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

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.

To be published in "Computational Gas-Solids Flows and Reacting Systems: Theory, Methods and Practice" IGI-Global, 2009

References in corpus (20)

• Enskog Theory for Polydisperse Granular Mixtures. I. Navier-Stokes order Transport
• Kinetic Temperatures for a Granular Mixture
• Enskog Theory for Polydisperse Granular Mixtures II. Sonine Polynomial Approximation
• Inherent Rheology of a Granular Fluid in Uniform Shear Flow
• Transport properties of dense dissipitive hard-sphere fluids for arbitrary energy loss models
• Modified Sonine approximation for the Navier-Stokes transport coefficients of a granular gas
• Breakdown of the Sonine expansion for the velocity distribution of Granular Gases
• A model for the atomic-scale structure of a dense, nonequilibrium fluid: the homogeneous cooling state of granular fluids
• Chapman-Enskog expansion about nonequilibrium states: the sheared granular fluid
• Brazil-nut effect versus reverse Brazil-nut effect in a moderately dense granular fluid
• The rheology of dense, polydisperse granular fluids under shear
• Mass transport of impurities in a moderately dense granular gas
• On the role of the Knudsen layer in rapid granular flows
• Exact Short Time Dynamics for Steeply Repulsive Potentials
• Shear Stress Correlations in Hard and Soft Sphere Fluids
• DSMC evaluation of the Navier-Stokes shear viscosity of a granular fluid
• Kinetic Theory and Hydrodynamics of Dense, Reacting Fluids far from Equilibrium
• Atomic-scale structure of hard-core fluids under shear flow
• Hard Sphere Dynamics for Normal and Granular Fluids
• Origins of Hydrodynamics for a Granular Gas