Hydrodynamic model for molecular gases in thermal nonequilibrium

We derive a hydrodynamic model for the internal energy excitation of molecular gases in thermal nonequilibrium based on kinetic theory. The co-existence of fast and slow collisions in the system results in thermal nonequilibrium between the translational and internal energy modes. A proper scaling for the Boltzmann equation that accounts for the different relaxation times is obtained from a dimensional analysis. The collisions are divided into fast and slow processes based on the magnitude of the net internal energy. As opposed to conventional perturbations methods, the fast collision operator is expanded in a small parameter used to define the threshold for the net energy for fast collisions. A lemma allows us to split the internal energy of all the levels into perturbed elastic and inelastic contributions for the fast collisions. The introduction of perturbed energy levels is crucial to separate the energy collision invariant into two types of fast collisional invariants. The gas particle population is shown to thermalize to a quasi-equilibrium state described by a Maxwell-Boltzmann distribution function with distinct translational energy temperature and internal energy temperature. The role of the fast collisions is the thermalization of the translational and internal energy modes. Euler equations for the conservation of the mass, momentum, translational energy, and internal energy are also derived. The role of the slow collisions is to contribute to the thermal relaxation of the translational and internal energy modes.