# American Institute of Mathematical Sciences

September  2011, 4(3): 669-700. doi: 10.3934/krm.2011.4.669

## Non equilibrium ionization in magnetized two-temperature thermal plasma

 1 Department of Engineering Science, University West Gustava Melins gata 2, 461 39 Trollhättan, Sweden 2 Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Université Pierre et Marie Curie B.C. 187, 4 Place Jussieu 75252 Paris Cedex 05, France

Received  October 2010 Revised  February 2011 Published  August 2011

A thermal plasma is studied accounting for both impact ionization, and an electromagnetic field. This plasma problem is modeled based on a system of Boltzmann type transport equations. Electron-neutral collisions are assumed to be much more frequently elastic than inelastic, to complete previous investigations of thermal plasma [4]-[6]. A viscous hydrodynamic/diffusion limit is derived in two stages doing an Hilbert expansion and using the Chapman-Enskog method. The resultant viscous fluid model is characterized by two temperatures, and non equilibrium ionization. Its diffusion coefficients depend on the magnetic field, and can be computed explicitely.
Citation: Isabelle Choquet, Brigitte Lucquin-Desreux. Non equilibrium ionization in magnetized two-temperature thermal plasma. Kinetic & Related Models, 2011, 4 (3) : 669-700. doi: 10.3934/krm.2011.4.669
##### References:
 [1] N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors,, J. Math. Phys., 37 (1996), 3306. doi: 10.1063/1.531567. Google Scholar [2] C. Cercignani, "The Boltzmann Equation and its Applications,", Applied Mathematical Sciences, 67 (1988). Google Scholar [3] S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,", Cambridge Mathematical Library, (1958). Google Scholar [4] I. Choquet and B. Lucquin-Desreux, Hydrodynamic limit for an arc discharge at atmospheric pressure,, J. of Stat. Phys., 119 (2005), 197. doi: 10.1007/s10955-004-2711-8. Google Scholar [5] I. Choquet, P. Degond and B. Lucquin-Desreux, A hierarchy of diffusion models for partially ionized plasmas,, Discrete and Continuous Dynamical Systems Series B, 8 (2007), 735. doi: 10.3934/dcdsb.2007.8.735. Google Scholar [6] I. Choquet, P. Degond and B. Lucquin-Desreux, A strong ionization model in plasma physics,, Mathematical and Computer Modelling, 49 (2009), 88. doi: 10.1016/j.mcm.2007.06.035. Google Scholar [7] P. Degond and B. Lucquin-Desreux, The asymptotics of collision operators for two species of particles of disparate masses,, Math. Models and Methods in the Appl. Science, 6 (1996), 405. doi: 10.1142/S0218202596000158. Google Scholar [8] P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases,, Transport Theory and Statistical Physics, 26 (1996), 595. doi: 10.1080/00411459608222915. Google Scholar [9] P. Degond, A. Nouri and C. Schmeiser, Macroscopic models for ionization in the presence of strong electric fields, Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998),, Transport Theory and Statistical Physics, 29 (2000), 551. doi: 10.1080/00411450008205891. Google Scholar [10] A. Fridman and L. A. Kennedy, "Plasma Physics and Engineering,", Taylor and Francis Group, (2004). Google Scholar [11] S. Ghorui, J. V. R. Heberlein and E. Pfender, Non-equilibrium modelling of oxygen-plasma cutting torch,, Journal of Pysics D: Applied Physics, 40 (2007), 1966. doi: 