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 & 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
[1]

Marzia Bisi, Giampiero Spiga. On a kinetic BGK model for slow chemical reactions. Kinetic & Related Models, 2011, 4 (1) : 153-167. doi: 10.3934/krm.2011.4.153
[2]

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

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

Charles Nguyen, Stephen Pankavich. A one-dimensional kinetic model of plasma dynamics with a transport field. Evolution Equations & Control Theory, 2014, 3 (4) : 681-698. doi: 10.3934/eect.2014.3.681
[5]

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

Franz Achleitner, Anton Arnold, Eric A. Carlen. On multi-dimensional hypocoercive BGK models. Kinetic & Related Models, 2018, 11 (4) : 953-1009. doi: 10.3934/krm.2018038
[7]

Silvia Caprino, Carlo Marchioro. On the plasma-charge model. Kinetic & Related Models, 2010, 3 (2) : 241-254. doi: 10.3934/krm.2010.3.241
[8]

Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic & Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006
[9]

Alexander V. Bobylev, Marzia Bisi, Maria Groppi, Giampiero Spiga, Irina F. Potapenko. A general consistent BGK model for gas mixtures. Kinetic & Related Models, 2018, 11 (6) : 1377-1393. doi: 10.3934/krm.2018054
[10]

Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009
[11]

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

Emiliano Cristiani, Elisa Iacomini. An interface-free multi-scale multi-order model for traffic flow. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6189-6207. doi: 10.3934/dcdsb.2019135
[13]

George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557
[14]

K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591
[15]

Byung-Hoon Hwang, Seok-Bae Yun. Stationary solutions to the boundary value problem for the relativistic BGK model in a slab. Kinetic & Related Models, 2019, 12 (4) : 749-764. doi: 10.3934/krm.2019029
[16]

Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012
[17]

Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541
[18]

Eduard Feireisl. On weak solutions to a diffuse interface model of a binary mixture of compressible fluids. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 173-183. doi: 10.3934/dcdss.2016.9.173
[19]

Shi Yan, Jun Liu, Haiyang Huang, Xue-Cheng Tai. A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation. Inverse Problems & Imaging, 2019, 13 (3) : 653-677. doi: 10.3934/ipi.2019030
[20]

Helge Holden, Nils Henrik Risebro. Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Networks & Heterogeneous Media, 2018, 13 (3) : 409-421. doi: 10.3934/nhm.2018018

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (20)
  • HTML views (5)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences