Hydrodynamic equations for a granular mixture from kinetic theory - fundamental considerations

TL;DR

This paper reviews the derivation of hydrodynamic equations for granular mixtures from kinetic theory, focusing on the fundamental concepts and methods to connect microscopic grain interactions to macroscopic flow descriptions.

Contribution

It provides a systematic derivation of granular hydrodynamics from kinetic theory, including the Boltzmann-Enskog equation and transport coefficients, extending to Navier-Stokes and beyond.

Findings

Derivation of macroscopic hydrodynamic equations from kinetic theory.

Explicit integral equations for transport coefficients.

Extension of hydrodynamics beyond Navier-Stokes limitations.

Abstract

In this review, a theoretical description is provided for the solid (granular) phase of the gas-solid flows that are the focus of this book. Emphasis is placed on the fundamental concepts involved in deriving a macroscopic hydrodynamic description for the granular material in terms of the hydrodynamic fields (species densities, flow velocity, and the granular temperature) from a prescribed "microscopic" interaction among the grains. To this end, the role of the interstitial gas phase, body forces such as gravity, and other coupling to the environment are suppressed and retained only via a possible non-conservative external force and implicit boundary conditions. The general notion of a kinetic equation is introduced to obtain macroscopic balance equations for the fields. Constitutive equations for the fluxes in these balance equations are obtained from special "normal" solutions to the kinetic equation, resulting in a closed set of hydrodynamic equations. This general constructive procedure is illustrated for the Boltzmann-Enskog kinetic equation describing a system of smooth, inelastic hard spheres. For weakly inhomogeneous fluid states the granular Navier-Stokes hydrodynamic equations are obtained, including exact integral equations for the transport coefficients. A method to obtain practical solutions to these integral equations is described. Finally, a brief discussion is given for hydrodynamics beyond the Navier-Stokes limitations.