American Institute of Mathematical Sciences

Advanced
December  2018, 11(6): 1377-1393. doi: 10.3934/krm.2018054

A general consistent BGK model for gas mixtures

Alexander V. Bobylev 1, , Marzia Bisi 2, , Maria Groppi 2, , Giampiero Spiga 2,, and  Irina F. Potapenko 3,

1. 

Keldysh Applied Mathematics Institute, Russian Academy of Sciences, Miusskaya Sq. 4, RU-125047 Moscow, Russia
2. 

Dip. di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy
3. 

Keldysh Applied Mathematics Institute, Russian Academy of Sciences, Miusskaya Sq. 4, RU-125047 Moscow, Russia

* Corresponding author: G. Spiga

Received  September 2017 Revised  December 2017 Published  June 2018

We propose a kinetic model of BGK type for a gas mixture of an arbitrary number of species with arbitrary collision law. The model features the same structure of the corresponding Boltzmann equations and fulfils all consistency requirements concerning conservation laws, equilibria, and H-theorem. Comparison is made to existing BGK models for mixtures, and the achieved improvements are commented on. Finally, possible application to the case of Coulomb interaction is briefly discussed.

Keywords: Kinetic theory, Boltzmann equation, BGK models, gas mixtures, gas mixtures, exchange rates.
Mathematics Subject Classification: Primary: 76P05; Secondary: 82C40.
Citation: Alexander V. Bobylev, Marzia Bisi, Maria Groppi, Giampiero Spiga, Irina F. Potapenko. A general consistent BGK model for gas mixtures. Kinetic and Related Models, 2018, 11 (6) : 1377-1393. doi: 10.3934/krm.2018054
References:
[1]

P. Andries, K. Aoki and B. Perthame, A consistent BGK-type model for gas mixtures, J. Stat. Phys., 106 (2002), 993-1018.  doi: 10.1023/A:1014033703134.

[2]

P. L. Bhatnagar, E. P. Gross and K. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-524. 

[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]

S. Brull, An ellipsoidal statistical model for gas mixtures, Commun. Math. Sci., 13 (2015), 1-13.  doi: 10.4310/CMS.2015.v13.n1.a1.

[6]

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

[7]

S. Brull, V. Pavan and J. Schneider, Derivation of a BGK model for mixtures, Europ. J. Mech. B/Fluids, 33 (2012), 74-86.  doi: 10.1016/j.euromechflu.2011.12.003.

[8]

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

[9]

V. Garzó, A. Santos and J. J. Brey, A kinetic model for a multicomponent gas, Phys. Fluids, 1 (1989), 380-383. 

[10]

E. Goldman and L. Sirovich, Equations for gas mixtures, Phys. Fluids, 10 (1967), 1928-1940. 

[11]

J. M. Greene, Improved Bhatnagar-Gross-Krook model for electron-ion collisions, Phys. Fluids, 16 (1973), 2022-2023.  doi: 10.1063/1.1694254.

[12]

M. Groppi, S. Rjasanow and G. Spiga, A kinetic relaxation approach to fast reactive mixtures: Shock wave structure, J. Stat. Mech. - Theory Exp., 2009 (2009), P10010.  doi: 10.1088/1742-5468/2009/10/P10010.

[13]

M. Groppi and G. Spiga, A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures, Phys. Fluids, 16 (2004), 4273-4284.  doi: 10.1063/1.1808651.

[14]

E. P. Gross and M. Krook, Model for collision processes in gases: Small-amplitude oscillations of charged two-component systems, Phys. Rev., 102 (1956), 593-604.  doi: 10.1103/PhysRev.102.593.

[15]

J. R. Haack, C. D. Hauck and M. S. Murillo, A conservative, entropic multispecies BGK model, J. Stat. Phys., 168 (2017), 826-856.  doi: 10.1007/s10955-017-1824-9.

[16]

J. R. Haack, C. D. Hauck and M. S. Murillo, Interfacial mixing in high energy-density matter with a multiphysics kinetic model, Phys. Rev. E, 96 (2017), 063310 (pp. 1–14).

