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The lattice Boltzmann method has been successfully applied for the simulation of flow through porous media in the creeping regime. Its technical properties, namely discretization, straightforward implementation and parallelization, are responsible for its popularity. However, flow through porous media is not restricted to near zero Reynolds numbers since inertial effects play a role in numerous natural and industrial processes. In this paper we investigate the capability of the lattice Boltzmann method to correctly describe flow in porous media at moderate Reynolds numbers. The selection of the lattice resolution, the collision kernel and the boundary conditions becomes increasingly important and the challenge is to keep artifacts due to compressibility effects at a minimum. The lattice Boltzmann results show an accurate quantitative agreement with Finite Element Method results and evidence the capability of the method to reproduce Darcy's law at low Reynolds numbers and Forchheimer's law at high Reynolds numbers., Comment: 9 pages, 7 figures

Static and dynamic properties of two-dimensional bidisperse dissipative particles are numerically studied near the jamming transition. We investigate the dependency of the critical scaling on the ratio of the different diameters and find a new scaling of the maximum overlap (not consistent with the scaling of the mean overlap). The ratio of kinetic and potential energies drastically slows down near the jamming transition, i.e. the relaxation time diverges at the jamming point., Comment: 6 pages, 4 figures. Proceedings of the 28th International Symposium on Rarefied Gas Dynamics, AIP Conf. Proc. (2012), in press

The Discrete Particle Method (DPM) is used to model granular flows down an inclined chute. We observe three major regimes: static piles, steady uniform flows and accelerating flows. For flows over a smooth base, other (quasi-steady) regimes are observed where the flow is either highly energetic and strongly layered in depth for small inclinations, or non-uniform and oscillating for larger inclinations. For steady uniform flows, depth profiles of density, velocity and stress have been obtained using an improved coarse-graining method, which allows accurate statistics even at the base of the flow. A shallow-layer model for granular flows is completed with macro-scale closure relations obtained from micro-scale DPM simulations of steady flows. We thus obtain relations for the effective basal friction, shape factor, mean density, and the normal stress anisotropy as functions of layer thickness, flow velocity and basal roughness. For collisional flows, the functional dependencies are well determined and have been obtained., Comment: Will be presented at PARTICLES 2011 - CIMNE

An expression for the stress tensor near an external boundary of a discrete mechanical system is derived explicitly in terms of the constituents' degrees of freedom and interaction forces. Starting point is the exact and general coarse graining formulation presented by Goldhirsch in [I. Goldhirsch, Gran. Mat., 12(3):239-252, 2010], which is consistent with the continuum equations everywhere but does not account for boundaries. Our extension accounts for the boundary interaction forces in a self-consistent way and thus allows the construction of continuous stress fields that obey the macroscopic conservation laws even within one coarse-graining width of the boundary. The resolution and shape of the coarse-graining function used in the formulation can be chosen freely, such that both microscopic and macroscopic effects can be studied. The method does not require temporal averaging and thus can be used to investigate time-dependent flows as well as static and steady situations. Finally, the fore-mentioned continuous field can be used to define 'fuzzy' (highly rough) boundaries. Two discrete particle method simulations are presented in which the novel boundary treatment is exemplified, including a chute flow over a base with roughness greater than a particle diameter.

We study the effect of polydispersity on the macroscopic physical properties of granular packings in two and three dimensions. A mean-field approach is developed to approximate the macroscale quantities as functions of the microscopic ones. We show that the trace of the fabric and stress tensors are proportional to the mean packing properties (e.g. packing fraction, average coordination number, and average normal force) and dimensionless correction factors, which depend only on the moments of the particle-size distribution. Similar results are obtained for the elements of the stiffness tensor of isotropic packings in the linear affine response regime. Our theoretical predictions are in good agreement with the simulation results., Comment: 13 pages, 13 figures, 2 tables

The pressure and the viscosity in two-dimensional sheared granular assemblies are investigated numerically. The behavior of both pressure and viscosity is smoothly changing qualitatively when starting from a mono-disperse hard-disk system without dissipation and moving towards a system of (i) poly-disperse, (ii) soft particles with (iii) considerable dissipation. In the rigid, elastic limit of mono-disperse systems, the viscosity is approximately inverse proportional to the area fraction difference from $\phi_{\eta} \simeq 0.7$, but the pressure is still finite at $\phi_{\eta}$. In moderately soft, dissipative and poly-disperse systems, on the other hand, we confirm the recent theoretical prediction that both scaled pressure (divided by the kinetic temperature $T$) and scaled viscosity (divided by $\sqrt{T}$) diverge at the same density, i.e., the jamming transition point $\phi_J > \phi_\eta$, with the exponents -2 and -3, respectively. Furthermore, we observe that the critical region of the jamming transition becomes invisible as the restitution coefficient approaches unity, i.e. for vanishing dissipation. In order to understand the conflict between these two different predictions on the divergence of the pressure and the viscosity, the transition from soft to hard particles is studied in detail and the dimensionless control parameters are defined as ratios of various time-scales. We introduce a dimensionless number, i.e. the ratio of dissipation rate and shear rate, that can identify the crossover from the scaling of very hard, i.e. rigid disks to the scaling in the soft, jamming regime., Comment: 23 pages, 20 figures, to appear in Progress of Theoretical Physics Supplement

In dense, static, polydisperse granular media under isotropic pressure, the probability density and the correlations of particle-wall contact forces are studied. Furthermore, the probability density functions of the populations of pressures measured with different sized circular pressure cells is examined. The questions answered are: (i) What is the number of contacts that has to be considered so that the measured pressure lies within a certain error margin from its expectation value? (ii) What is the statistics of the pressure probability density as function of the size of the pressure cell? Astonishing non-random correlations between contact forces are evidenced, which range at least 10 to 15 particle diameter. Finally, an experiment is proposed to tackle and better understand this issue., Comment: 10 pages, 12 figures

The linear stability analysis of an uniform shear flow of granular materials is revisited using several cases of a Navier-Stokes'-level constitutive model in which we incorporate the global equation of states for pressure and thermal conductivity (which are accurate up-to the maximum packing density $\nu_{m}$) and the shear viscosity is allowed to diverge at a density $\nu_\mu$ ($< \nu_{m}$), with all other transport coefficients diverging at $\nu_{m}$. It is shown that the emergence of shear-banding instabilities (for perturbations having no variation along the streamwise direction), that lead to shear-band formation along the gradient direction, depends crucially on the choice of the constitutive model. In the framework of a dense constitutive model that incorporates only collisional transport mechanism, it is shown that an accurate global equation of state for pressure or a viscosity divergence at a lower density or a stronger viscosity divergence (with other transport coefficients being given by respective Enskog values that diverge at $\nu_m$) can induce shear-banding instabilities, even though the original dense Enskog model is stable to such shear-banding instabilities. For any constitutive model, the onset of this shear-banding instability is tied to a {\it universal} criterion in terms of constitutive relations for viscosity and pressure, and the sheared granular flow evolves toward a state of lower "dynamic" friction, leading to the shear-induced band formation, as it cannot sustain increasing dynamic friction with increasing density to stay in the homogeneous state. A similar criterion of a lower viscosity or a lower viscous-dissipation is responsible for the shear-banding state in many complex fluids., Comment: 26 pages

A study of the transport coefficients of a system of elastic hard disks, based on the use of Helfand-Einstein expressions is reported. The self-diffusion, the viscosity, and the heat conductivity are examined with averaging techniques especially appropriate for the use in event-driven molecular dynamics algorithms with periodic boundary conditions. The density and size dependence of the results is analyzed, and comparison with the predictions from Enskog's theory is carried out. In particular, the behavior of the transport coefficients in the vicinity of the fluid-solid transition is investigated and a striking power law divergence of the viscosity in this region is obtained, while all other examined transport coefficients show a drop in that density range., Comment: submitted to PRE

