Polytropic gas modelling at kinetic and macroscopic levels. (English) Zbl 1479.76089
Summary: In this paper, we consider the kinetic model of continuous type describing a polyatomic gas in two different settings corresponding to a different choice of the functional space used to define macroscopic quantities. Such a model introduces a single continuous variable supposed to capture all the phenomena related to the more complex structure of a polyatomic molecule. In particular, we provide a direct comparison of these two settings, and show their equivalence after the distribution function is rescaled and the cross section is reformulated. We then focus on the kinetic model for which the rigorous existence and uniqueness result in the space homogeneous case is recently proven [I. M. Gamba and the second author, “On the Cauchy problem for Boltzmann equation modelling a polyatomic gas”, Preprint, arXiv:2005.01017]. Using the cross section proposed in that analysis together with the maximum entropy principle, we establish macroscopic models of six and fourteen fields. In the case of six moments, we calculate the exact, nonlinear, production term and prove its total agreement with extended thermodynamics. Moreover, for the fourteen moments model, we provide new expressions for relaxation times and transport coefficients in a linearized setting, that yield both matching with the experimental data for dependence of the shear viscosity upon temperature and a satisfactory agreement with the theoretical value of the Prandtl number.
MSC:
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 35Q20 | Boltzmann equations |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
Keywords:
non-equilibrium kinetic gas theory; polyatomic gas; Boltzmann equation; moment equation; existence; uniqueness; cross section model; maximum entropy principleReferences:
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