American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Approximate explicit stationary solutions to a Vlasov equation for planetary rings
  • KRM Home
  • This Issue
  • Next Article
    Escaping the trap of 'blocking': A kinetic model linking economic development and political competition
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 and 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.

[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. 
[1]

Michael Herty, Gabriella Puppo, Sebastiano Roncoroni, Giuseppe Visconti. The BGK approximation of kinetic models for traffic. Kinetic and Related Models, 2020, 13 (2) : 279-307. doi: 10.3934/krm.2020010
[2]

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
[3]

Young-Pil Choi, Seok-Bae Yun. A BGK kinetic model with local velocity alignment forces. Networks and Heterogeneous Media, 2020, 15 (3) : 389-404. doi: 10.3934/nhm.2020024
[4]

Carlota M. Cuesta, Sabine Hittmeir, Christian Schmeiser. Weak shocks of a BGK kinetic model for isentropic gas dynamics. Kinetic and Related Models, 2010, 3 (2) : 255-279. doi: 10.3934/krm.2010.3.255
[5]

Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic and Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787
[6]

Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic and Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047
[7]

Charles Nguyen, Stephen Pankavich. A one-dimensional kinetic model of plasma dynamics with a transport field. Evolution Equations and Control Theory, 2014, 3 (4) : 681-698. doi: 10.3934/eect.2014.3.681
[8]

Zhen Cheng, Wenjun Wang. The Cauchy problem of a two-phase flow model for a mixture of non-interacting compressible fluids. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4155-4176. doi: 10.3934/cpaa.2021151
[9]

Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Fast algorithms for the approximation of a traffic flow model on networks. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 427-448. doi: 10.3934/dcdsb.2006.6.427
[10]

Franz Achleitner, Anton Arnold, Eric A. Carlen. On multi-dimensional hypocoercive BGK models. Kinetic and Related Models, 2018, 11 (4) : 953-1009. doi: 10.3934/krm.2018038
[11]

Silvia Caprino, Carlo Marchioro. On the plasma-charge model. Kinetic and Related Models, 2010, 3 (2) : 241-254. doi: 10.3934/krm.2010.3.241
[12]

Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic and Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006
[13]

Seung-Yeal Ha, Doron Levy. Particle, kinetic and fluid models for phototaxis. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 77-108. doi: 10.3934/dcdsb.2009.12.77
[14]

Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039
[15]

Fucai Li, Yue Li. Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3583-3604. doi: 10.3934/cpaa.2021122
[16]

Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic and Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009
[17]

Alexander V. Bobylev, Marzia Bisi, Maria Groppi, Giampiero Spiga, Irina F. Potapenko. A general consistent BGK model for gas mixtures. Kinetic and Related Models, 2018, 11 (6) : 1377-1393. doi: 10.3934/krm.2018054
[18]

Byung-Hoon Hwang, Ho Lee, Seok-Bae Yun. Relativistic BGK model for massless particles in the FLRW spacetime. Kinetic and Related Models, 2021, 14 (6) : 949-959. doi: 10.3934/krm.2021031
[19]

Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks and Heterogeneous Media, 2021, 16 (1) : 69-90. doi: 10.3934/nhm.2020034
[20]

Emiliano Cristiani, Elisa Iacomini. An interface-free multi-scale multi-order model for traffic flow. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6189-6207. doi: 10.3934/dcdsb.2019135

2021 Impact Factor: 1.398

Tools

Metrics

  • PDF downloads (163)
  • HTML views (68)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy