August 2018, 11(4): 821-858. doi: 10.3934/krm.2018033

On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases

1. 

CEA-CESTA, 15 avenue des sablières, CS 60001 33116 Le Barp Cedex, France

2. 

University of Parma, Dept. of Mathematics, Physics and Computer Sciences, Parco Area delle Scienze 53/A, I-43124 Parma, Italy

3. 

University of Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France

4. 

University Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu - Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, F-75013, Paris, France

* Corresponding author: L. Desvillettes

Received  August 2017 Revised  November 2017 Published  April 2018

In this paper, we propose a formal derivation of the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic gases. We use a direct extension of the model devised in [8,16] for treating the internal energy with only one continuous parameter. This model is based on the Borgnakke-Larsen procedure [6]. We detail the dissipative terms related to the interaction between the gradients of temperature and the gradients of concentrations (Dufour and Soret effects), and present a complete explicit computation in one case when such a computation is possible, that is when all cross sections in the Boltzmann equation are constants.

Citation: Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic & Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033
References:
[1]

P. AndriesP. Le TallecJ. P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B/Fluids, 19 (2000), 813-830. doi: 10.1016/S0997-7546(00)01103-1.

[2]

C. Bardos, Une interprétation des relations existant entre les équations de Boltzmann, de Navier-Stokes et d'Euler à l'aide de l'entropie, Math. Aplic. Comp., 6 (1987), 97-117.

[3]

M. Bisi and M. J. Cáceres, A BGK relaxation model for polyatomic gas mixtures, Commun. Math. Sci., 14 (2016), 297-325. doi: 10.4310/CMS.2016.v14.n2.a1.

[4]

M. Bisi, M. Groppi and G. Spiga, Kinetic Bhatnagar-Gross-Krook model for fast reactive mixtures and its hydrodynamic limit, Phys. Rev. E, 81 (2010), 036327 (pp. 1-9). doi: 10.1103/PhysRevE.81.036327.

[5]

M. BisiT. Ruggeri and G. Spiga, Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and Extended Thermodynamics, Kinet. Relat. Models, 11 (2018), 71-95. doi: 10.3934/krm.2018004.

[6]

C. Borgnakke and P. S. Larsen, Statistical collision model for Monte-Carlo simulation of polyatomic mixtures, Journ. Comput. Phys., 18 (1975), 405-420. doi: 10.1016/0021-9991(75)90094-7.

[7]

L. BoudinB. GrecM. Pavic and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures, Kinet. Relat. Models, 6 (2013), 137-157. doi: 10.3934/krm.2013.6.137.

[8]

J. F. BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, Eur. J. Mech. B/Fluids, 13 (1994), 237-254.

[9]

M. Briant and E. Daus, The Boltzmann equation for a multi-species mixture close to global equilibrium, Archive Rational Mech. Anal., 222 (2016), 1367-1443. doi: 10.1007/s00205-016-1023-x.

[10]

S. Brull and J. Schneider, On the ellipsoidal statistical model for polyatomic gases, Cont. Mech. Thermodyn., 20 (2009), 489-508. doi: 10.1007/s00161-009-0095-3.

[11]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[12]

E. DausA. JüngelC. Mouhot and N. Zamponi, Hypocoercivity for a linearized multispecies Boltzmann system, SIAM J. Math. Anal., 48 (2016), 538-568. doi: 10.1137/15M1017934.

[13]

L. Desvillettes, Sur un modèle de type Borgnakke-Larsen conduisant à des lois d'energie non-linéaires en température pour les gaz parfaits polyatomiques, Annales de la Faculté des Sciences de Toulouse, Série 6, 6 (1997), 257-262. doi: 10.5802/afst.864.

[14]

L. Desvillettes, Convergence towards the thermodynamical equilibrium, in Trends in Applications of Mathematics to Mechanics, Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall, Boca Raton, 106 (2000), 115-126.

[15]

L. Desvillettes and F. Golse, A remark concerning the Chapman{Enskog asymptotics, in Advances in Kinetic Theory and Computing, Series on Advances in Mathematics for Applied Sciences, World Scientific Publications, Singapour, 22 (1994), 191-203.

[16]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B/Fluids, 24 (2005), 219-236. doi: 10.1016/j.euromechflu.2004.07.004.

[17]

B. Dubroca and L. Mieussens, A conservative and entropic discrete-velocity model for rarefied polyatomic gases, in CEMRACS 1999 (Orsay), 10 of ESAIM Proc., Soc. Math. Appl. Indust., Paris, (1999), 127-139. doi: 10.1051/proc:2001012.

[18]

A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, Lecture Notes in Physics Monographs, M 24,1994.

[19]

A. Ern and V. Giovangigli, The kinetic equilibrium regime, Physica A, 260 (1998), 49-72.

[20]

H. FunaganeS. TakataK. Aoki and K. Kugimoto, Poiseuille flow and thermal transpiration of a rarefied polyatomic gas through a circular tube with applications to microflows, Boll. Unione Mat. Ital. (9), 4 (2011), 19-46.