10.1088/0022-3727/40/7/020. Google Scholar [12] A. Gleizes, B. Chervy and J. J. Gonzales, Calculation of a two-temperature plasma composition: bases and application to SF$_6$,, J. Phys. D: Appl. Phys, 32 (1999), 2060. doi: 10.1088/0022-3727/32/16/315. Google Scholar [13] E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws,", Applied Mathematical Sciences, 118 (1996). Google Scholar [14] J. J. Gonzales, R. Girard and A. Gleizes, Decay and post-arc phases of a SF$_6$ arc plasma: a thermal and chemical non-equilibrium model,, Journal of Physics D: Applied Physics, 33 (2000), 2759. doi: 10.1088/0022-3727/33/21/314. Google Scholar [15] R. G. Jahn, "Physics of Electric Propulsion,", McGraw-Hill, (1968). Google Scholar [16] L. D. Landau, E. M. Lifschitz and L. P. Pitaevskii, "Course of Theoretical Physics, Vol 10: Physical Kinetics,", Pergamon Press, (1981). Google Scholar [17] B. Lucquin-Desreux, Fluid limit for magnetized plasmas,, Transp. Theory in Stat. Phys., 27 (1998), 99. doi: 10.1080/00411459808205811. Google Scholar [18] B. Lucquin-Desreux, Diffusion of electrons by multicharged ions,, Mathematical Models and Methods in Applied Sciences, 10 (2000), 409. Google Scholar [19] A. B. Murphy, Diffusion in equilibrium mixtures of ionized gases,, Physical Review E, 48 (1993), 3594. doi: 10.1103/PhysRevE.48.3594. Google Scholar [20] L. Onsager, Reciprocal relations in irreversible processes,, Physical Review, 38 (1931), 2265. doi: 10.1103/PhysRev.38.2265. Google Scholar [21] A. V. Potapov, Chemical equilibrium of multitemperature systems,, High Temp., 4 (1966), 55. Google Scholar [22] Y. P. Raizer, "Gas Discharge Physics,", Springer-Verlag, (1991). Google Scholar [23] J. D. Ramshaw and C. H. Chang, Multicomponent diffusion in two-temperature magnetohydrodynamics,, Physical Review E, 53 (1996), 6382. doi: 10.1103/PhysRevE.53.6382. Google Scholar [24] V. Rat, P. André, J. Aubreton, M.-F. Elchinger, P. Fauchais and A. Lefort, Transport properties in a two-temperature plasma: Theory and applications,, Physical Review E, 64 (2001). Google Scholar [25] S. E. Selezneva and M. I. Boulos, Supersonic induction plasma jet modeling,, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 180 (2009), 306. doi: 10.1016/S0168-583X(01)00436-0. Google Scholar [26] Y. Tanaka, T. Michishita and Y. Uesugi, Hydrodynamic chemical non-equilibrium model of a pulsed arc discharge in dry air at atmospheric pressure,, Plasma Sources Science Technology, 14 (2005), 134. doi: 10.1088/0963-0252/14/1/016. Google Scholar [27] J. P. Trelles, J. V. R. Heberlein and E. Pfender, Non-equilibrium modelling of arc plasma torches,, Journal of Pysics D: Applied Physics, 40 (2007), 5937. doi: 10.1088/0022-3727/40/19/024. Google Scholar [28] S. Vacquié, L'arc électrique,, in Eyrolles collection, (2000). Google Scholar [29] M. C. M. van de Sanden, P. P. J. M. Schram, A. G. Peeters, J. A. M. van der Mullen and G. M. W. Kroesen, Thermodynamic generalization of the Saha equation for a two-temperature plasma,, Physical Review A, 40 (1989), 5273. doi: 10.1103/PhysRevA.40.5273. Google Scholar [30] J. Wendelstorf, "Ab Initio Modelling of Thermal Plasma Gas Discharges (Electric Arcs),", Ph.D. thesis, (2000). Google Scholar [31] S. Xue, P. Proulx and M. I. Boulos, Turbulence modeling of inductively coupled plasma flows,, in, (2003), 993. Google Scholar