We study a homogeneously driven granular gas of inelastic hard particles with rough surfaces subject to Coulomb friction. The stationary state as well as the full dynamic evolution of the translational and rotational granular temperatures are investigated as a function of the three parameters of the friction model. Four levels of approximation to the (velocity-dependent) tangential restitution are introduced and used to calculate translational and rotational temperatures in a mean field theory. When comparing these theoretical results to numerical simulations of a randomly driven mono-layer of particles subject to Coulomb friction, we find that already the simplest model leads to qualitative agreement, but only the full Coulomb friction model is able to reproduce/predict the simulation results quantitatively for all magnitudes of friction. In addition, the theory predicts two relaxation times for the decay to the stationary state. One of them corresponds to the equilibration between the translational and rotational degrees of freedom. The other one, which is slower in most cases, is the inverse of the common relaxation rate of translational and rotational temperatures., Comment: 23 pages, 17 figures

We report molecular dynamics results for the contact values of the radial distribution functions of binary additive mixtures of hard disks. The simulation data are compared with theoretical predictions from expressions proposed by Jenkins and Mancini [J. Appl. Mech. \textbf{54}, 27 (1987)] and Santos et al. [J. Chem. Phys. \textbf{117}, 5785 (2002)]. Both theories agree quantitatively within a very small margin, which renders the former still a very useful and simple tool to work with. The latter (higher-order and self-consistent) theory provides a small qualitative correction for low densities and is superior especially in the high-density domain., Comment: 9 pages, 6 figures

The first normal stress difference (${\mathcal N}_1$) and the microstructure in a dense sheared granular fluid of smooth inelastic hard-disks are probed using event-driven simulations. While the anisotropy in the second moment of fluctuation velocity, which is a Burnett-order effect, is known to be the progenitor of normal stress differences in {\it dilute} granular fluids, we show here that the collisional anisotropies are responsible for the normal stress behaviour in the {\it dense} limit. As in the elastic hard-sphere fluids, ${\mathcal N}_1$ remains {\it positive} (if the stress is defined in the {\it compressive} sense) for dilute and moderately dense flows, but becomes {\it negative} above a critical density, depending on the restitution coefficient. This sign-reversal of ${\mathcal N}_1$ occurs due to the {\it microstructural} reorganization of the particles, which can be correlated with a preferred value of the {\it average} collision angle $\theta_{av}=\pi/4 \pm \pi/2$ in the direction opposing the shear. We also report on the shear-induced {\it crystal}-formation, signalling the onset of fluid-solid coexistence in dense granular fluids. Different approaches to take into account the normal stress differences are discussed in the framework of the relaxation-type rheological models., Comment: 21 pages, 13 figures

A discrete model for the sintering of polydisperse, inhomogeneous arrays of cylinders is presented with empirical contact force-laws, taking into account plastic deformations, cohesion, temperature dependence (melting), and long-time effects. Samples are prepared under constant isotropic load, and are sintered for different sintering times. Increasing both external load and sintering time eads to a stronger, stiffer sample after cooling down. The material behavior is interpreted from both microscopic and macroscopic points of view. Among the interesting results is the observation, that the coordination number, even though it has the tendency to increase, sometimes slightly decreases, whereas the density continuously increases during sintering -- this is interpreted as an indicator of reorganization effects in the packing. Another result of this study is the finding, that strongly attractive contacts occur during cool-down of the sample and leave a sintered block of material with almost equally strong attractive and repulsive contact forces., Comment: 18 pages, 22 figures