[21]

V. Giovangigli, Multicomponent Flow Modeling, MESST Series, Birkhauser Boston, 1999. doi: 10.1007/978-1-4612-1580-6.

[22]

F. Golse, Fluid dynamic limits of the kinetic theory of gases, in From Particle Systems to Partial Differential Equations (Particle Systems and PDEs, Braga, Portugal, December 2012), Springer Proceedings in Mathematics & Statistics, C. Bernardin, P. Goncalves Eds., 75 (2014), 3-91. doi: 10.1007/978-3-642-54271-8_1.

[23]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J.Math.Chem., 26 (1999), 197-219.

[24]

S. KawashimaA. Matsumura and T. Nishida, On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Commun. Math. Phys., 70 (1979), 97-124. doi: 10.1007/BF01982349.

[25]

S. Kosuge, K. Aoki and T. Goto, Shock wave structure in polyatomic gases: Numerical analysis using a model Boltzmann equation in AIP Conf. Proc., 1786 (2016), 180004. doi: 10.1063/1.4967673.

[26]

P. Le Tallec, A hierarchy of hyperbolic models linking Boltzmann to Navier Stokes equations for polyatomic gases, ZAMM Z. Angew. Math. Mech., 80 (2000), 779-789. doi: 10.1002/1521-4001(200011)80:11/12<779::AID-ZAMM779>3.0.CO;2-I.

[27]

F. R. McCourt, J. J. Beenakker, W. E. Köhler and I. Kuscer, Non Equilibrium Phenomena in Polyatomic Gases. Volume Ⅰ: Dilute Gases, Clarendon Press, Oxford, 1990.

[28]

L. MonchickK. S. Yun and E. A. Mason, Formal kinetic theory of transport phenomena in polyatomic gas mixtures, J. Chem. Phys., 39 (1963), 654-669. doi: 10.1063/1.1734304.

[29]

E. Nagnibeda and E. Kustova, Non-Equilibrium Reacting Gas Flows, Springer Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01390-4.

[30]

M. PavićT. Ruggeri and S. Simić, Maximum entropy principle for polyatomic gases, Physica A, 392 (2013), 1302-1317. doi: 10.1016/j.physa.2012.12.006.

[31]

T. Ruggeri and M. Sugiyama, Rational Extended Thermodynamics Beyond the Monatomic Gas, Springer International Publishing, Switzerland, 2015. doi: 10.1007/978-3-319-13341-6.

[32]

S. TakataH. Funagane and K. Aoki, Fluid modeling for the Knudsen compressor: Case of polyatomic gases, Kinet. Relat. Models, 3 (2010), 353-372. doi: 10.3934/krm.2010.3.353.

[33]

L. Waldmann and E. Trübenbacher, Formale kinetische Theorie von Gasgemischen aus anregbaren Molekülen, Zeitschr. Naturforschg, 17a (1962), 363-376.

[34]

V. M. Zhdanov, Transport Processes in Multicomponent Plasmas, Taylor and Francis, London, 2002. doi: 10.1088/0741-3335/44/10/701.

show all references

References:
[1]

P. AndriesP. Le TallecJ. P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B/Fluids, 19 (2000), 813-830. doi: 10.1016/S0997-7546(00)01103-1.

[2]

C. Bardos, Une interprétation des relations existant entre les équations de Boltzmann, de Navier-Stokes et d'Euler à l'aide de l'entropie, Math. Aplic. Comp., 6 (1987), 97-117.

[3]

M. Bisi and M. J. Cáceres, A BGK relaxation model for polyatomic gas mixtures, Commun. Math. Sci., 14 (2016), 297-325. doi: 10.4310/CMS.2016.v14.n2.a1.

[4]

M. Bisi, M. Groppi and G. Spiga, Kinetic Bhatnagar-Gross-Krook model for fast reactive mixtures and its hydrodynamic limit, Phys. Rev. E, 81 (2010), 036327 (pp. 1-9). doi: 10.1103/PhysRevE.81.036327.

[5]

M. BisiT. Ruggeri and G. Spiga, Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and Extended Thermodynamics, Kinet. Relat. Models, 11 (2018), 71-95. doi: 10.3934/krm.2018004.

[6]

C. Borgnakke and P. S. Larsen, Statistical collision model for Monte-Carlo simulation of polyatomic mixtures, Journ. Comput. Phys., 18 (1975), 405-420. doi: 10.1016/0021-9991(75)90094-7.

[7]

L. BoudinB. GrecM. Pavic and F. Salvarani, Diffusion asymptotics of a kinetic model for gaseous mixtures, Kinet. Relat. Models, 6 (2013), 137-157. doi: 10.3934/krm.2013.6.137.

[8]

J. F. BourgatL. DesvillettesP. Le Tallec and B. Perthame, Microreversible collisions for polyatomic gases and Boltzmann's theorem, Eur. J. Mech. B/Fluids, 13 (1994), 237-254.

[9]

M. Briant and E. Daus, The Boltzmann equation for a multi-species mixture close to global equilibrium, Archive Rational Mech. Anal., 222 (2016), 1367-1443. doi: 10.1007/s00205-016-1023-x.