show all references

##### References:
 [1] N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors,, J. Math. Phys., 37 (1996), 3306. doi: 10.1063/1.531567. Google Scholar [2] C. Cercignani, "The Boltzmann Equation and its Applications,", Applied Mathematical Sciences, 67 (1988). Google Scholar [3] S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,", Cambridge Mathematical Library, (1958). Google Scholar [4] I. Choquet and B. Lucquin-Desreux, Hydrodynamic limit for an arc discharge at atmospheric pressure,, J. of Stat. Phys., 119 (2005), 197. doi: 10.1007/s10955-004-2711-8. Google Scholar [5] I. Choquet, P. Degond and B. Lucquin-Desreux, A hierarchy of diffusion models for partially ionized plasmas,, Discrete and Continuous Dynamical Systems Series B, 8 (2007), 735. doi: 10.3934/dcdsb.2007.8.735. Google Scholar [6] I. Choquet, P. Degond and B. Lucquin-Desreux, A strong ionization model in plasma physics,, Mathematical and Computer Modelling, 49 (2009), 88. doi: 10.1016/j.mcm.2007.06.035. Google Scholar [7] P. Degond and B. Lucquin-Desreux, The asymptotics of collision operators for two species of particles of disparate masses,, Math. Models and Methods in the Appl. Science, 6 (1996), 405. doi: 10.1142/S0218202596000158. Google Scholar [8] P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases,, Transport Theory and Statistical Physics, 26 (1996), 595. doi: 10.1080/00411459608222915. Google Scholar [9] P. Degond, A. Nouri and C. Schmeiser, Macroscopic models for ionization in the presence of strong electric fields, Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998),, Transport Theory and Statistical Physics, 29 (2000), 551. doi: 10.1080/00411450008205891. Google Scholar [10] A. Fridman and L. A. Kennedy, "Plasma Physics and Engineering,", Taylor and Francis Group, (2004). Google Scholar [11] S. Ghorui, J. V. R. Heberlein and E. Pfender, Non-equilibrium modelling of oxygen-plasma cutting torch,, Journal of Pysics D: Applied Physics, 40 (2007), 1966. doi: 10.1088/0022-3727/40/7/020. Google Scholar [12] A. Gleizes, B. Chervy and J. J. Gonzales, Calculation of a two-temperature plasma composition: bases and application to SF$_6$,, J. Phys. D: Appl. Phys, 32 (1999), 2060. doi: 10.1088/0022-3727/32/16/315. Google Scholar [13] E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws,", Applied Mathematical Sciences, 118 (1996). Google Scholar [14] J. J. Gonzales, R. Girard and A. Gleizes, Decay and post-arc phases of a SF$_6$ arc plasma: a thermal and chemical non-equilibrium model,, Journal of Physics D: Applied Physics, 33 (2000), 2759. doi: 10.1088/0022-3727/33/21/314. Google Scholar [15] R. G. Jahn, "Physics of Electric Propulsion,", McGraw-Hill, (1968). Google Scholar [16] L. D. Landau, E. M. Lifschitz and L. P. Pitaevskii, "Course of Theoretical Physics, Vol 10: Physical Kinetics,", Pergamon Press, (1981). Google Scholar [17] B. Lucquin-Desreux, Fluid limit for magnetized plasmas,, Transp. Theory in Stat. Phys., 27 (1998), 99. doi: 10.1080/00411459808205811. Google Scholar [18] B. Lucquin-Desreux, Diffusion of electrons by multicharged ions,, Mathematical Models and Methods in Applied Sciences, 10 (2000), 409. Google Scholar [19] A. B. Murphy, Diffusion in equilibrium mixtures of ionized gases,, Physical Review E, 48 (1993), 3594. doi: 10.1103/PhysRevE.48.3594. Google Scholar [20] L. Onsager, Reciprocal relations in irreversible processes,, Physical Review, 38 (1931), 2265. doi: 10.1103/PhysRev.38.2265. Google Scholar [21] A. V. Potapov, Chemical equilibrium of multitemperature systems,, High Temp., 4 (1966), 55. Google Scholar [22] Y. P. Raizer, "Gas Discharge Physics,", Springer-Verlag, (1991). Google Scholar [23] J. D. Ramshaw and C. H. Chang, Multicomponent diffusion in two-temperature magnetohydrodynamics,, Physical Review E, 53 (1996), 6382. doi: 10.1103/PhysRevE.53.6382. Google Scholar [24] V. Rat, P. André, J. Aubreton, M.-F. Elchinger, P. Fauchais and A. Lefort, Transport properties in a two-temperature plasma: Theory and applications,, Physical Review E, 64 (2001). Google Scholar [25] S. E. Selezneva and M. I. Boulos, Supersonic induction plasma jet modeling,, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 180 (2009), 306. doi: 10.1016/S0168-583X(01)00436-0. Google Scholar [26] Y. Tanaka, T. Michishita and Y. Uesugi, Hydrodynamic chemical non-equilibrium model of a pulsed arc discharge in dry air at atmospheric pressure,, Plasma Sources Science Technology, 14 (2005), 134. doi: 10.1088/0963-0252/14/1/016. Google Scholar [27] J. P. Trelles, J. V. R. Heberlein and E. Pfender, Non-equilibrium modelling of arc plasma torches,, Journal of Pysics D: Applied Physics, 40 (2007), 5937. doi: 10.1088/0022-3727/40/19/024. Google Scholar [28] S. Vacquié, L'arc électrique,, in Eyrolles collection, (2000). Google Scholar [29] M. C. M. van de Sanden, P. P. J. M. Schram, A. G. Peeters, J. A. M. van der Mullen and G. M. W. Kroesen, Thermodynamic generalization of the Saha equation for a two-temperature plasma,, Physical Review A, 40 (1989), 5273. doi: 10.1103/PhysRevA.40.5273. Google Scholar [30] J. Wendelstorf, "Ab Initio Modelling of Thermal Plasma Gas Discharges (Electric Arcs),", Ph.D. thesis, (2000). Google Scholar [31] S. Xue, P. Proulx and M. I. Boulos, Turbulence modeling of inductively coupled plasma flows,, in, (2003), 993. Google Scholar
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