Volume 4, Issue 1, March 2018, Page: 27-37
Travelling Waves Solution of the Unsteady Flow Problem of a Collisional Plasma Bounded by a Moving Plate
Taha Zakaraia Abdel Wahid, Department of Basic Sciences, El-Gezeera High Institute for Engineering and Technology, Cairo, Egypt; Mathematic Department, Faculty of Science, Menofia University, Shebin El-Kom, Egypt
Received: Sep. 21, 2017;       Accepted: Oct. 26, 2017;       Published: Mar. 15, 2018
DOI: 10.11648/j.fm.20180401.14      View  1538      Downloads  57
Abstract
The extension of the previous paper [Can. J. Phys. Vol. 88, (2010), 501–511] has been made. Therefore, the effect of the neutral atoms collisions with electrons and with positive ions is taken into consideration, which was ignored, for the sake of simplicity, in the earlier work. Thus, we will have multi-collision terms (electron–electron, electron–ion, electron– neutral) instead of one term, as was studied before for the sake of facilitation. These collision terms are needed to obtain the real physical situation. The new procedures will increase the ability of the research applications. This study is based on the solution of the BGK (Bhatnager–Gross–Krook) model of the nonlinear partial differential Boltzmann equations coupled with Maxwell’s partial differential equations. The initial-boundary value problem of the Rayleigh flow problem applied to the system of the plasma (positive ions + electrons+ neutral atoms), bounded by a moving plate, is solved. For this purpose, the traveling wave solution method is used to get the exact solution of the nonlinear partial differential equations system. The ratios between the different contributions of the internal energy changes are predicted via the extended Gibbs equation for both dia-magnetic and para-magnetic plasma. The results are applied to a typical model of laboratory argon plasma. 3D-Graphics illustrating the calculated variables are drawn to predict their behavior and the results are discussed.
Keywords
Rayleigh Flow Problem, Charged Gas, Boltzmann Equation, Maxwell Equations, Exact Solution, Boltzmann H-Theorem, Internal Energy, Extended Gibbs Formula
To cite this article
Taha Zakaraia Abdel Wahid, Travelling Waves Solution of the Unsteady Flow Problem of a Collisional Plasma Bounded by a Moving Plate, Fluid Mechanics. Vol. 4, No. 1, 2018, pp. 27-37. doi: 10.11648/j.fm.20180401.14
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
J. K. Singh, Naveen Joshi and S. Ghousia Begum” Unsteady Magnetohydrodynamic Couette-Poiseuille Flow within Porous Plates Filled with Porous Medium in the Presence of a Moving Magnetic Field with Hall and Ion-slip Effects” International Journal Of Heat And Technology, Vol. 34, No. 1, March, 2016, pp. 89-97.
[2]
Griener, M., Cavedon, M., Eich, T., Fuchert, G., Herrmann, A., Lunt, T., et al. “ Fast piezoelectric valve offering controlled gas injection in magnetically confined fusion plasmas for diagnostic and fuelling purposes.” Rev Sci Instrum. 2017 Mar; 88 (3):033509. doi: 10.1063/1.4978629.
[3]
Gérard Belmont, Roland Grappin, Fabrice Mottez, Filippo Pantellini, Guy Pelletier” Collisionless Plasmas in Astrophysics.” WILEY-VCHVerlag GmbH& Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany, 2014.
[4]
V. A. Godyak "Electron Energy Distribution Function Control In Gas Discharge Plasmas" Physics Of Plasmas 20, 101611 (1-17), (2013); doi: 10.1063/1.4823075.
[5]
N. J. Mason, "Atomic and Molecular Collision Data for Plasma Modeling", 20Th ESCAMPIG, July 2010, Novi Sad, Serbia.
[6]
A. M. Abourabia and T. Z. Abdel Wahid " The unsteady Boltzmann kinetic Equation and non-equilibrium thermodynamics of an electron gas for the Rayleigh flow problem "Can. J. Phys. vol 88, (2010), 501–511.
[7]
V. P. Shidlovskiy. Introduction to dynamics of rarefied gases. Elsevier, N. Y. (1967), 78–85.
[8]
V. M. Zhdanov and V. I. Roldughin. Phys.-Uspekhi, 41, (1998), 349. doi:10.1070/PU1998v041n04ABEH000383.
[9]
C. Cercignani. The Boltzmann Equation and its application. Springer, New York, USA. 1988.
[10]
G. Lebon, D. Jou, and J. Casas-Va`zquez. Understanding non-equilibrium thermodynamics: foundations, applications, frontiers. Springer-Verlag, Berlin, Heidelberg, Germany. (2008).
[11]
Taha Zakaraia Abdel Wahid " Kinetic and Irreversible Thermodynamic study of Plasma and Neutral Gases.", LAMBERT Academic Publishing, Germany.,(2014). ISBN: 978-3-659-62296-0.
