  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.
