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

A general consistent BGK model for gas mixtures

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.

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. AndriesK. 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. BhatnagarE. 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. BrullV. 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. GarzóA. 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. GroppiS. 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. HaackC. 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. KlingenbergM. 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. AndriesK. 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. BhatnagarE. 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. BrullV. 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. GarzóA. 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. GroppiS. 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. HaackC. 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. KlingenbergM. 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. 

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