American Institute of Mathematical Sciences

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March  2011, 4(1): 153-167. doi: 10.3934/krm.2011.4.153

On a kinetic BGK model for slow chemical reactions

Marzia Bisi 1, and  Giampiero Spiga 1,

1. 

Dipartimento di Matematica, Università di Parma, V.le G.P. Usberti 53/A, 43124 Parma, Italy, Italy

Received  July 2010 Revised  October 2010 Published  January 2011

A recently proposed consistent BGK-type approach for chemically reacting gas mixtures is discussed, which accounts for the correct rates of transfer for mass, momentum and energy, and recovers the exact conservation equations and collision equilibria, including mass action law. In particular, the hydrodynamic limit is derived by a Chapman-Enskog procedure, and compared to existing results for the reactive and non-reactive cases.
Keywords: Kinetic theory; Chemical reaction; BGK model..
Mathematics Subject Classification: 82C40, 76P0.
Citation: Marzia Bisi, Giampiero Spiga. On a kinetic BGK model for slow chemical reactions. Kinetic & Related Models, 2011, 4 (1) : 153-167. doi: 10.3934/krm.2011.4.153
References:
[1]

M. Abramowitz and I. A. Stegun (Eds.), "Handbook of Mathematical Functions,'' Dover, New York, 1965. Google Scholar

[2]

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

[3]

K. Aoki, Y. Sone and T. Yamada, Numerical analysis of gas flows condensing on its plane condensed phase on the basis of kinetic theory, Phys. Fluids A, 2 (1990), 1867-1878. doi: 10.1063/1.857661.  Google Scholar

[4]

P. L. Bhatnagar, E. P. Gross and K. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-524. doi: 10.1103/PhysRev.94.511.  Google Scholar

[5]

M. Bisi, M. Groppi and G. Spiga, Grad's distribution functions in the kinetic equations for a chemical reaction, Continuum Mech. Thermodyn., 14 (2002), 207-222. doi: 10.1007/s001610100066.  Google Scholar

[6]

M. Bisi, M. Groppi and G. Spiga, Fluid-dynamic equations for reacting gas mixtures, Applications of Mathematics, 50 (2005), 43-62. doi: 10.1007/s10492-005-0003-5.  Google Scholar

[7]

M. Bisi, M. Groppi and G. Spiga, Kinetic problems in rarefied gas mixtures, in "Proceedings of 26th International Symposium on Rarefied Gas Dynamics'' (Kyoto, Japan, July 21-25, 2008), T. Abe Ed., A.I.P., New York, 2009, pp. 87-92. Google Scholar

[8]

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. doi: 10.1103/PhysRevE.81.036327.  Google Scholar

[9]

A. V. Bobylev, The theory of the spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Review C, 7 (1988), 112-229.  Google Scholar

[10]

C. Cercignani, "The Boltzmann Equation and its Applications,'' Springer Verlag, New York, 1988.  Google Scholar

[11]

C. Cercignani, "Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations,'' Cambridge University Press, Cambridge, 2000.  Google Scholar

[12]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-uniform Gases,'' University Press, Cambridge, 1970.  Google Scholar

[13]

J. F. Clarke and M. McChesney, "The Dynamics of Real Gases,'' Butterworths, London, 1964. Google Scholar

[14]

F. Conforto, R. Monaco, F. Schürrer and I. Ziegler, Steady detonation waves via the Boltzmann equation for a reacting mixture, J. Phys. A, 36 (2003), 5381-5398. doi: 10.1088/0305-4470/36/20/303.  Google Scholar

[15]

S. R. De Groot and P. Mazur, "Non-equilibrium Thermodynamics,'' North Holland, Amsterdam, 1962.  Google Scholar

[16]

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

[17]

G. Dixon-Lewis, Flame structure and flame reaction kinetics. II. Transport phenomena in multicomponent systems, Proc. R. Soc. Lond. A, 307 (1968), 111-135. doi: 10.1098/rspa.1968.0178.  Google Scholar

[18]

J. H. Ferziger and H. G. Kaper, "Mathematical Theory of Transport Processes in Gases,'' North Holland, Amsterdam, 1972. Google Scholar

[19]

V. Garzó, A. Santos and J. J. Brey, A kinetic model for a multicomponent gas, Phys. of Fluids A: Fluid Dynamics, 1 (1989), 380-383. doi: 10.1063/1.857458.  Google Scholar

[20]

V. Giovangigli, "Multicomponent Flow Modeling,'' Birkhäuser Verlag, Boston, 1999.  Google Scholar

[21]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219. doi: 10.1023/A:1019194113816.  Google Scholar

[22]

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

[23]

R. J. Kee, M. E. Coltrin and P. Glarborg, "Chemically Reacting Flow: Theory and Practice,'' Wiley, New York, 2003. doi: 10.1002/0471461296.  Google Scholar

[24]

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

[25]

R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chemical Engineering Science, 52 (1997), 861-911. doi: 10.1016/S0009-2509(96)00458-7.  Google Scholar

[26]

K. K. Kuo, "Principles of Combustion,'' Wiley, New York, 2005. Google Scholar

[27]

