Extended Thermodynamics of dense gases with many moments – The macroscopic approach

• An iterative procedure to find over-systems of a model for dense gases is proposed.
• It is proved that the integrability conditions give restrictions only on some additive arbitrary tensors.
• It is proved that the kinetic approach gives a particular solution of the present one.
• It is shown that only the model with 11 and 24 moments generate additive arbitrary tensors which propagate in the over-systems.

To understand the non-equilibrium phenomena beyond the assumption of local equilibrium in rarefied polyatomic gases, a new version of extended thermodynamics has been recently introduced. It proposes two hierarchies of balance equations; the first of these is the usual one, while the second is an “energy block”. In this framework, we investigate here the model with an arbitrary but fixed number of moments and obtain its closure up to every order with respect to equilibrium. This is done in the context of the macroscopic approach. This is not a trivial aspect; in fact, if we impose the conditions up to a given order with respect to equilibrium, there is always the risk to obtain restrictions on the same expressions by imposing the equations at an higher order! An article has been published to prove that this eventuality does not occur. Similarly, if we impose the conditions for a model with a given number of moments, there is always the risk to obtain restrictions on the same expressions by imposing the equations for the model with an higher number of moments! Here this risk is confirmed, but controlled.