We present experiments along with molecular dynamics (MD) simulations of a two-dimensional (2D) granular material in a Couette cell undergoing slow shearing. The grains are disks confined between an inner, rotating wheel and a fixed outer ring. The simulation results are compared to experimental studies and quantitative agreement is found. Tracking the positions and orientations of individual particles allows us to obtain density distributions, velocity and particle rotation rate for the system. The key issue of this paper is to show the extent to which {\em quantitative} agreement between an experiment and MD simulations is possible. Besides many differences in model-details and the experiment, the qualitative features are nicely reproduced. We discuss the quantitative agreement/disagreement, give possible reasons, and outline further research perspectives., Comment: 8 pages, 6 figures

Extensions to kinetic theory and hydrodynamic models are proposed that account for the existence of multi-particle contacts. In the presence of multi-particle contacts (involving elastic, reversible, potential contact energy), dissipation of the translational (kinetic) energy is reduced and a class of different models lead to deviations from the classical inelastic hard sphere (IHS) homogeneous cooling state (HCS), as examined here. The theoretical results are found to be in perfect agreement with the numerical simulations., Comment: 4 pages 3 figures

We find it absurd that Walliser [1] essentially used the same analysis and obtained identical results as reported in [3], yet arrived at different conclusions. Namely, based on an incomplete theory and using erroneous arguments, he not only disputes the original results [2], but also claims them wrong. A more complete theory and much more detailed studies were published in [3], from which we concluded that such results support the mechanism of segregation introduced in ref. [2]. We want to make it clear that Walliser obtained partial results of ref. [3] and arrived at the opposite conclusion. In the following we discuss his comment and its relevance, but at the same time point out what went wrong with his arguments.

Canul-Chay et al. [1] have conducted the segregation experiment of binary granular mixtures in a bed and subjected the bed to vibration. They report that the reverse Brazil-Nut phenomenon (RBNP) was never observed in their experiments and thus conclude that such does not exist. However, we have clearly demonstrated in [2] by Molecular Dynamics simulations that the reverse Brazil-Nut problem not only exists, but can be determined from the solid-liquid phase boundary by a scaling theory for the crossover from RBNP. Hence, instead of disputing Canul-Chay et al.'s sound experimental results [1], in this reply we want to draw the reader's attention to the experimental set up and the interpretation of the experimental results.

We propose a mean field (MF) theory for a homogeneously driven granular gas of inelastic particles with Coulomb friction. The model contains three parameters, a normal restitution coefficient $r_n$, a maximum tangential restitution coefficient $r_t^m$, and a Coulomb friction coefficient $\mu$. The parameters can be tuned to explore a wide range of physical situations. In particular, the model contains the frequently used $\mu \to \infty$ limit as a special case. The MF theory is compared with the numerical simulations of a randomly driven monolayer of spheres for a wide range of parameter values. If the system is far away from the clustering instability ($r_n \approx 1$), we obtain a good agreement between mean field and simulations for $\mu=0.5$ and $r_t^m=0.4$, but for much smaller values of $r_n$ the agreement is less good. We discuss the reasons of this discrepancy and possible refinements of our computational scheme., Comment: 6 pages, 3 figures (10 *.eps files), elsart style (macro included), in Proceedings of the International Conference "Statistical Mechanics and Strongly Correlated Systems", University of Rome "La Sapienza" (Italy), 27-29 September 1999

In the Brazil nut problem (BNP), hard spheres with larger diameters rise to the top. There are various explanations (percolation, reorganization, convection), but a broad understanding or control of this effect is by no means acheived. A theory is presented for the crossover from the BNP to the reverse Brazil nut problem (RBNP) based on a competition between the percolation effect and the condensation of hard spheres. The crossover condition is determined, and the theoretical predictions are compared to Molecular Dynamics simulations in two and three dimensions., Comment: 9 pages which includes 7 figures

We study a two-dimensional granular gas of inelastic spheres subject to multiplicative driving proportional to a power $|v(\vec{x})|^{\delta}$ of the local particle velocity $v(\vec{x})$. The steady state properties of the model are examined for different values of $\delta$, and compared with the homogeneous case $\delta=0$. A driving linearly proportional to $v(\vec{x})$ seems to reproduce some experimental observations which could not be reproduced by a homogeneous driving. Furthermore, we obtain that the system can be homogenized even for strong dissipation, if a driving inversely proportional to, Comment: 4 pages, 5 figures (accepted as Phys. Rev. Lett.)

