Polytropic gas modelling at kinetic and macroscopic levels

Abstract

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. 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.

Boltzmann collision operator for polyatomic gases in agreement with experimental data and DSMC method

“…They are all of limited agreement with theoretical restrictions and experimental data. These results were improved in [17,18] using the collision kernel proposed in the rigorous analysis [5]. In particular, [17] shows an agreement with the theoretical value of the Prandtl number given by Eucken's formula [19] with an error of up to 12%.…”

“…These results were improved in [17,18] using the collision kernel proposed in the rigorous analysis [5]. In particular, [17] shows an agreement with the theoretical value of the Prandtl number given by Eucken's formula [19] with an error of up to 12%. This error was eliminated in [18] by studying 17-moment equations and by introducing frozen collisions that do not change microscopic internal energies and consequently enriching the collision kernel from [5] with additional free parameters.…”

Compactness property of the linearized Boltzmann operator for a diatomic single gas model

“…We give now some models that satisfy assumptions (9), which have been recently studied in the literature (see [19,16,4]).…”

“…Moreover, W in ( 24) is clearly microreversible, and the measure dEdGdvdIdv ′ dI ′ is obviously invariant. We remark that ω and −ω refer to the same σ, and thus the Jacobian (16) has to be taken twice in (23).…”

A kinetic model of polyatomic gas with resonant collisions

“…The internal energy of the pair is itself allocated between both molecules. This procedure and its consequences on the underlying Boltzmann equation have been widely investigated during the last thirty years, see for example [11,15,16,17] which offer further physical discussions about the model, and [26,6,18,3], where various mathematical properties of the Borgnakke-Larsen approach are studied.…”