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A consistent kinetic model for a two-component mixture with an application to plasma
1. | Dept. of Mathematics at Würzburg University, Emil Fischer Str. 40, Würzburg, 97074, Germany |
2. | Università degli Studi dell'Insubria, Via Valleggio, Como, 22100, Italy |
We consider a non reactive multi component gas mixture.We propose a class of models, which can be easily generalized to multiple species. The two species mixture is modelled by a system of kinetic BGK equations featuring two interaction terms to account for momentum and energy transfer between the species. We prove consistency of our model: conservation properties, positivity of the solutions for the space homogeneous case, positivity of all temperatures, H-theorem and convergence to a global equilibrium in the space homogeneous case in the form of a global Maxwell distribution. Thus, we are able to derive the usual macroscopic conservation laws. In particular, by considering a mixture composed of ions and electrons, we derive the macroscopic equations of ideal MHD from our model.
References:
[1] |
P. Andries, K. Aoki and B. Perthame,
A consistent BGK-type model for gas mixtures, Journal of Statistical Physics, 106 (2002), 993-1018.
doi: 10.1023/A:1014033703134. |
[2] |
P. Asinari,
Asymptotic analysis of multiple-relaxation-time lattice Boltzmann schemes for mixture modeling, Computers and Mathematics with Applications, 55 (2008), 1392-1407.
doi: 10.1016/j.camwa.2007.08.006. |
[3] |
P. M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press, 2006.
doi: 10.1017/CBO9780511807183.![]() ![]() |
[4] |
M. Bennoune, M. Lemou and L. Mieussens,
Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, Journal of Computational Physics, 227 (2008), 3781-3803.
doi: 10.1016/j.jcp.2007.11.032. |
[5] |
F. Bernard, A. Iollo and G. Puppo,
Accurate asymptotic preserving boundary conditions for
kinetic equations on Cartesian grids, Journal of Scientific Computing, 65 (2015), 735-766.
doi: 10.1007/s10915-015-9984-8. |
[6] |
C. Besse, P. Degond, F. Deluzet, J. Claudel, G. Gallice and C. Tessieras,
A model hierarchy for ionospheric plasma modelling, Mathematical Models and Methods in Applied Sciences, 14 (2004), 393-415.
doi: 10.1142/S0218202504003283. |
[7] |
S. Brull,
An ellipsoidal statistical model for gas mixtures, Communications in Mathematical Sciences, 13 (2015), 1-13.
doi: 10.4310/CMS.2015.v13.n1.a1. |
[8] |
S. Brull, V. Pavan and J. Schneider,
Derivation of a BGK model for mixtures, European Journal of Mechanics B/Fluids, 33 (2012), 74-86.
doi: 10.1016/j.euromechflu.2011.12.003. |
[9] |
C. Cercignani,
Rarefied Gas Dynamics, From Basic Concepts to Actual Calculations, Cambridge University Press, 2000. |
[10] |
C. Cercignani, The Boltzmann Equation and its Applications, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9.![]() ![]() ![]() |
[11] |
A. Crestetto, N. Crouseilles and M. Lemou,
Kinetic/fluid micro-macro numerical schemes
for Vlasov-Poisson-BGK equation using particles, Kinetic and Related Models, 5 (2012), 787-816.
doi: 10.3934/krm.2012.5.787. |
[12] |
G. Dimarco, L. Mieussens and V. Rispoli,
An asymptotic preserving automatic domain decomposition method for the Vlasov-Poisson-BGK system with applications to plasmas, Journal of Computational Physics, 274 (2014), 122-139.
doi: 10.1016/j.jcp.2014.06.002. |
[13] |
G. Dimarco and L. Pareschi,
Numerical methods for kinetic equations, Acta Numerica, 23 (2014), 369-520.
doi: 10.1017/S0962492914000063. |
[14] |
F. Filbet and S. Jin,
A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, Journal of Computational Physics, 229 (2010), 7625-7648.
doi: 10.1016/j.jcp.2010.06.017. |
[15] |
V. Garzó, A. Santos and J. J. Brey,
A kinetic model for a multicomponent gas, Physics of Fluids, 1 (1989), 380-383.
|
[16] |
M. Groppi, S. Monica and G. Spiga,
A kinetic ellipsoidal BGK model for a binary gas mixture, epljournal, 96 (2011), 64002.
doi: 10.1209/0295-5075/96/64002. |
[17] |
E. P. Gross and M. Krook,
Model for collision processes in gases: Small-amplitude oscillations of charged two-component systems, Physical Review, 102 (1956), 593.
doi: 10.1103/PhysRev.102.593. |
[18] |
B. Hamel,
Kinetic model for binary gas mixtures, Physics of Fluids, 8 (1965), 418-425.
doi: 10.1063/1.1761239. |
[19] |
M. Monteferrante, S. Melchionna and U. M. B. Marconi,
Lattice Boltzmann method for mixtures at variable Schmidt number, Journal of Chemical Physics, 141 (2014), 014102.
doi: 10.1063/1.4885719. |
[20] |
S. Pieraccini and G. Puppo,
Implicit-explicit schemes for BGK kinetic equations, Journal of Scientific Computing, 32 (2007), 1-28.
doi: 10.1007/s10915-006-9116-6. |
[21] |
C. E. Pico Ortiz, L. O. E. dos Santos and P. C. Philippi, Thermal lattice Boltzmann BGK model for ideal binary mixtures, 19th International Congress of Mechanical Engineering, 2007. |
[22] |
V. Sofonea and R. Sekerka,
BGK models for diffusion in isothermal binary fluid systems, Physica, 299 (2001), 494-520.
doi: 10.1016/S0378-4371(01)00246-1. |
[23] |
H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005.
![]() ![]() |
show all references
References:
[1] |
P. Andries, K. Aoki and B. Perthame,
A consistent BGK-type model for gas mixtures, Journal of Statistical Physics, 106 (2002), 993-1018.
doi: 10.1023/A:1014033703134. |
[2] |
P. Asinari,
Asymptotic analysis of multiple-relaxation-time lattice Boltzmann schemes for mixture modeling, Computers and Mathematics with Applications, 55 (2008), 1392-1407.
doi: 10.1016/j.camwa.2007.08.006. |
[3] |
P. M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press, 2006.
doi: 10.1017/CBO9780511807183.![]() ![]() |
[4] |
M. Bennoune, M. Lemou and L. Mieussens,
Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, Journal of Computational Physics, 227 (2008), 3781-3803.
doi: 10.1016/j.jcp.2007.11.032. |
[5] |
F. Bernard, A. Iollo and G. Puppo,
Accurate asymptotic preserving boundary conditions for
kinetic equations on Cartesian grids, Journal of Scientific Computing, 65 (2015), 735-766.
doi: 10.1007/s10915-015-9984-8. |
[6] |
C. Besse, P. Degond, F. Deluzet, J. Claudel, G. Gallice and C. Tessieras,
A model hierarchy for ionospheric plasma modelling, Mathematical Models and Methods in Applied Sciences, 14 (2004), 393-415.
doi: 10.1142/S0218202504003283. |
[7] |
S. Brull,
An ellipsoidal statistical model for gas mixtures, Communications in Mathematical Sciences, 13 (2015), 1-13.
doi: 10.4310/CMS.2015.v13.n1.a1. |
[8] |
S. Brull, V. Pavan and J. Schneider,
Derivation of a BGK model for mixtures, European Journal of Mechanics B/Fluids, 33 (2012), 74-86.
doi: 10.1016/j.euromechflu.2011.12.003. |
[9] |
C. Cercignani,
Rarefied Gas Dynamics, From Basic Concepts to Actual Calculations, Cambridge University Press, 2000. |
[10] |
C. Cercignani, The Boltzmann Equation and its Applications, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9.![]() ![]() ![]() |
[11] |
A. Crestetto, N. Crouseilles and M. Lemou,
Kinetic/fluid micro-macro numerical schemes
for Vlasov-Poisson-BGK equation using particles, Kinetic and Related Models, 5 (2012), 787-816.
doi: 10.3934/krm.2012.5.787. |
[12] |
G. Dimarco, L. Mieussens and V. Rispoli,
An asymptotic preserving automatic domain decomposition method for the Vlasov-Poisson-BGK system with applications to plasmas, Journal of Computational Physics, 274 (2014), 122-139.
doi: 10.1016/j.jcp.2014.06.002. |
[13] |
G. Dimarco and L. Pareschi,
Numerical methods for kinetic equations, Acta Numerica, 23 (2014), 369-520.
doi: 10.1017/S0962492914000063. |
[14] |
F. Filbet and S. Jin,
A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, Journal of Computational Physics, 229 (2010), 7625-7648.
doi: 10.1016/j.jcp.2010.06.017. |
[15] |
V. Garzó, A. Santos and J. J. Brey,
A kinetic model for a multicomponent gas, Physics of Fluids, 1 (1989), 380-383.
|
[16] |
M. Groppi, S. Monica and G. Spiga,
A kinetic ellipsoidal BGK model for a binary gas mixture, epljournal, 96 (2011), 64002.
doi: 10.1209/0295-5075/96/64002. |
[17] |
E. P. Gross and M. Krook,
Model for collision processes in gases: Small-amplitude oscillations of charged two-component systems, Physical Review, 102 (1956), 593.
doi: 10.1103/PhysRev.102.593. |
[18] |
B. Hamel,
Kinetic model for binary gas mixtures, Physics of Fluids, 8 (1965), 418-425.
doi: 10.1063/1.1761239. |
[19] |
M. Monteferrante, S. Melchionna and U. M. B. Marconi,
Lattice Boltzmann method for mixtures at variable Schmidt number, Journal of Chemical Physics, 141 (2014), 014102.
doi: 10.1063/1.4885719. |
[20] |
S. Pieraccini and G. Puppo,
Implicit-explicit schemes for BGK kinetic equations, Journal of Scientific Computing, 32 (2007), 1-28.
doi: 10.1007/s10915-006-9116-6. |
[21] |
C. E. Pico Ortiz, L. O. E. dos Santos and P. C. Philippi, Thermal lattice Boltzmann BGK model for ideal binary mixtures, 19th International Congress of Mechanical Engineering, 2007. |
[22] |
V. Sofonea and R. Sekerka,
BGK models for diffusion in isothermal binary fluid systems, Physica, 299 (2001), 494-520.
doi: 10.1016/S0378-4371(01)00246-1. |
[23] |
H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005.
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