Origins of Hydrodynamics for a Granular Gas

TL;DR

This paper discusses the theoretical foundations for deriving hydrodynamic equations for granular gases, focusing on the nonlinear Boltzmann equation, and demonstrates the existence of hydrodynamic modes through two different approaches.

Contribution

It provides a formal derivation of nonlinear hydrodynamic equations for granular gases using the Chapman-Enskog method and analyzes hydrodynamic modes via spectral analysis of perturbations.

Findings

Derivation of Navier-Stokes order hydrodynamic equations with explicit transport coefficients.

Identification of hydrodynamic modes in the spectrum of the linearized dynamics.

Formal demonstration of the internal consistency of the hydrodynamic description.

Abstract

The basis for a hydrodynamic description of granular gases is discussed for a low density gas of smooth, inelastic hard spheres. The more fundamental mesoscopic description is taken to be the nonlinear Boltzmann kinetic equation. Two arguments are presented in favor of a hydrodynamics for granular gases. The first one is the concept of a "normal" solution and its explicit approximate construction via the Chapman-Enskog method. The second is the demonstration of hydrodynamic modes in the spectrum of the generator for the dynamics of small spatial perturbations of the homogeneous reference state. In the first case, a derivation of the nonlinear hydrodynamic equations is given to Navier-Stokes order, with explicit expressions for the transport coefficients. The approach is formal and the context of the derivation is left unspecified, although internal mathematically consistency is demonstrated. The second method is more restricted, leading only to linearized hydrodynamics, but with the potential to define more sharply the context of hydrodynamics.