An entropy criterion for the validity of Navier-Stokes order continuum theory for gas-solid flow: Kinetic theory analysis

•Kinetic theory was used to analyze the entropy of gas-solid systems.•The local thermodynamic equilibrium postulate and N-S continuum models are self-consistent.•Entropy criterion for the validity of N-S order continuum model was proposed.•The criterion is a function of the gradient of hydrodynamic variables and particle inelasticity.

Navier-Stokes order continuum theory (NS theory) established on the basis of local thermodynamic equilibrium (LTE) postulate has been widely used to study gas-solid flows. Qualitatively, the postulate is valid for systems sufficiently close to equilibrium, however, continuum theory provides no information on how small these deviations from equilibrium should be in order to guarantee the validity of LTE. To this end, kinetic theory analysis was carried out to formulate an entropy criterion, it was shown that (i) the entropy density up to the NS order is exactly the same as the one at the LTE condition, meaning that the LTE postulate is only valid for linear non-equilibrium regime, although NS theory and the LTE postulate are logically self-consistent; (ii) an entropy criterion characterizing the relative importance of nonlinear effects is derived, which should be small in order to make sure that the LTE postulate is valid and also sets the boundary of validity of NS theory; and (iii) the proposed entropy criterion, Is≪1, is not only a function of granular temperature gradient and velocity gradient, but also a function of 1-e2 characterizing the inelasticity of particle-particle collisions. It is more complex and systematic than the traditional criterion based on the Knudsen number, Kn≪1.