Dilute Granular Flows in a Medium

Some aspects of the dissipative Boltzmann equation for granular matter diffusing in a background medium are investigated. Both collisions with field particles (supposed to be elastic) and inelastic collisions of the grains between themselves (with nonconstant restitution coefficient) are taken into account, leading to the simultaneous presence of two different collision operators. The closure problem for the macroscopic transport equations in a collision‐dominated regime is mainly addressed. Two different approximations, one based on local equilibrium and the other on a Grad‐type expansion, are worked out. Results relevant to collision equilibria and to the hydrodynamic limit coincide to first‐order accuracy in the small parameter (Navier‐Stokes level). Explicit expressions for the equilibrium temperature and of the limiting drift‐diffusion equation for granular density are given.

Keywords:

• Kinetic theory
• Granular flows
• Boltzmann equation
• Macroscopic closure

Acknowledgments

This work was performed in the frame of the activities sponsored by MIUR (Project “Mathematical Problems of Kinetic Theories”), INDAM, GNFM, the University of Parma (Italy), and the European Network HYKE “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis.”