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 and 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), 3306-3333. doi: 10.1063/1.531567.

[2]

C. Cercignani, "The Boltzmann Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988.

[3]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," Cambridge Mathematical Library, Cambridge, 1958.

[4]

I. Choquet and B. Lucquin-Desreux, Hydrodynamic limit for an arc discharge at atmospheric pressure, J. of Stat. Phys., 119 (2005), 197-239. doi: 10.1007/s10955-004-2711-8.

[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), 735-772. doi: 10.3934/dcdsb.2007.8.735.

[6]

I. Choquet, P. Degond and B. Lucquin-Desreux, A strong ionization model in plasma physics, Mathematical and Computer Modelling, 49 (2009), 88-113. doi: 10.1016/j.mcm.2007.06.035.

[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), 405-436. doi: 10.1142/S0218202596000158.

[8]

P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases, Transport Theory and Statistical Physics, 26 (1996), 595-633. doi: 10.1080/00411459608222915.

[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), 551-561. doi: 10.1080/00411450008205891.

[10]

A. Fridman and L. A. Kennedy, "Plasma Physics and Engineering," Taylor and Francis Group, 2004.

[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), 1966-1976. doi: 10.1088/0022-3727/40/7/020.

[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), 2060-2067. doi: 10.1088/0022-3727/32/16/315.

[13]

E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws," Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996.

[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), 2759-2768. doi: 10.1088/0022-3727/33/21/314.

[15]

R. G. Jahn, "Physics of Electric Propulsion," McGraw-Hill, 1968.

[16]

L. D. Landau, E. M. Lifschitz and L. P. Pitaevskii, "Course of Theoretical Physics, Vol 10: Physical Kinetics," Pergamon Press, 1981.

[17]

B. Lucquin-Desreux, Fluid limit for magnetized plasmas, Transp. Theory in Stat. Phys., 27 (1998), 99-135. doi: 10.1080/00411459808205811.

[18]

B. Lucquin-Desreux, Diffusion of electrons by multicharged ions, Mathematical Models and Methods in Applied Sciences, 10 (2000), 409-440.

[19]

A. B. Murphy, Diffusion in equilibrium mixtures of ionized gases, Physical Review E, 48 (1993), 3594-3602. doi: 10.1103/PhysRevE.48.3594.

[20]

L. Onsager, Reciprocal relations in irreversible processes, Physical Review, 38 (1931), 2265-2279. doi: 10.1103/PhysRev.38.2265.

[21]

A. V. Potapov, Chemical equilibrium of multitemperature systems, High Temp., 4 (1966), 55-58.

[22]

Y. P. Raizer, "Gas Discharge Physics," Springer-Verlag, 1991.

[23]

J. D. Ramshaw and C. H. Chang, Multicomponent diffusion in two-temperature magnetohydrodynamics, Physical Review E, 53 (1996), 6382-6388. doi: 10.1103/PhysRevE.53.6382.

[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), 026409 (20p).

[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), 306-311. doi: 10.1016/S0168-583X(01)00436-0.

[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), 134-151. doi: 10.1088/0963-0252/14/1/016.

[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), 5937-5952. doi: 10.1088/0022-3727/40/19/024.

[28]

S. Vacquié, L'arc électrique, in Eyrolles collection "Sciences et Technique de l'Ingénieur," CNRS Editions, 2000.

[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), 5273-276. doi: 10.1103/PhysRevA.40.5273.

[30]

J. Wendelstorf, "Ab Initio Modelling of Thermal Plasma Gas Discharges (Electric Arcs)," Ph.D. thesis, Braunschweig University, Germany, 2000.

[31]

S. Xue, P. Proulx and M. I. Boulos, Turbulence modeling of inductively coupled plasma flows, in "Thermal Spray 2003: Advancing the Science and Applying the Technology" (Ed. C. Moreau and B. Marple), ASM Int., (2003), 993-999.

show all references

References:
[1]

N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306-3333. doi: 10.1063/1.531567.

