The Enskog equation for confined elastic hard spheres

A kinetic equation for a system of elastic hard spheres or disks confined by a hard wall of arbitrary shape is derived. It is a generalization of the modified Enskog equation in which the effects of the confinement are taken into account and it is supposed to be valid up to moderate densities. From the equation, balance equations for the hydrodynamic fields are derived, identifying the collisional transfer contributions to the pressure tensor and heat flux. A Lyapunov functional, $\mathcal{H}[f]$, is identified. For any solution of the kinetic equation, $\mathcal{H}$ decays monotonically in time until the system reaches the inhomogeneous equilibrium distribution, that is a Maxwellian distribution with a the density field consistent with equilibrium statistical mechanics

Homogeneous hydrodynamics of a collisional model of confined granular gases

The hydrodynamic equation governing the homogeneous time evolution of the temperature in a model of confined granular gas is studied by means of the Enskog equation. The existence of a normal solution of the kinetic equation is assumed as a condition for hydrodynamics. Dimensional analysis implies a scaling of the distribution function that is used to determine it in the first Sonine approximation, with a coefficient that evolves in time through its dependence on the temperature. The theoretical predictions are compared with numerical results obtained by the direct simulation Monte Carlo method, and a good agreement is found. The relevance of the normal homogeneous distribution function to derive inhomogeneous hydrodynamic equations, for instance using the Champan-Enskog algorithm, is indicated.Comment: Accepted in Phys. Rev.

Hydrodynamics for a model of a confined quasi-two-dimensional granular gas

The hydrodynamic equations for a model of a confined quasi-two-dimensional gas of smooth inelastic hard spheres are derived from the Boltzmann equation for the model, using a generalization of the Chapman-Enskog method. The heat and momentum fluxes are calculated to Navier-Stokes order, and the associated transport coefficients are explicitly determined as functions of the coefficient of normal restitution and the velocity parameter involved in the definition of the model. Also an Euler transport term contributing to the energy transport equation is considered. This term arises from the gradient expansion of the rate of change of the temperature due to the inelasticity of collisions, and vanishes for elastic systems. The hydrodynamic equations are particularized for the relevant case of a system in the homogeneous steady state. The relationship with previous works is analyzed

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 an 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 non-equilibrium 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

Towards an HH-theorem for granular gases

The $H$-theorem, originally derived at the level of Boltzmann non-linear kinetic equation for a dilute gas undergoing elastic collisions, strongly constrains the velocity distribution of the gas to evolve irreversibly towards equilibrium. As such, the theorem could not be generalized to account for dissipative systems: the conservative nature of collisions is an essential ingredient in the standard derivation. For a dissipative gas of grains, we construct here a simple functional $\mathcal H$ related to the original $H$, that can be qualified as a Lyapunov functional. It is positive, and results backed by three independent simulation approaches (a deterministic spectral method, the stochastic Direct Simulation Monte Carlo technique, and Molecular Dynamics) indicate that it is also non-increasing. Both driven and unforced cases are investigated

Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres

The local balance equations for the density, momentum, and energy of a dilute gas of elastic or inelastic hard spheres, strongly confined between two parallel hard plates are obtained. The starting point is a Boltzmann-like kinetic equation, recently derived for this system. As a consequence of the confinement, the pressure tensor and the heat flux contain, in addition to the terms associated to the motion of the particles, collisional transfer contributions, similar to those that appear beyond the dilute limit. The complexity of these terms, and of the kinetic equation itself, compromise the potential of the equation to describe the rich phenomenology observed in this kind of systems. For this reason, a simpler model equation based on the Boltzmann equation is proposed. The model is formulated to keep the main properties of the underlying equation, and it is expected to provide relevant information in more general states than the original equation. As an illustration, the solution describing a macroscopic state with uniform temperature, but a density gradient perpendicular to the plates is considered. This is the equilibrium state for an elastic system, and the inhomogeneous cooling state for the case of inelastic hard spheres. The results are in good agreement with previous results obtained directly from the Boltzmann equation

Confined granular gases under the influence of vibrating walls

The dynamics of a system composed of inelastic hard spheres or disks that are confined between two parallel vertically vibrating walls is studied (the vertical direction is defined as the direction perpendicular to the walls). The distance between the two walls is supposed to be larger than twice the diameter of the particles so that the particles can pass over each other, but still much smaller than the dimensions of the walls. Hence, the system can be considered to be quasi-two-dimensional (quasi-one-dimensional) in the hard spheres (disks) case. For dilute systems, a closed evolution equation for the one-particle distribution function is formulated that takes into account the effects of the confinement. Assuming the system is spatially homogeneous, the kinetic equation is solved approximating the distribution function by a two-temperatures (horizontal and vertical) gaussian distribution. The obtained evolution equations for the partial temperatures are solved, finding a very good agreement with Molecular Dynamics simulation results for a wide range of the parameters (inelasticity, height and density) for states whose projection over a plane parallel to the walls is homogeneous. In the stationary state, where the energy lost in collisions is compensated by the energy injected by the walls, the pressure tensor in the horizontal direction is analyzed and its relation with an instability of the homogenous state observed in the simulations is discussed.Comment: 32 pages, 14 figures, submitted to Journal of Statistical Mechanics: Theory and Experimen

Dynamics of an inelastic tagged particle under strong confinement

The dynamics of a tagged particle immersed in a fluid of particles of the same size but different mass is studied when the system is confined between two hard parallel plates separated a distance smaller than twice the diameter of the particles. The collisions between particles are inelastic while the collisions of the particles with the hard walls inject energy in the direction perpendicular to the wall, so that stationary states can be reached in the long-time limit. The velocity distribution of the tagged particle verifies a Boltzmann-Lorentz-like equation that is solved assuming that it is a spatially homogeneous gaussian distribution with two different temperatures (one associated to the motion parallel to the wall and another associated to the perpendicular direction). It is found that the temperature perpendicular to the wall diverges when the tagged particle mass approaches a critical mass from below, while the parallel temperature remains finite. Molecular Dynamics simulation results agree very well with the theoretical predictions for tagged particle masses below the critical mass. The measurements of the velocity distribution function of the tagged particle confirm that it is gaussian if the mass is not close to the critical mass, while it deviates from gaussianity when approaching the critical mass. Above the critical mass, the velocity distribution function is very far from a gaussian, being the marginal distribution in the perpendicular direction bimodal and with a much larger variance than the one in the parallel direction.Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía (Spain) throughGrant. US-1380729theMinisterio de Ciencia e Innovación (Spain) throughGrant PID2021- 126348NB-100 FEDER fund