[17]

B. B. Hamel, Kinetic model for binary gas mixtures, Phys. Fluids, 8 (1965), 418-425.  doi: 10.1063/1.1761239.

[18]

C. Klingenberg, M. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture with an application to plasma, Kinet. Relat. Models, 10 (2017), 445-465.  doi: 10.3934/krm.2017017.

[19]

M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, 1969. doi: 10.1007/978-1-4899-6381-9.

[20]

G. M. Kremer, M. Pandolfi Bianchi and A. J. Soares, A relaxation kinetic model for transport phenomena in a reactive flow, Phys. Fluids, 18 (2006), 037104, 15pp. doi: 10.1063/1.2185691.

[21]

L. D. Landau, Kinetic equation for the Coulomb interaction, Phys. Z. Sowjetunion, 10 (1936), 154-164. 

[22]

L. D. Landau and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford, 1969.

[23]

E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Butterworth–Heinemann, 1981.

[24]

T. F. Morse, Kinetic model equations for a gas mixture, Phys. Fluids, 7 (1964), 2012-2013.  doi: 10.1063/1.1711112.

[25]

L. Sirovich, Kinetic modeling of gas mixtures, Phys. Fluids, 5 (1962), 908-918.  doi: 10.1063/1.1706706.

[26]

P. Welander, On the temperature jump in a rarefied gas, Ark. Fys., 7 (1954), 507-533. 

show all references

References:
[1]

P. Andries, K. Aoki and B. Perthame, A consistent BGK-type model for gas mixtures, J. Stat. Phys., 106 (2002), 993-1018.  doi: 10.1023/A:1014033703134.

[2]

P. L. Bhatnagar, E. P. Gross and K. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-524. 

[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]

S. Brull, An ellipsoidal statistical model for gas mixtures, Commun. Math. Sci., 13 (2015), 1-13.  doi: 10.4310/CMS.2015.v13.n1.a1.

[6]

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

[7]

S. Brull, V. Pavan and J. Schneider, Derivation of a BGK model for mixtures, Europ. J. Mech. B/Fluids, 33 (2012), 74-86.  doi: 10.1016/j.euromechflu.2011.12.003.

[8]

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

[9]

V. Garzó, A. Santos and J. J. Brey, A kinetic model for a multicomponent gas, Phys. Fluids, 1 (1989), 380-383. 

[10]

E. Goldman and L. Sirovich, Equations for gas mixtures, Phys. Fluids, 10 (1967), 1928-1940. 

[11]

J. M. Greene, Improved Bhatnagar-Gross-Krook model for electron-ion collisions, Phys. Fluids, 16 (1973), 2022-2023.  doi: 10.1063/1.1694254.

[12]

M. Groppi, S. Rjasanow and G. Spiga, A kinetic relaxation approach to fast reactive mixtures: Shock wave structure, J. Stat. Mech. - Theory Exp., 2009 (2009), P10010.  doi: 10.1088/1742-5468/2009/10/P10010.

[13]

M. Groppi and G. Spiga, A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures, Phys. Fluids, 16 (2004), 4273-4284.  doi: 10.1063/1.1808651.

[14]

E. P. Gross and M. Krook, Model for collision processes in gases: Small-amplitude oscillations of charged two-component systems, Phys. Rev., 102 (1956), 593-604.  doi: 10.1103/PhysRev.102.593.

[15]

J. R. Haack, C. D. Hauck and M. S. Murillo, A conservative, entropic multispecies BGK model, J. Stat. Phys., 168 (2017), 826-856.  doi: 10.1007/s10955-017-1824-9.

[16]

J. R. Haack, C. D. Hauck and M. S. Murillo, Interfacial mixing in high energy-density matter with a multiphysics kinetic model, Phys. Rev. E, 96 (2017), 063310 (pp. 1–14).

[17]

B. B. Hamel, Kinetic model for binary gas mixtures, Phys. Fluids, 8 (1965), 418-425.  doi: 10.1063/1.1761239.

[18]

C. Klingenberg, M. Pirner and G. Puppo, A consistent kinetic model for a two-component mixture with an application to plasma, Kinet. Relat. Models, 10 (2017), 445-465.  doi: 10.3934/krm.2017017.

[19]

M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, 1969. doi: 10.1007/978-1-4899-6381-9.

