Hydrodynamic singularities and clustering in a freely cooling inelastic gas

We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional coarse-grained flow. We observe that at a late stage of the instability the shear stress becomes negligibly small, and the gas flows solely by inertia. As a result the flow formally develops a finite time singularity, as the velocity gradient and the gas density diverge at some location. We argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute inelastic gases.Comment: 4 pages, 4 figure

Inelastic Collapse of Three Particles

A system of three particles undergoing inelastic collisions in arbitrary spatial dimensions is studied with the aim of establishing the domain of ``inelastic collapse''---an infinite number of collisions which take place in a finite time. Analytic and simulation results show that for a sufficiently small restitution coefficient, $0\leq r<7-4\sqrt{3}\approx 0.072$, collapse can occur. In one dimension, such a collapse is stable against small perturbations within this entire range. In higher dimensions, the collapse can be stable against small variations of initial conditions, within a smaller $r$ range, $0\leq r<9-4\sqrt{5}\approx 0.056$.Comment: 6 pages, figures on request, accepted by PR

Close-packed floating clusters: granular hydrodynamics beyond the freezing point?

Monodisperse granular flows often develop regions with hexagonal close packing of particles. We investigate this effect in a system of inelastic hard spheres driven from below by a "thermal" plate. Molecular dynamics simulations show, in a wide range of parameters, a close-packed cluster supported by a low-density region. Surprisingly, the steady-state density profile, including the close-packed cluster part, is well described by a variant of Navier-Stokes granular hydrodynamics (NSGH). We suggest a simple explanation for the success of NSGH beyond the freezing point.Comment: 4 pages, 5 figures. To appear in Phys. Rev. Let

Towards granular hydrodynamics in two-dimensions

We study steady-state properties of inelastic gases in two-dimensions in the presence of an energy source. We generalize previous hydrodynamic treatments to situations where high and low density regions coexist. The theoretical predictions compare well with numerical simulations in the nearly elastic limit. It is also seen that the system can achieve a nonequilibrium steady-state with asymmetric velocity distributions, and we discuss the conditions under which such situations occur.Comment: 8 pages, 9 figures, revtex, references added, also available from http://arnold.uchicago.edu/?ebn

Dynamics and stress in gravity driven granular flow

We study, using simulations, the steady-state flow of dry sand driven by gravity in two-dimensions. An investigation of the microscopic grain dynamics reveals that grains remain separated but with a power-law distribution of distances and times between collisions. While there are large random grain velocities, many of these fluctuations are correlated across the system and local rearrangements are very slow. Stresses in the system are almost entirely transfered by collisions and the structure of the stress tensor comes almost entirely from a bias in the directions in which collisions occur.Comment: 4 pages, 3 eps figures, RevTe

Granular Collapse as a Percolation Transition

Inelastic collapse is found in a two-dimensional system of inelastic hard disks confined between two walls which act as an energy source. As the coefficient of restitution is lowered, there is a transition between a state containing small collapsed clusters and a state dominated by a large collapsed cluster. The transition is analogous to that of a percolation transition. At the transition the number of clusters n_s of size s scales as $n_s \sim s^{-\tau}$ with $\tau \approx 2.7$.Comment: 10 pages revtex, 5 figures, accepted by Phys Rev E many changes and corrections from previous submissio

On the validity of the Boltzmann equation to describe low density granular systems

The departure of a granular gas in the instable region of parameters from the initial homogeneous cooling state is studied. Results from Molecular Dynamics and from Direct Monte Carlo simulation of the Boltzmann equation are compared. It is shown that the Boltzmann equation accurately predicts the low density limit of the system. The relevant role played by the parallelization of the velocities as time proceeds and the dependence of this effect on the density is analyzed in detail

A nonlinear hydrodynamical approach to granular materials

We propose a nonlinear hydrodynamical model of granular materials. We show how this model describes the formation of a sand pile from a homogeneous distribution of material under gravity, and then discuss a simulation of a rotating sandpile which shows, in qualitative agreement with experiment, a static and dynamic angle of repose.Comment: 17 pages, 14 figures, RevTeX4; minor changes to wording and some additional discussion. Accepted by Phys. Rev.

Velocity correlations in granular materials

A system of inelastic hard disks in a thin pipe capped by hot walls is studied with the aim of investigating velocity correlations between particles. Two effects lead to such correlations: inelastic collisions help to build localized correlations, while momentum conservation and diffusion produce long ranged correlations. In the quasi-elastic limit, the velocity correlation is weak, but it is still important since it is of the same order as the deviation from uniformity. For system with stronger inelasticity, the pipe contains a clump of particles in highly correlated motion. A theory with empirical parameters is developed. This theory is composed of equations similar to the usual hydrodynamic laws of conservation of particles, energy, and momentum. Numerical results show that the theory describes the dynamics satisfactorily in the quasi-elastic limit, however only qualitatively for stronger inelasticity.Comment: 12 pages (REVTeX), 15 figures (Postscript). submitted to Phys. Rev.

Velocity and density profiles of granular flow in channels using lattice gas automaton

We have performed two-dimensional lattice-gas-automaton simulations of granular flow between two parallel planes. We find that the velocity profiles have non-parabolic distributions while simultaneously the density profiles are non-uniform. Under non-slip boundary conditions, deviation of velocity profiles from the parabolic form of newtonian fluids is found to be characterized solely by ratio of maximal velocity at the center to the average velocity, though the ratio depends on the model parameters in a complex manner. We also find that the maximal velocity ($u_{max}$) at the center is a linear function of the driving force (g) as $u_{max} = \alpha g - \delta$ with non-zero $\delta$ in contrast with newtonian fluids. Regarding density profiles, we observe that densities near the boundaries are higher than those in the center. The width of higher densities (above the average density) relative to the channel width is a decreasing function of a variable which scales with the driving force (g), energy dissipation parameter ($\epsilon$) and the width of the system (L) as $g^{\mu} L^{\nu}/\epsilon$ with exponents $\mu = 1.4 \pm 0.1$ and $\nu = 0.5 \pm 0.1$. A phenomenological theory based on a scaling argument is presented to interpret these findings.Comment: Latex, 15 figures, to appear in PR