[12]
A. M. Abourabia and Taha Zakaraia Abdel Wahid " Solution of The Krook Kinetic Equation Model and Non-Equilibrium Thermodynamics of a Rarefied Gas Affected by a Nonlinear Thermal Radiation Field", J. Non-Equilibrium Thermodynamic, 36 (2011), 75–98.
[13]
A. M. Abourabia and Taha Zakaraia Abdel Wahid " Kinetic and thermodynamic treatment for the Rayleigh flow problem of an inhomogeneous charged gas mixture", J. Non-Equilibrium Thermodynamic, 37, 1, (2012), 1–25.
[14]
A. M. Abourabia and Taha Zakaraia Abdel Wahid " Kinetic and thermodynamic treatments of a neutral binary gas mixture affected by a nonlinear thermal radiation field." Can. J. Phys. 90: 137–149 (2012).
[15]
Taha Zakaraia Abdel Wahid " Kinetic and thermodynamic treatment for the exact solution of the unsteady Rayleigh flow problem of a rarefied homogeneous charged gas." J. Non-Equilibrium Thermodynamic, 37, 2, (2012), 119–141.
[16]
Taha Zakaraia Abdel Wahid and S. K. Elagan " Kinetic treatment for the exact solution of the unsteady Rayleigh flow problem of a rarefied homogeneous charged gas bounded by an oscillating plate." Can. J. Phys. 90: 987–998 (2012).
[17]
Taha Zakaraia Abdel Wahid " Exact solution of the unsteady Krook kinetic model and nonequilibrium thermodynamic study for a rarefied gas affected by a nonlinear thermal radiation field. ", Canadian Journal of Physics (2013); 91(3):201-210.
[18]
Taha Zakaraia Abdel Wahid " Travelling Waves Solution of the Unsteady Flow Problem of a Rarefied Non-Homogeneous Charged Gas Bounded by an Oscillating Plate." Mathematical Problems in Engineering 2013; 2013(ID 503729):1-13.
[19]
Taha Zakaraia Abdel Wahid " Travelling Wave Solution of the Unsteady BGK Model for a Rarefied Gas Affected by a Thermal Radiation Field.", Sohag Journal of Mathematics 2, No. 2, 75-87 (2015).
[20]
Taha Zakaraia Abdel Wahid "Travelling waves solution of the unsteady problem of binary gas mixture affected by a nonlinear thermal radiation field. ", American Journal of Physics and Applications. Vol. 2, No. 6, 2014, pp. 121-134. (doi: 10.11648/j.ajpa.20140206.13).
[21]
D. D. Millar " Ion Currents, Ion-Neutral Collisions And Plasma Transport Phenomena " Aust. J. Phys.,29, (1976), 249-261.
[22]
E. M. Purcell. Electricity and magnetism. 3rd ed. McGraw-Hill Book Co., Singapore. 1965.
[23]
J. D. Huba. NRL plasma formulary. Navel Research Laboratory, Washington, D. C. (20011).
[24]
S. I. Braginskii " Transport processes in a plasma " Reviews of Plasma Physics, Volume 1. Authorized translation from the Russian by Herbert Lashinsky, University of Maryland, USA. Edited by M. A. Leontovich. Published by Consultants Bureau, New York, (1965), p.205.
[25]
L. Lees. J. Soc. Ind. Appl. Math. 13, (1965), 278. doi:10.1137/0113017.
[26]
H. Grad. Commun. Pure Appl. Math. 2, (1949), 331. doi:10.1002/cpa.3160020403.
[27]
Isabelle Choquet, Pierre Degond, Brigitte Lucquin-Desreux " A strong ionization model in plasma physics" Mathematical and Computer Modelling 49 (2009) 88-113.
[28]
J. Gratton, S. M. Mahajan, and F. Minotti "Non Newtonian Gravity Creeping Flow" International Centre for Theoretical Physics, Trieste (Italy), (1988), 1-17.
[29]
G. Nugroho, A. M. S. Ali, and Z. A. Abdul Karim “Towards a new simple analytical formulation of Navier-Stokes Equations” World Acd. of Sci., Eng. And Tec.,(2009).
[30]
C. O. Bennett and J. E. Myers. Moment, heat and mass transfer. 3rd ed. McGraw-Hill Book Co., Singapore. (1988).
[31]
I. H. Hutchinson “Introduction to Plasma Physics” Electronic book available on net [http://silas.psfc.mit.edu/introplasma/index.html ](2001).
[32]
P. Van der Linde. Periodica Polytechnica Ser. Chem. Eng., 12, (1998), 97.
[33]
S. Yonemura and K. Nanbu. Jpn. J. Physiol. 40, (2001), 7052. doi:10.1143/JJAP.40.7052.
[34]
Taha Zakaraia Abdel Wahid “The Effect of Lorentz and Centrifugal Forces on Gases and Plasma.” LAMBERT Academic Publishing, Germany, September 2017, ISBN: 978-620-2-05504-8.
[35]
A. G. Sitenko. Electromagnetic fluctuation in plasma. Academic Press, New York. (1967).
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