  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.
