Moment Equations for Polyatomic Gases
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Moment Equations for Polyatomic Gases ˇ c · Srboljub Simi´c Milana Pavi´c-Coli´
Received: 25 November 2013 / Accepted: 3 April 2014 © Springer Science+Business Media Dordrecht 2014
Abstract The aim of this paper is to analyze the moment equations for polyatomic gases whose internal degrees of freedom are modeled by a continuous internal energy function. The closure problem is resolved using the maximum entropy principle. The macroscopic equations are divided in two hierarchies—“momentum” and “energy” one. As an example, the system of 14 moments equations is studied. The main new result is determination of the production terms which contain two parameters. They can be adapted to fit the expected values of Prandtl number and/or temperature dependence of the viscosity. The ratios of relaxation times are also discussed. Keywords Kinetic theory of gases · Polyatomic gases · Moment equations · Transport coefficients Mathematics Subject Classification 82C40 · 82D05 · 82C05
1 Introduction Modeling a non-equilibrium processes in polyatomic gases can be performed in a variety of ways. The choice of the model strongly depends on the problem itself, as well as needed
This work was supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, within the project “Mechanics of nonlinear and dissipative systems—contemporary models, analysis and applications”, Project No. ON174016. ˇ c M. Pavi´c-Coli´ Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia e-mail: [email protected] ˇ c M. Pavi´c-Coli´ CMLA, ENS Cachan, Cachan, France
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S. Simi´c ( ) Department of Mechanics, University of Novi Sad, Novi Sad, Serbia e-mail: [email protected]
ˇ c, S. Simi´c M. Pavi´c-Coli´
level of accuracy. The main difference with respect to the monatomic case appears in modeling internal degrees of freedom of the molecule. This can be done using classical, semiclassical, or quantum-mechanical approach [15]. This paper is devoted to the macroscopic modeling of polyatomic gases by means of the moment equations. In the monatomic case, they can be derived by three different approaches. The first one is the Grad’s method [11] based upon expansion of velocity distribution function in terms of the Hermite polynomials. Closure is achieved through compatibility of the moments of polynomial approximation with macroscopic variables. The second one is the application of maximum entropy principle [3, 4, 7, 10, 16, 20], where the velocity distribution function comes out as a solution of the variational problem with constraints. Finally, the third approach is developed within macroscopic theory of a non-equilibrium processes—extended thermodynamics [21]. It fills the gap between macroscopic level (classical TIP) and mesoscopic level (kinetic theory of gases) and resolves the closure problem using the entropy principle. In the polyatomic case, the problem was partially treated by several approaches [14, 17, 25], but there is no complete picture, yet. In this study we give a contribut
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