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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

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