Entropy, entropy flux and entropy supply rate of granular fluids

The aim of this work is to analyze the entropy, entropy flux and entropy supply rate of granular fluids within the frameworks of the Boltzmann equation and continuum thermodynamics. It is shown that the entropy inequality for a granular gas that follows from the Boltzmann equation differs from the one of a simple fluid due to the presence of a term which can be identified as the rate of entropy supply density. From the knowledge of a non-equilibrium distribution function -- valid for for processes closed to equilibrium and quasi-elastic restitution coefficients -- it is obtained that the rate of entropy supply density is equal to the rate of internal energy production density divided by the temperature and the entropy flux is equal to the heat flux vector divided by the temperature. A thermodynamic theory of a granular fluid is also developed whose objective is the determination of the basic fields of mass density, momentum density and internal energy density. The constitutive laws are restricted by the principle of material frame indifference and by the entropy principle. Through the exploitation of the entropy principle with Lagrange multipliers, it is shown that the results obtained from the kinetic theory for granular gases concerning the rate of entropy supply density and entropy flux are valid in general for processes close to equilibrium of granular fluids, where linearized constitutive equations hold.