Based on experiments of the compaction of granular materials under periodic shear of a packing of glass beads, a minimal model for the dynamics of the packing density as a function of time is proposed. First, a random ``energy landscape'' is created by a random walk (RW) in energy. Second, an ensemble of RWs is performed for various temperatures in different temperature-time sequences. We identify the minimum (mean) of the energy landscape with the maximum (random) density. The temperature scaled by the step-size of the energy landscape determines the dynamics of the system and can be regarded as the agitation or shear amplitude. The model reproduces qualitatively both tapping and shear experiments. {\bf Key-words}: Compaction, crystallization, Sinai-diffusion, random walks, quenched disorder, Comment: 8 pages, 4 figures - presented at: Compaction, Innsbruck, Austria Feb. 2000

One of the essential questions in the area of granular matter is, how to obtain macroscopic tensorial quantities like stress and strain from ``microscopic'' quantities like the contact forces in a granular assembly. Different averaging strategies are introduced, tested, and used to obtain volume fractions, coordination numbers, and fabric properties. We derive anew the non-trivial relation for the stress tensor that allows a straightforward calculation of the mean stress from discrete element simulations and comment on the applicability. Furthermore, we derive the ``elastic'' (reversible) mean displacement gradient, based on a best-fit hypothesis. Finally, different combinations of the tensorial quantities are used to compute some material properties. The bulk modulus, i.e. the stiffness of the granulate, is a linear function of the trace of the fabric tensor which itself is proportional to the density and the coordination number. The fabric, the stress and the strain tensors are {\em not} co-linear so that a more refined analysis than a classical elasticity theory is required., Comment: 12 pages, 13 figures

In a freely cooling granular material fluctuations in density and temperature cause position dependent energy loss. Due to strong local dissipation, pressure and energy drop rapidly and material moves from `hot' to `cold' regions, leading to even stronger dissipation and thus causing the density instability. The assumption of `molecular chaos' is valid only in the homogeneous cooling regime. As soon as the density instability occurs, the impact parameter is not longer uniformly distributed. The pair-correlation and the structure functions show that the molecular chaos assumption --- together with reasonable excluded volume modeling --- is important for short distances and irrelevant on large length scales. In this study, the probability distribution of the collision frequency is examined for pipe flow and for freely cooling granular materials as well. Uncorrelated events lead to a Poisson distribution for the collision frequencies. In contrast, the fingerprint of the cooperative phenomena discussed here is a power-law decay of the probability for many collisions per unit time. Keywords: discrete element method, event driven simulations, clustering instability, arching, shock waves, power-law distribution, cooperative phenomena., Comment: 27 pages 14 figs (2 color)

The inelastic hard sphere model of granular material is simple, easily accessible to theory and simulation, and captures much of the physics of granular media. It has three drawbacks, all related to the approximation that collisions are instantaneous: 1) The number of collisions per unit time can diverge, i.e. the ``inelastic collapse'' can occur. 2) All interactions are binary, multiparticle contacts cannot occur and 3) no static limit exists. We extend the inelastic hard sphere model by defining a duration of contact t_c such that dissipation is allowed only if the time between contacts is larger than t_c. We name this generalized model the ``TC model'' and discuss it using examples of dynamic and static systems. The contact duration used here does not change the instantaneous nature of the hard sphere contacts, but accounts for a reduced dissipation during ``multiparticle contacts''. Kinetic and elastic energies are defined as well as forces and stresses in the system. Finally, we present event-driven numerical simulations of situations far beyond the inelastic collapse, possible only with the TC model., Comment: 15 pages, Latex, 14 bw.ps figures + 2 col.ps figures, to be published in Granular Matter 1(3) 1998