On the ellipsoidal statistical model for polyatomic gases

Continuum Mechanics and Thermodynamics - Tập 20 - Trang 489-508 - 2009
Stéphane Brull1, Jacques Schneider2
1Institut des mathématiques de Toulouse, UMR 5219, Université Paul Sabatier, Toulouse cedex 9, France
2ANLA, University of Toulon, La Garde, France

Tóm tắt

The aim of this article is to construct a BGK-type model for polyatomic gases which gives in the hydrodynamic limit the proper transport coefficient. Its construction relies upon a systematic procedure: minimizing Boltzmann entropy under suitable moments constraints (Levermore in J Stat Phys 83:1021–1065, 1996; Brull and Schneider in Cont Mech Thermodyn 20(2):63–74, 2008). The obtained model corresponds to the ellipsoidal statistical model introduced in Andries et al. (Eur J Mech B Fluids 19:813–830, 2000). We also study the return to equilibrium of its solutions in the homogeneous case.

Tài liệu tham khảo

Andries P., Le Tallec P., Perlat J.P., Perthame B.: Entropy condition for the ES BGK model of Boltzmann equation for mono and polyatomic gases. Eur. J. Mech. B Fluids 19, 813–830 (2000)

Arkeryd L.: Stability in L 1 for the spatially homogeneous equation. Arch. Rat. Mech. Anal. 103(2), 151–167 (1988)

Bahi, Y.: Contribution la simulation numrique des coulements en gaz rarfis. Phd Thesis, University Pierre et Marie Curie, Paris (1997)

Bhatnagar P.L., Gross E.P., Krook M.: A model for collision processes in gases. Phys. Rev. 94, 511 (1954)

Borgnakke C., Larsen P.S.: Statistical collision model for Monte–Carlo simulation of polyatomic gas mixtures. J. Comp. Phys. 18, 405–420 (1975)

Bourgat J.F., Desvillettes L., Le Tallec P., Perthame B.: Microreversible collisions for polyatomic gases and Boltzmann’s theorem. Eur. J. Mech. B Fluids 13(2), 237–254 (1994)

Brull S., Schneider J.: A new approach of the ellipsoidal statistical model. Cont. Mech. Thermodyn. 20(2), 63–74 (2008)

Cercignani, C.: The Boltzmann Equation and Its Applications, pp. 40–103. Scottish Academic Press, Edinburgh (1988)

Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases, 3rd Edn. Cambridge Mathematical Library (1970)

Csiszár I.: I-divergence geometry of probability distributions and minimization problems Sanov property. Ann. Probab. 3, 146–158 (1975)

Collet J.F.: Extensive Lyapounov functionals for moment-preserving evolution equations. C. R. A. S. Ser. I 334, 429–434 (2002)

Desvillettes L.: Sur un Modèle de type Borgnakke-Larsen Conduisant des lois d’Energie Non-linéaires en Température pour les Gaz Parfaits Polyatomiques. Ann. Fac. Sci. Toulouse Sér. 6(2), 257–262 (1997)

Goldstein, D., Sturtevant, B., Broadwell, J.E.: Investigation of the motion of discrete-velocity gases, In: Muntz, E.P., Weaver, D.P., Campbell, D.H. (Eds.) Rarefied gas dynamics: theoretical and computational techniques, progress in astronautics and aeronautics, vol. 118, pp. 100–117 (1989)

Holway, L.H.: Kinetic theory of shock structure using an ellipsoidal distribution function. In: Rarefied Gas Dynamics [Proceedings Fourth International Symposium, University Toronto (1964)], vol. I. Academie Press, New York, pp. 193–215 (1966)

Kusker I.: A model for rotational energy exchange in polyatomic gases. Physica A 158, 784–800 (1989)

Le Tallec P.: A hierarchy of hyperbolic models linking Boltzmann to Navier Stokes equations for polyatomic gases. ZAMM 80(11–12), 779–790 (2000)

Levermore C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065 (1996)

Michel P., Schneider J.: Approximation simultanée de réels par des nombres rationnels et noyau de collision de l’équation de Boltzmann. CRAS-série 1 Maths. 330(9), 857–862 (2000)

Mieussens, L.: Modèles à vitesses discrètes et méthodes numériques pour l’équation de Boltzmann BGK. Phd Thesis (1999)

Mieussens L.: Discrete velocity models and implicit scheme for the BGK equation rarefied gas dynamic. M3AS 10(8), 1121–1149 (2000)

Junk M.: Domain of definition of Levermore’s five-moment system. J. Stat. Phys. 93, 1143–1167 (1998)

Junk M.: Maximum entropy for reduced moment problems. M3AS 10, 1121–1149 (2000)

Rogier F., Schneider J.: A direct method for solving the Boltzmann equation. TTSP 23(1–3), 313–338 (1994)

Pullin D.I.: Kinetic models for polyatomic molecules with phenomenological energy exchange. Phys. Fluids 21, 209–216 (1978)

Schneider J.: Entropic approximation in kinetic theory. M2AN 38(3), 541–561 (2004)

Palczewski A., Schneider J., Bobylev A.: A consistency result for a discrete-velocity model of the Boltzmann equation. SIAM. J. Numer. Anal. 34(5), 1865–1883 (1997)

Toscani G.: Remarks on entropy and equilibrium states. Appl. Math. Lett. 12(7), 19–25 (1999)

Villani C.: Fisher information estimates for Boltzmann’s collision operator. J. Maths Pures Appl. 77, 821–837 (1998)

Wennberg, B.: Stability and Exponential Convergence for the Boltzmann Equation. Phd thesis, Chalmers University Tech. (1993)

Thông tin
Thông tin xuất bản

Continuum Mechanics and Thermodynamics

Tập 20

489-508

Thông tin tác giả