System Of Inelastic Hard Spheres Research Articles

Memory effects in the relaxation of a confined granular gas.

The accuracy of a model to describe the horizontal dynamics of a confined quasi-two-dimensional system of inelastic hard spheres is discussed by comparing its predictions for the relaxation of the temperature in a homogenous system with molecular dynamics simulation results for the original system. A reasonably good agreement is found. Next the model is used to investigate the peculiarities of the nonlinear evolution of the temperature when the parameter controlling the energy injection is instantaneously changed while the system was relaxing. This can be considered as a nonequilibrium generalization of the Kovacs effect. It is shown that, in the low-density limit, the effect can be accurately described by using a simple kinetic theory based on the first Sonine approximation for the one-particle distribution function. Some possible experimental implications are indicated.

Shock propagation in granular flow subjected to an external impact

We analyze a recent experiment [Boudet, Cassagne, and Kellay, Phys. Rev. Lett. 103, 224501 (2009)] in which the shock created by the impact of a steel ball on a flowing monolayer of glass beads is studied quantitatively. We argue that radial momentum is conserved in the process and hence show that in two dimensions the shock radius increases in time t as a power law t{1/3}. This is confirmed in event driven simulations of an inelastic hard sphere system. The experimental data are compared with the theoretical prediction and are shown to compare well at intermediate times. At long times the experimental data exhibit a crossover to a different scaling behavior. We attribute this to the problem becoming effectively three dimensional due to the accumulation of particles at the shock front and propose a simple hard sphere model that incorporates this effect. Simulations of this model capture the crossover seen in the experimental data.

Collision statistics in sheared inelastic hard spheres

The dynamics of sheared inelastic-hard-sphere systems is studied using nonequilibrium molecular-dynamics simulations and direct simulation Monte Carlo. In the molecular-dynamics simulations Lees-Edwards boundary conditions are used to impose the shear. The dimensions of the simulation box are chosen to ensure that the systems are homogeneous and that the shear is applied uniformly. Various system properties are monitored, including the one-particle velocity distribution, granular temperature, stress tensor, collision rates, and time between collisions. The one-particle velocity distribution is found to agree reasonably well with an anisotropic Gaussian distribution, with only a slight overpopulation of the high-velocity tails. The velocity distribution is strongly anisotropic, especially at lower densities and lower values of the coefficient of restitution, with the largest variance in the direction of shear. The density dependence of the compressibility factor of the sheared inelastic-hard-sphere system is quite similar to that of elastic-hard-sphere fluids. As the systems become more inelastic, the glancing collisions begin to dominate over more direct, head-on collisions. Examination of the distribution of the times between collisions indicates that the collisions experienced by the particles are strongly correlated in the highly inelastic systems. A comparison of the simulation data is made with direct Monte Carlo simulation of the Enskog equation. Results of the kinetic model of Montanero [J. Fluid Mech. 389, 391 (1999)] based on the Enskog equation are also included. In general, good agreement is found for high-density, weakly inelastic systems.

Velocity correlations in dense granular shear flows: Effects on energy dissipation and normal stress

We study the effect of precollisional velocity correlations on granular shear flow by molecular dynamics simulations of an inelastic hard sphere system. Comparison of the simulations with kinetic theory reveals that the theory overestimates both the energy dissipation rate and the normal stress in the dense flow region. We find that the relative normal velocity of colliding particles is smaller than that expected from random collisions, and the discrepancies in the dissipation and the normal stress can be adjusted by introducing the idea of the collisional temperature, from which we conclude that the velocity correlation neglected in the kinetic theory is responsible for the discrepancies. Our analysis of the distributions of the precollisional velocity suggests that the correlation grows through multiple inelastic collisions during the time scale of the inverse of the shear rate. As for the shear stress, the discrepancy is also found in the dense region, but it depends strongly on the particle inelasticity.

The inelastic hard dimer gas: A nonspherical model for granular matter

We study a two-dimensional gas of inelastic smooth hard dimers. Since the collisions between dimers are dissipative, being characterized by a coefficient of restitution alpha<1, and no external driving force is present, the energy of the system decreases in time and no stationary state is achieved. However, the resulting nonequilibrium state of the system displays several interesting properties in close analogy with systems of inelastic hard spheres, whose relaxational dynamics has been thoroughly explored. We generalize to inelastic systems a recently method introduced [G. Ciccotti and G. Kalibaeva, J. Stat. Phys. 115, 701 (2004)] to study the dynamics of rigid elastic bodies made up of different spheres held together by rigid bonds. Each dimer consists of two hard disks of diameter d, whose centers are separated by a fixed distance a. By describing the rigid bonds by means of holonomic constraints and deriving the appropriate collision rules between dimers, we reduce the dynamics to a set of equations which can be solved by means of event-driven simulation. After deriving the algorithm we study the decay of the total kinetic energy, and of the ratio between the rotational and the translational kinetic energy of inelastic dimers. We show numerically that the celebrated Haff's homogeneous cooling law t(-2), describing how the kinetic energy of an inelastic hard-sphere system with a constant coefficient of restitution decreases in time, holds even in the case of these nonspherical particles. We fully characterize this homogeneous decay process in terms of appropriate decay constants and confirm numerically the scaling behavior of the velocity distributions.

Effects of Velocity Correlation on Early Stage of Free Cooling Process of Inelastic Hard Sphere System

The free cooling process in the inelastic hard sphere system is studied by analysing the data from large scale molecular dynamics simulations on a three dimensional system. The initial energy decay, the velocity distribution function, and the velocity correlation functions are calculated to be compared with theoretical predictions. The energy decay rate in the homogeneous cooling state is slightly but distinctively smaller than that expected from the independent collision assumption. The form of the one particle velocity distribution is found not to be stationary. These contradict to the predictions of the kinetic theory based on the Enskog-Boltzmann equation and suggest that the velocity correlation is already important in the early stage of homogeneous cooling state. The energy decay rate is analysed in terms of the velocity correlation.

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.

Torroidal convection roll in a three-dimensional granular system

We study the behavior of a system of inelastic hard spheres in a gravitational field and contained in a cylinder by means of a molecular dynamics simulation with an event-driven algorithm. If energy is supplied to the system by a vibrating bottom wall, the system becomes fluidized and a convection roll, as evidenced by the time-averaged velocity field, appears. The convection is strongly influenced by the value of the particle–side-wall restitution coefficient: for elastic particle–side-wall collisions, the sense of the convection is reversed and the convection intensity is significantly reduced compared with the case of elastic collisions.

Dynamics of rarefied granular gases.

This paper presents quite general bidimensional gas-dynamic equations--derived from kinetic theory-which include the fourth cumulant kappa(r,t) as a dynamic field. The dynamics describes a low-density system of inelastic hard spheres (disks) with normal restitution coefficient r. Two illustrative examples are given and the role of kappa in them is discussed. Our general gas-dynamic equations would deal with 9 hydrodynamic fields (which corresponds to 14 in three-dimension). These fields are the standard hydrodynamic fields plus the components p(ij) of the traceless part of the pressure tensor, the energy flux vector Q and the fourth cumulant kappa. The present formulation requires no constitutive equations. The two examples are: the well-known homogeneous cooling state and a system, with and without gravity, steadily heated by two parallel walls. In the first case, the dynamics yield a description of the homogeneous cooling state consistent with known results adding extra details mainly about the transient time behavior. The steadily heated system kept in a static state gives rise to quite simple but nontrivial equations. In the case with gravity, it is shown that when kappa is included as a dynamic field, the formalism leads to a non-Fourier law already to first order in dissipation. Setting gravity g=0 a perturbative solution is shown and favorably compared with observations obtained from molecular dynamics (MD). In both cases, with and without gravity, kappa is not homogeneous. An analytic extension suggests a divergent situation for a small negative value of q, which originates in the unavoidable extension of the formalism to exothermic collisions associated with a restitution coefficient larger than one. This divergent behavior is observed in MD.

Pseudo-thermal cooling in a bounded system of viscoelastic hard-spheres

Using computer simulations, the cooling rate of a system of inelastic hard spheres in the Couette geometry has been investigated after stopping the relative motion of boundaries. The energy of the system is initially balanced between the incoming kinetic energy from the moving boundaries and the dissipated kinetic energy due to the successive collisions of inelastic particles. When the relative motion of moving boundaries is suddenly stopped the pseudo-thermal energy of the system decays. However, the rate of decay is found to be slower than that introduced for a shear-free ensemble. Moreover, the simulations clearly show that the existence of pseudo-thermal energy is a necessary condition for diffusion in a particulate system. The simulation results may elucidate the physical origin of some phenomena such as clustering and freezing in hard-sphere systems, and the effect of velocity gradients on these phenomena.

The granular phase diagram

The kinetic energy distribution function satisfying the Boltzmann equation is studied analytically and numerically for a system of inelastic hard spheres in the case of binary collisions. Analytically, this function is shown to have a similarity form in the simple cases of uniform or steady-state flows. This determines the region of validity of hydrodynamic description. The latter is used to construct the phase diagram of granular systems, and discriminate between clustering instability and inelastic collapse. The molecular dynamics results support analytical results, but also exhibit a novel fluctuational breakdown of mean-field descriptions.