from the Coulomb criterion. In addition, we show that a linear hydrostaticlike region at the top of the packings unexplained by the Janssen theory arises because most of the particle-wall tangential forces in this region are far from the Coulomb yield criterion. The distributions of particle-particle and particle-wall contact forces P(f) exhibit exponential-like decay at large forces in agreement with previous studies.

Hard-sphere limit of soft-sphere model for granular materials: stiffness dependence of steady granular flow

Dynamical behavior of steady granular flow is investigated numerically in the inelastic hard-sphere limit of the soft-sphere model. We find distinctively different limiting behaviors for the two flow regimes, i.e., the collisional flow and the frictional flow. In the collisional flow, the hard-sphere limit is straightforward; the number of collisions per particle per unit time converges to a finite value and the total contact time fraction with other particles goes to zero. For the frictional flow, however, we demonstrate that the collision rate diverges as the power of the particle stiffness so that the time fraction of the multiple contacts remains finite even in the hard-sphere limit, although the contact time fraction for the binary collisions tends to zero.

Statistics of the contact network in frictional and frictionless granular packings

Simulated granular packings with different particle friction coefficient mu are examined. The distribution of the particle-particle and particle-wall normal and tangential contact forces P(f) are computed and compared with existing experimental data. Here f identical with F/(-)F is the contact force F normalized by the average value (-)F. P(f) exhibits exponential-like decay at large forces, a plateau/peak near f=1, with additional features at forces smaller than the average that depend on mu. Additional information beyond the one-point force distribution functions is provided in the form of the force-force spatial distribution function and the contact point radial distribution function. These quantities indicate that correlations between forces are only weakly dependent on friction and decay rapidly beyond approximately three particle diameters. Distributions of particle-particle contact angles show that the contact network is not isotropic and only weakly dependent on friction. High force-bearing structures, or force chains, do not play a dominant role in these three-dimensional, unloaded packings.

Stability of monomer-dimer piles

We present an experimental and theoretical study of piles consisting of monodisperse spherical grains mixed with a weight fraction nu(d) of dimer grains made by the rigid bonding of two such spherical grains. The maximum static angle of stability tantheta(c) of the pile increases from 0.45 to 1.1 and the grain packing fraction Phi decreases from 0.58 to 0.52 as nu(d) is increased from 0 to 1. The stability of these piles appears to be controlled by the grains sitting on the surface, which roll out of their local "traps" as the tilt angle is increased. We attribute the increase in tan theta(c)(nu(d)) to the enhanced stability of dimers on the surface, such that at higher tilt angles, there are sufficiently many stable surface traps available to accommodate the reduced density of monomers on the surface. A full characterization of the grain-scale roughness of the surface is required to quantitatively account for the changes in theta(c) with nu(d).

Hydrodynamic model for a dynamical jammed-to-flowing transition in gravity driven granular media

Granular material on an inclined plane will flow like a fluid if the angle theta the plane makes with the horizontal is large enough. We study chute flow down a plane using a hydrodynamic model previously used to describe granular Couette flow. Our model predicts a jammed-to-flowing transition as theta is increased even though it does not include solid friction, which might seem necessary to stabilize a state without flow. The transition is driven by coupling between mean and fluctuating velocity. In agreement with experiments and simulations, it predicts flow for layers with a thickness H larger than a critical value H(stop)(theta) and absence of flow for H<H(stop)(theta).

Origin of a repose angle: kinetics of rearrangement for granular materials

A microstructural theory of dense granular materials is presented, based on two main ideas: first, that macroscopic shear results from activated local rearrangements at a mesoscopic scale; second, that the update frequency of microscopic processes is determined by granular temperature. In a shear cell, the resulting constitutive equations account for Bagnold's scaling and for the existence of a Coulomb criterion. In a granular flow down an inclined plane, they account for the rheology observed in experiments [Phys. Fluids 11, 542 (1999)]] and for temperature and velocity profiles measured numerically [Europhys. Lett. 56, 214 (2001)]] [Phys. Rev. E 64, 051302 (2001)]].

Continuum theory of partially fluidized granular flows

A continuum theory of partially fluidized granular flows is developed. The theory is based on a combination of the equations for the flow velocity and shear stresses coupled with the order-parameter equation which describes the transition between the flowing and static components of the granular system. We apply this theory to several important granular problems: avalanche flow in deep and shallow inclined layers, rotating drums, and shear granular flows between two plates. We carry out quantitative comparisons between the theory and experiment.

Analogies between granular jamming and the liquid-glass transition

Based on large-scale, three-dimensional chute flow simulations of granular systems, we uncover strong analogies between the jamming of the grains and the liquid-glass transition. The angle of inclination theta in the former transition appears as an analog of temperature T in the latter. The transition is manifested in the development of a plateau in the contact normal force distribution P(f) at small forces, the splitting of the second peak in the pair-correlation function g(r), and increased fluctuations of the system energy. The static state also exhibits history dependence, akin to the quench-rate dependence of structural properties of glasses, due to the hyperstaticity of the contact network.

Geometry of frictionless and frictional sphere packings

We study static packings of frictionless and frictional spheres in three dimensions, obtained via molecular dynamics simulations, in which we vary particle hardness, friction coefficient, and coefficient of restitution. Although frictionless packings of hard spheres are always isostatic (with six contacts) regardless of construction history and restitution coefficient, frictional packings achieve a multitude of hyperstatic packings that depend on system parameters and construction history. Instead of immediately dropping to four, the coordination number reduces smoothly from z=6 as the friction coefficient mu between two particles is increased.