[10]

S. Brull and J. Schneider, On the ellipsoidal statistical model for polyatomic gases, Cont. Mech. Thermodyn., 20 (2009), 489-508. doi: 10.1007/s00161-009-0095-3.

[11]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[12]

E. DausA. JüngelC. Mouhot and N. Zamponi, Hypocoercivity for a linearized multispecies Boltzmann system, SIAM J. Math. Anal., 48 (2016), 538-568. doi: 10.1137/15M1017934.

[13]

L. Desvillettes, Sur un modèle de type Borgnakke-Larsen conduisant à des lois d'energie non-linéaires en température pour les gaz parfaits polyatomiques, Annales de la Faculté des Sciences de Toulouse, Série 6, 6 (1997), 257-262. doi: 10.5802/afst.864.

[14]

L. Desvillettes, Convergence towards the thermodynamical equilibrium, in Trends in Applications of Mathematics to Mechanics, Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall, Boca Raton, 106 (2000), 115-126.

[15]

L. Desvillettes and F. Golse, A remark concerning the Chapman{Enskog asymptotics, in Advances in Kinetic Theory and Computing, Series on Advances in Mathematics for Applied Sciences, World Scientific Publications, Singapour, 22 (1994), 191-203.

[16]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B/Fluids, 24 (2005), 219-236. doi: 10.1016/j.euromechflu.2004.07.004.

[17]

B. Dubroca and L. Mieussens, A conservative and entropic discrete-velocity model for rarefied polyatomic gases, in CEMRACS 1999 (Orsay), 10 of ESAIM Proc., Soc. Math. Appl. Indust., Paris, (1999), 127-139. doi: 10.1051/proc:2001012.

[18]

A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, Lecture Notes in Physics Monographs, M 24,1994.

[19]

A. Ern and V. Giovangigli, The kinetic equilibrium regime, Physica A, 260 (1998), 49-72.

[20]

H. FunaganeS. TakataK. Aoki and K. Kugimoto, Poiseuille flow and thermal transpiration of a rarefied polyatomic gas through a circular tube with applications to microflows, Boll. Unione Mat. Ital. (9), 4 (2011), 19-46.

[21]

V. Giovangigli, Multicomponent Flow Modeling, MESST Series, Birkhauser Boston, 1999. doi: 10.1007/978-1-4612-1580-6.

[22]

F. Golse, Fluid dynamic limits of the kinetic theory of gases, in From Particle Systems to Partial Differential Equations (Particle Systems and PDEs, Braga, Portugal, December 2012), Springer Proceedings in Mathematics & Statistics, C. Bernardin, P. Goncalves Eds., 75 (2014), 3-91. doi: 10.1007/978-3-642-54271-8_1.

[23]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J.Math.Chem., 26 (1999), 197-219.

[24]

S. KawashimaA. Matsumura and T. Nishida, On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Commun. Math. Phys., 70 (1979), 97-124. doi: 10.1007/BF01982349.

[25]

S. Kosuge, K. Aoki and T. Goto, Shock wave structure in polyatomic gases: Numerical analysis using a model Boltzmann equation in AIP Conf. Proc., 1786 (2016), 180004. doi: 10.1063/1.4967673.

[26]

P. Le Tallec, A hierarchy of hyperbolic models linking Boltzmann to Navier Stokes equations for polyatomic gases, ZAMM Z. Angew. Math. Mech., 80 (2000), 779-789. doi: 10.1002/1521-4001(200011)80:11/12<779::AID-ZAMM779>3.0.CO;2-I.

[27]

F. R. McCourt, J. J. Beenakker, W. E. Köhler and I. Kuscer, Non Equilibrium Phenomena in Polyatomic Gases. Volume Ⅰ: Dilute Gases, Clarendon Press, Oxford, 1990.

[28]

L. MonchickK. S. Yun and E. A. Mason, Formal kinetic theory of transport phenomena in polyatomic gas mixtures, J. Chem. Phys., 39 (1963), 654-669. doi: 10.1063/1.1734304.

[29]

E. Nagnibeda and E. Kustova, Non-Equilibrium Reacting Gas Flows, Springer Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01390-4.

[30]

M. PavićT. Ruggeri and S. Simić, Maximum entropy principle for polyatomic gases, Physica A, 392 (2013), 1302-1317. doi: 10.1016/j.physa.2012.12.006.

[31]

T. Ruggeri and M. Sugiyama, Rational Extended Thermodynamics Beyond the Monatomic Gas, Springer International Publishing, Switzerland, 2015. doi: 10.1007/978-3-319-13341-6.

[32]

S. TakataH. Funagane and K. Aoki, Fluid modeling for the Knudsen compressor: Case of polyatomic gases, Kinet. Relat. Models, 3 (2010), 353-372. doi: 10.3934/krm.2010.3.353.

[33]

L. Waldmann and E. Trübenbacher, Formale kinetische Theorie von Gasgemischen aus anregbaren Molekülen, Zeitschr. Naturforschg, 17a (1962), 363-376.

[34]

V. M. Zhdanov, Transport Processes in Multicomponent Plasmas, Taylor and Francis, London, 2002. doi: 10.1088/0741-3335/44/10/701.

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