Effective two-dimensional model for granular matter with phase separation.

Granular systems confined in vertically vibrated shallow horizontal boxes (quasi-two-dimensional geometry) present a liquid-to-solid phase transition when the frequency of the periodic forcing is increased. An effective model, where grains move and collide in two-dimensions is presented, which reproduces the aforementioned phase transition. The key element is that besides the two-dimensional degrees of freedom, each grain has an additional variable ɛ that accounts for the kinetic energy stored in the vertical motion in the real quasi-two-dimensional motion. This energy grows monotonically during free flight, mimicking the energy gained by collisions with the vibrating walls and, at collisions, this energy is instantaneously transferred to the horizontal degrees of freedom. As a result, the average values of ɛ and the kinetic temperature are decreasing functions of the local density, giving rise to an effective pressure that can present van der Waals loops. A kinetic theory approach predicts the conditions that must satisfy the energy growth function to obtain the phase separation, which are verified with molecular dynamics simulations. Notably, the effective equation of state and the critical points computed considering the velocity-time-of-flight correlations differ only slightly from those obtained by simple kinetic theory calculations that neglect those correlations.

Effect of the vibration profile on shallow granular systems.

We describe the collective behaviour of a system of many inelastic spherical particles inside a box which is being periodically vibrated. The box is shallow, with large horizontal dimensions, while the height is less than two particle diameters. The vibrations are not symmetric: the time the box is moving up is, in general, different from the time it is moving down. The limit cycles of isolated grains are largely affected by the asymmetry of the vibration mode, increasing the size in phase space of the chaotic regions. When many grains are placed in the box, the phase separation between dense, solid-like regions, coexisting with fluid-like regions takes place at smaller global densities for asymmetric vibration profiles. Besides, the order parameter of the transition takes larger values when asymmetric forcing is used.

Shear viscosity of a model for confined granular media.

The shear viscosity in the dilute regime of a model for confined granular matter is studied by simulations and kinetic theory. The model consists on projecting into two dimensions the motion of vibrofluidized granular matter in shallow boxes by modifying the collision rule: besides the restitution coefficient that accounts for the energy dissipation, there is a separation velocity that is added in each collision in the normal direction. The two mechanisms balance on average, producing stationary homogeneous states. Molecular dynamics simulations show that in the steady state the distribution function departs from a Maxwellian, with cumulants that remain small in the whole range of inelasticities. The shear viscosity normalized with stationary temperature presents a clear dependence with the inelasticity, taking smaller values compared to the elastic case. A Boltzmann-like equation is built and analyzed using linear response theory. It is found that the predictions show an excellent agreement with the simulations when the correct stationary distribution is used but a Maxwellian approximation fails in predicting the inelasticity dependence of the viscosity. These results confirm that transport coefficients depend strongly on the mechanisms that drive them to stationary states.

Hydrodynamic modes in a confined granular fluid.

Confined granular fluids, placed in a shallow box that is vibrated vertically, can achieve homogeneous stationary states due to energy injection mechanisms that take place throughout the system. These states can be stable even at high densities and inelasticities allowing for a detailed analysis of the hydrodynamic modes that govern the dynamics of granular fluids. By analyzing the decay of the time correlation functions it is shown that there is a crossover from a quasielastic regime in which energy evolves as a slow mode to an inelastic regime with energy slaved to the other conserved fields. The two regimes have well differentiated transport properties and in the inelastic regime the dynamics can be described by a reduced hydrodynamics with modified longitudinal viscosity and sound speed. The crossover between the two regimes takes place at a wave vector that is proportional to the inelasticity. A two-dimensional granular model, with collisions that mimic the energy transfers that take place in a confined system, is studied by means of microscopic simulations. The results show excellent agreement with the theoretical framework and allow validation of hydrodynamiclike models.

Sudden chain energy transfer events in vibrated granular media.

In a mixture of two species of grains of equal size but different mass, placed in a vertically vibrated shallow box, there is spontaneous segregation. Once the system is at least partly segregated and clusters of the heavy particles have formed, there are sudden peaks of the horizontal kinetic energy of the heavy particles, that is otherwise small. Together with the energy peaks the clusters rapidly expand and the segregation is partially lost. The process repeats once segregation has taken place again, either randomly or with some regularity in time depending on the experimental or numerical parameters. An explanation for these events is provided based on the existence of a fixed point for an isolated particle bouncing with only vertical motion. The horizontal energy peaks occur when the energy stored in the vertical motion is partly transferred into horizontal energy through a chain reaction of collisions between heavy particles.

Rise of a Brazil nut: a transition line.

Using molecular dynamics we study the behavior of a large particle immersed in a bed of smaller ones. The system is bidimensional, consisting of many rough inelastic hard disks of equal size plus a larger one: the intruder. All possible parameters of the system are kept fixed except for two dimensionless parameters determining the frequency and amplitude of the vibrating base. A systematic exploration of this parameter space leads to determining a transition line separating a zone in which the Brazil nut effect is observed and one in which it is not. The results strongly suggest that, in the region of the parameter space in which the study is made, there is a minimum amplitude and a maximum frequency for the Brazil nut effect to take place. These results compare well with isolated results from other authors.

Friction and convection in a vertically vibrated granular system.

The effect of friction in the thermal convection instability of granular fluids is studied by means of molecular dynamics simulations. It is found that the transitions between different convective states (zero, one, and two rolls) are primarily governed by the average energy loss per collisions and not by the friction and restitution coefficients separately, and can be roughly described in terms of a single effective restitution coefficient. The average energy loss per collisions, for a fixed value of the restitution coefficient, shows a maximum for a friction coefficient kappa approximately 0.3. The presence of this maximum manifests itself as a reentrant behavior in the transition lines in parameter space when the value of the friction coefficient is increased beyond 0.3.

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.