R. Monaco, M. Pandolfi Bianchi and A. J. Soares, BGK-type models in strong reaction and kinetic chemical equilibrium regimes, J. Phys. A: Math. Gen., 38 (2005), 10413-10431. doi: 10.1088/0305-4470/38/48/012.  Google Scholar

[28]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Physica A, 272 (1999), 563-573. doi: 10.1016/S0378-4371(99)00336-2.  Google Scholar

[29]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Birkhäuser Verlag, Boston, 2002.  Google Scholar

[30]

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

show all references

References:
[1]

M. Abramowitz and I. A. Stegun (Eds.), "Handbook of Mathematical Functions,'' Dover, New York, 1965. Google Scholar

[2]

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

[3]

K. Aoki, Y. Sone and T. Yamada, Numerical analysis of gas flows condensing on its plane condensed phase on the basis of kinetic theory, Phys. Fluids A, 2 (1990), 1867-1878. doi: 10.1063/1.857661.  Google Scholar

[4]

P. L. Bhatnagar, E. P. Gross and K. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-524. doi: 10.1103/PhysRev.94.511.  Google Scholar

[5]

M. Bisi, M. Groppi and G. Spiga, Grad's distribution functions in the kinetic equations for a chemical reaction, Continuum Mech. Thermodyn., 14 (2002), 207-222. doi: 10.1007/s001610100066.  Google Scholar

[6]

M. Bisi, M. Groppi and G. Spiga, Fluid-dynamic equations for reacting gas mixtures, Applications of Mathematics, 50 (2005), 43-62. doi: 10.1007/s10492-005-0003-5.  Google Scholar

[7]

M. Bisi, M. Groppi and G. Spiga, Kinetic problems in rarefied gas mixtures, in "Proceedings of 26th International Symposium on Rarefied Gas Dynamics'' (Kyoto, Japan, July 21-25, 2008), T. Abe Ed., A.I.P., New York, 2009, pp. 87-92. Google Scholar

[8]

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. doi: 10.1103/PhysRevE.81.036327.  Google Scholar

[9]

A. V. Bobylev, The theory of the spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Review C, 7 (1988), 112-229.  Google Scholar

[10]

C. Cercignani, "The Boltzmann Equation and its Applications,'' Springer Verlag, New York, 1988.  Google Scholar

[11]

C. Cercignani, "Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations,'' Cambridge University Press, Cambridge, 2000.  Google Scholar

[12]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-uniform Gases,'' University Press, Cambridge, 1970.  Google Scholar

[13]

J. F. Clarke and M. McChesney, "The Dynamics of Real Gases,'' Butterworths, London, 1964. Google Scholar

[14]

F. Conforto, R. Monaco, F. Schürrer and I. Ziegler, Steady detonation waves via the Boltzmann equation for a reacting mixture, J. Phys. A, 36 (2003), 5381-5398. doi: 10.1088/0305-4470/36/20/303.  Google Scholar

[15]

S. R. De Groot and P. Mazur, "Non-equilibrium Thermodynamics,'' North Holland, Amsterdam, 1962.  Google Scholar

[16]

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

[17]

G. Dixon-Lewis, Flame structure and flame reaction kinetics. II. Transport phenomena in multicomponent systems, Proc. R. Soc. Lond. A, 307 (1968), 111-135. doi: 10.1098/rspa.1968.0178.  Google Scholar

[18]

J. H. Ferziger and H. G. Kaper, "Mathematical Theory of Transport Processes in Gases,'' North Holland, Amsterdam, 1972. Google Scholar

[19]

V. Garzó, A. Santos and J. J. Brey, A kinetic model for a multicomponent gas, Phys. of Fluids A: Fluid Dynamics, 1 (1989), 380-383. doi: 10.1063/1.857458.  Google Scholar

[20]

V. Giovangigli, "Multicomponent Flow Modeling,'' Birkhäuser Verlag, Boston, 1999.  Google Scholar

[21]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219. doi: 10.1023/A:1019194113816.  Google Scholar

[22]

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

[23]

R. J. Kee, M. E. Coltrin and P. Glarborg, "Chemically Reacting Flow: Theory and Practice,'' Wiley, New York, 2003. doi: 10.1002/0471461296.  Google Scholar

[24]

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

[25]

R. Krishna and J. A. Wesselingh, The Maxwell-Stefan approach to mass transfer, Chemical Engineering Science, 52 (1997), 861-911. doi: 10.1016/S0009-2509(96)00458-7.  Google Scholar

[26]

K. K. Kuo, "Principles of Combustion,'' Wiley, New York, 2005. Google Scholar

[27]

R. Monaco, M. Pandolfi Bianchi and A. J. Soares, BGK-type models in strong reaction and kinetic chemical equilibrium regimes, J. Phys. A: Math. Gen., 38 (2005), 10413-10431. doi: 10.1088/0305-4470/38/48/012.  Google Scholar

[28]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Physica A, 272 (1999), 563-573. doi: 10.1016/S0378-4371(99)00336-2.  Google Scholar

[29]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Birkhäuser Verlag, Boston, 2002.  Google Scholar

[30]

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

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