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

Abstract

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.

Homogeneous dynamics in a vibrated granular monolayer

“…For the studied values of the parameters, it is found that Eq. (47) has no solution for ∆Tz ∆T > 0 and has one solution for ∆Tz ∆T < 0. In any case, even if the curves cross each other, it can happen that the initially hotter system crosses the stationary value and reaches the stationary state less quickly.…”

Kinetic equation and nonequilibrium entropy for a quasi-two-dimensional gas

“…Recently, the interest in the slit geometry with two parallel, hard plates of a single component system of hard spheres, with a separation between the two plates smaller than two particle diameters, has largely increased, due to a series of experimental observations in systems of macroscopic particles [9][10][11][12]. Although it is quite sure that the inelasticity of collisions, inherent to the macroscopic character of the spheres, and the subsequently needed energy injection to reach and maintain a steady state, play a crucial role in many of the observed features [13][14][15], it is clear that before considering the consequences of these factors, the idealized system of elastic hard spheres must be addressed. The central issue is whether it is possible to formulate a macroscopic, hydrodynamic-like theory to describe the two-dimensional dynamics of the system when observed from above or from below and, if the answer is affirmative, which is the form of the equations and the expressions of the coefficients appearing in them.…”

Kinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres

“…The ideas of the method are the same as for ordinary granular gases [18]. The obtained hydrodynamic equations for the number density n(r, t), velocity flow u(r, t), and granular temperature T(r, t) read [19] …”

“…All these coefficients depend on the temperature, and also on the coefficient of normal restitution α and the characteristic speed ∆. Their expressions can be found in [19], and in a more concise way in the Appendix of [16]. They are given by the normal solutions-in the kinetic theory sense-of first-order partial differential equations.…”