On a kinetic BGK model for slow chemical reactions

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	A. V. Bobylev, The theory of the spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Review C, 7 (1988), 112-229.
[10]	C. Cercignani, "The Boltzmann Equation and its Applications,'' Springer Verlag, New York, 1988.
[11]	C. Cercignani, "Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations,'' Cambridge University Press, Cambridge, 2000.
[12]	S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-uniform Gases,'' University Press, Cambridge, 1970.
[13]	J. F. Clarke and M. McChesney, "The Dynamics of Real Gases,'' Butterworths, London, 1964.
[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.
[15]	S. R. De Groot and P. Mazur, "Non-equilibrium Thermodynamics,'' North Holland, Amsterdam, 1962.
[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.
[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.
[18]	J. H. Ferziger and H. G. Kaper, "Mathematical Theory of Transport Processes in Gases,'' North Holland, Amsterdam, 1972.
[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.
[20]	V. Giovangigli, "Multicomponent Flow Modeling,'' Birkhäuser Verlag, Boston, 1999.
[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.
[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.
[23]	R. J. Kee, M. E. Coltrin and P. Glarborg, "Chemically Reacting Flow: Theory and Practice,'' Wiley, New York, 2003. doi: 10.1002/0471461296.
[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.
[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.
[26]	K. K. Kuo, "Principles of Combustion,'' Wiley, New York, 2005.
[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.
[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.
[29]	Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Birkhäuser Verlag, Boston, 2002.
[30]	P. Welander, On the temperature jump in a rarefied gas, Ark. Fys., 7 (1954), 507-533.

show all references

References:

[1]	M. Abramowitz and I. A. Stegun (Eds.), "Handbook of Mathematical Functions,'' Dover, New York, 1965.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	A. V. Bobylev, The theory of the spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Review C, 7 (1988), 112-229.
[10]	C. Cercignani, "The Boltzmann Equation and its Applications,'' Springer Verlag, New York, 1988.
[11]	C. Cercignani, "Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations,'' Cambridge University Press, Cambridge, 2000.
[12]	S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-uniform Gases,'' University Press, Cambridge, 1970.
[13]	J. F. Clarke and M. McChesney, "The Dynamics of Real Gases,'' Butterworths, London, 1964.
[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.
[15]	S. R. De Groot and P. Mazur, "Non-equilibrium Thermodynamics,'' North Holland, Amsterdam, 1962.
[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.
[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.
[18]	J. H. Ferziger and H. G. Kaper, "Mathematical Theory of Transport Processes in Gases,'' North Holland, Amsterdam, 1972.
[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.
[20]	V. Giovangigli, "Multicomponent Flow Modeling,'' Birkhäuser Verlag, Boston, 1999.
[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.
[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.
[23]	R. J. Kee, M. E. Coltrin and P. Glarborg, "Chemically Reacting Flow: Theory and Practice,'' Wiley, New York, 2003. doi: 10.1002/0471461296.
[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.
[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.
[26]	K. K. Kuo, "Principles of Combustion,'' Wiley, New York, 2005.
[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.
[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.
[29]	Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Birkhäuser Verlag, Boston, 2002.
[30]	P. Welander, On the temperature jump in a rarefied gas, Ark. Fys., 7 (1954), 507-533.

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