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A numerical model of the Boltzmann equation related to the discontinuous Galerkin method
On a kinetic BGK model for slow chemical reactions
1. | Dipartimento di Matematica, Università di Parma, V.le G.P. Usberti 53/A, 43124 Parma, Italy, Italy |
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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