[20]

G. M. Kremer, M. Pandolfi Bianchi and A. J. Soares, A relaxation kinetic model for transport phenomena in a reactive flow, Phys. Fluids, 18 (2006), 037104, 15pp. doi: 10.1063/1.2185691.

[21]

L. D. Landau, Kinetic equation for the Coulomb interaction, Phys. Z. Sowjetunion, 10 (1936), 154-164. 

[22]

L. D. Landau and E. M. Lifshitz, Mechanics, Pergamon Press, Oxford, 1969.

[23]

E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Butterworth–Heinemann, 1981.

[24]

T. F. Morse, Kinetic model equations for a gas mixture, Phys. Fluids, 7 (1964), 2012-2013.  doi: 10.1063/1.1711112.

[25]

L. Sirovich, Kinetic modeling of gas mixtures, Phys. Fluids, 5 (1962), 908-918.  doi: 10.1063/1.1706706.

[26]

P. Welander, On the temperature jump in a rarefied gas, Ark. Fys., 7 (1954), 507-533. 

[1]

Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic and Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003
[2]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 4035-4067. doi: 10.3934/dcdss.2020458
[3]

Sebastiano Boscarino, Seung Yeon Cho, Maria Groppi, Giovanni Russo. BGK models for inert mixtures: Comparison and applications. Kinetic and Related Models, 2021, 14 (5) : 895-928. doi: 10.3934/krm.2021029
[4]

Carlota M. Cuesta, Sabine Hittmeir, Christian Schmeiser. Weak shocks of a BGK kinetic model for isentropic gas dynamics. Kinetic and Related Models, 2010, 3 (2) : 255-279. doi: 10.3934/krm.2010.3.255
[5]

Niclas Bernhoff. Boundary layers and shock profiles for the discrete Boltzmann equation for mixtures. Kinetic and Related Models, 2012, 5 (1) : 1-19. doi: 10.3934/krm.2012.5.1
[6]

Yuanchang Sun, Lisa M. Wingen, Barbara J. Finlayson-Pitts, Jack Xin. A semi-blind source separation method for differential optical absorption spectroscopy of atmospheric gas mixtures. Inverse Problems and Imaging, 2014, 8 (2) : 587-610. doi: 10.3934/ipi.2014.8.587
[7]

Niclas Bernhoff. Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic and Related Models, 2017, 10 (4) : 925-955. doi: 10.3934/krm.2017037
[8]

Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic and Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004
[9]

Laurent Boudin, Bérénice Grec, Milana Pavić, Francesco Salvarani. Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinetic and Related Models, 2013, 6 (1) : 137-157. doi: 10.3934/krm.2013.6.137
[10]

Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219
[11]

Raffaele Esposito, Mario Pulvirenti. Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas. Kinetic and Related Models, 2010, 3 (2) : 281-297. doi: 10.3934/krm.2010.3.281
[12]

Karsten Matthies, George Stone, Florian Theil. The derivation of the linear Boltzmann equation from a Rayleigh gas particle model. Kinetic and Related Models, 2018, 11 (1) : 137-177. doi: 10.3934/krm.2018008
[13]

Michael Herty, Gabriella Puppo, Sebastiano Roncoroni, Giuseppe Visconti. The BGK approximation of kinetic models for traffic. Kinetic and Related Models, 2020, 13 (2) : 279-307. doi: 10.3934/krm.2020010
[14]

Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic and Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185
[15]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025
[16]

Vladimir Djordjić, Milana Pavić-Čolić, Nikola Spasojević. Polytropic gas modelling at kinetic and macroscopic levels. Kinetic and Related Models, 2021, 14 (3) : 483-522. doi: 10.3934/krm.2021013
[17]

Karsten Matthies, George Stone. Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3299-3355. doi: 10.3934/dcds.2018143
[18]

Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic and Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009
[19]

Masashi Ohnawa. Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities. Kinetic and Related Models, 2012, 5 (4) : 857-872. doi: 10.3934/krm.2012.5.857
[20]

François Golse. The Boltzmann-Grad limit for the Lorentz gas with a Poisson distribution of obstacles. Kinetic and Related Models, 2022, 15 (3) : 517-534. doi: 10.3934/krm.2022001

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (594)
  • HTML views (243)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy