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June  2017, 10(2): 445-465. doi: 10.3934/krm.2017017

A consistent kinetic model for a two-component mixture with an application to plasma

Christian Klingenberg 1, , Marlies Pirner 1, and  Gabriella Puppo 2,

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

Received  January 2016 Revised  June 2016 Published  November 2016

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.

Keywords: Multi-fluid mixture, kinetic model, BGK approximation, plasma flow.
Mathematics Subject Classification: Primary: 82B40, 35Q20; Secondary: 82D10.
Citation: Christian Klingenberg, Marlies Pirner, Gabriella Puppo. A consistent kinetic model for a two-component mixture with an application to plasma. Kinetic & Related Models, 2017, 10 (2) : 445-465. doi: 10.3934/krm.2017017
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.  Google Scholar

[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.  Google Scholar

[3] P. M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511807183.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10] C. Cercignani, The Boltzmann Equation and its Applications, Springer-Verlag, New York, 1988.  doi: 10.1007/978-1-4612-1039-9.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[13]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numerica, 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar

[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.  Google Scholar

[15]

V. Garzó, A. Santos and J. J. Brey, A kinetic model for a multicomponent gas, Physics of Fluids, 1 (1989), 380-383.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

B. Hamel, Kinetic model for binary gas mixtures, Physics of Fluids, 8 (1965), 418-425.  doi: 10.1063/1.1761239.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[23] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3] P. M. Bellan, Fundamentals of Plasma Physics, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511807183.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10] C. Cercignani, The Boltzmann Equation and its Applications, Springer-Verlag, New York, 1988.  doi: 10.1007/978-1-4612-1039-9.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[13]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numerica, 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar

[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.  Google Scholar

[15]

V. Garzó, A. Santos and J. J. Brey, A kinetic model for a multicomponent gas, Physics of Fluids, 1 (1989), 380-383.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

B. Hamel, Kinetic model for binary gas mixtures, Physics of Fluids, 8 (1965), 418-425.  doi: 10.1063/1.1761239.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[23] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005.   Google Scholar
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