[2]

C. Cercignani, "The Boltzmann Equation and its Applications," Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988.

[3]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," Cambridge Mathematical Library, Cambridge, 1958.

[4]

I. Choquet and B. Lucquin-Desreux, Hydrodynamic limit for an arc discharge at atmospheric pressure, J. of Stat. Phys., 119 (2005), 197-239. doi: 10.1007/s10955-004-2711-8.

[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), 735-772. doi: 10.3934/dcdsb.2007.8.735.

[6]

I. Choquet, P. Degond and B. Lucquin-Desreux, A strong ionization model in plasma physics, Mathematical and Computer Modelling, 49 (2009), 88-113. doi: 10.1016/j.mcm.2007.06.035.

[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), 405-436. doi: 10.1142/S0218202596000158.

[8]

P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases, Transport Theory and Statistical Physics, 26 (1996), 595-633. doi: 10.1080/00411459608222915.

[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), 551-561. doi: 10.1080/00411450008205891.

[10]

A. Fridman and L. A. Kennedy, "Plasma Physics and Engineering," Taylor and Francis Group, 2004.

[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), 1966-1976. doi: 10.1088/0022-3727/40/7/020.

[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), 2060-2067. doi: 10.1088/0022-3727/32/16/315.

[13]

E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws," Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996.

[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), 2759-2768. doi: 10.1088/0022-3727/33/21/314.

[15]

R. G. Jahn, "Physics of Electric Propulsion," McGraw-Hill, 1968.

[16]

L. D. Landau, E. M. Lifschitz and L. P. Pitaevskii, "Course of Theoretical Physics, Vol 10: Physical Kinetics," Pergamon Press, 1981.

[17]

B. Lucquin-Desreux, Fluid limit for magnetized plasmas, Transp. Theory in Stat. Phys., 27 (1998), 99-135. doi: 10.1080/00411459808205811.

[18]

B. Lucquin-Desreux, Diffusion of electrons by multicharged ions, Mathematical Models and Methods in Applied Sciences, 10 (2000), 409-440.

[19]

A. B. Murphy, Diffusion in equilibrium mixtures of ionized gases, Physical Review E, 48 (1993), 3594-3602. doi: 10.1103/PhysRevE.48.3594.

[20]

L. Onsager, Reciprocal relations in irreversible processes, Physical Review, 38 (1931), 2265-2279. doi: 10.1103/PhysRev.38.2265.

[21]

A. V. Potapov, Chemical equilibrium of multitemperature systems, High Temp., 4 (1966), 55-58.

[22]

Y. P. Raizer, "Gas Discharge Physics," Springer-Verlag, 1991.

[23]

J. D. Ramshaw and C. H. Chang, Multicomponent diffusion in two-temperature magnetohydrodynamics, Physical Review E, 53 (1996), 6382-6388. doi: 10.1103/PhysRevE.53.6382.

[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), 026409 (20p).

[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), 306-311. doi: 10.1016/S0168-583X(01)00436-0.

[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), 134-151. doi: 10.1088/0963-0252/14/1/016.

[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), 5937-5952. doi: 10.1088/0022-3727/40/19/024.

[28]

S. Vacquié, L'arc électrique, in Eyrolles collection "Sciences et Technique de l'Ingénieur," CNRS Editions, 2000.

[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), 5273-276. doi: 10.1103/PhysRevA.40.5273.

[30]

J. Wendelstorf, "Ab Initio Modelling of Thermal Plasma Gas Discharges (Electric Arcs)," Ph.D. thesis, Braunschweig University, Germany, 2000.

[31]

S. Xue, P. Proulx and M. I. Boulos, Turbulence modeling of inductively coupled plasma flows, in "Thermal Spray 2003: Advancing the Science and Applying the Technology" (Ed. C. Moreau and B. Marple), ASM Int., (2003), 993-999.

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