Travelling Waves Solution of the Unsteady Flow Problem of a Rarefied Nonhomogeneous Charged Gas Bounded by an Oscillating Plate

Abstract

The extension of the previous paper (Abdel Wahid and Elagan, 2012) has been made for a nonhomogeneous charged rarefied gas mixture (two-component plasma) instead of a single electron gas. Therefore, the effect of the positive ion collisions with electrons and with each other is taken into consideration, which was ignored, as an approximation, in the earlier work. Thus, we will have four collision terms (electron-electron, electron-ion, ion-ion, and ion-electron) instead of one term, as was studied before. These collision terms are added together with a completely additional system of differential equations for ions. This study is based on the solution of the Bhatnager-Gross-Krook (BGK) model of the Boltzmann kinetic equation coupled with Maxwell’s equations. The initial-boundary value problem of the Rayleigh flow problem applied to the system of the two-component plasma (positive ions + electrons), bounded by an oscillating plate, is solved. This situation, for the best of my knowledge, is presented from the molecular viewpoint for the first time. For this purpose, the traveling wave solution method is used to get the exact solution of the nonlinear partial differential equations. In addition, the accurate formula of the whole four-collision frequency terms is presented. The distinction and comparisons between the perturbed and the equilibrium velocity distribution functions are illustrated. Definitely, the equilibrium time for electrons and for ions is calculated. The relation between those times and the relaxation time is deduced for both species of the mixture. The ratios between the different contributions of the internal energy changes are predicted via the extended Gibbs equation for both diamagnetic and paramagnetic plasmas. The results are applied to a typical model of laboratory argon plasma.

1. Introduction

A development of the previous paper [1] is introduced. The nonstationary Krook kinetic equation model for a rarefied charged binary gas mixture is solved, instead of the single gas. Analytically, the Bhatnager-Gross-Krook (BGK) model of the kinetic equation is applied. The travelling wave solution method is used to get the exact solution of the nonlinear partial differential equations. These equations were produced from applying the moment method to the unsteady BGK equation. Now we should solve eight nonlinear partial differential equations, which represent an arduous task. This situation, for the best of our knowledge, is presented in this paper for the first time. The unsteady solution for a binary charged gas mixture (two-component plasma) gives the problem a great generality and more applications. Taking into consideration the effect of the positive ions on the behavior of electrons, which was not observed in the previous work [1], this effect becomes a pressing mater in some physical situation; see, for example, [2]. The problem is investigated, with the new circumstance, to follow the behavior of the macroscopic properties of the gas such as the mean velocity, the shear stress, and the viscosity coefficient. The important quantities together with both the induced electric and magnetic fields are investigated for the two-component mixture, with respect to both distance and time. This new study is done to examine the behavior of electrons and positive ions in the microscale. This is performed using the kinetic Boltzmann equation coupled with Maxwell’s equation. An important novel thermodynamic treatment of the system is concluded. The behavior of the nonequilibrium thermodynamic distribution functions for both positive ions and electrons, in different cases, is illustrated. Such treatment holds for the first time for a nonhomogeneous charged gas mixture bounded by oscillating plate. The calculated velocities are substituted into the corresponding two-stream Maxwellian distribution functions permitting us to investigate the nonequilibrium thermodynamic properties of the system (gas particles + the oscillating plate). The entropy, entropy flux, entropy production, thermodynamic forces, and kinetic coefficients are obtained. The verification of the Boltzmann H-theorem, Le Chatelier principle, and the second law of thermodynamic is examined. The ratios between the different contributions of the internal energy change based upon the total derivatives of the extensive parameters estimated via the Gibbs formula.

The behavior of charged gases in nonequilibrium states has received considerable attention from the standpoint of understanding the characteristics of nonequilibrium phenomena [3]. The kinetic theory has contributed not only to the understanding of nonequilibrium transport phenomena in gases, but also to the development of general nonequilibrium statistical physics. It is well accepted that the Boltzmann equation [3–8] is one of the most reliable kinetic models for describing nonequilibrium phenomena in gas phase. Following its success and usefulness, The Boltzmann equation is widely used in order to describe various gas-phase transport phenomena such as plasma gases, granular gases, polyatomic gases, relativistic gases, thermal radiation affected by gases, and chemically reacting gases [9–15]. The kinetic equation of gas flow based on the Boltzmann equation has obvious peculiarities in comparison with the macroscopic description found by using the Navier-Stokes equations; see [4, 13].

For planar flows, the oscillating Couette problem has many analogies with Stokes’ second problem, which involves a flat plate oscillating in an unbounded medium. Stokes [16] first solved this problem in 1851 and later it was solved by Rayleigh [17] in 1911. Stokes’ second problem is one of the simplest configurations to study the behavior of a nonequilibrium gas responding to a plate oscillating in its own plane [18]. A comprehensive study has been carried out with the linearized Bhatnagar-Gross-Krook (BGK) equation over a wide range of Knudsen numbers [19]. The effect of the oscillation frequency on the amplitudes and phases of the velocity and shear stress was reported. In order to improve our understanding of time-periodic, shear driven gas flows, oscillatory Couette flow provides an ideal test case. It has been studied extensively using kinetic theory (Sharipov and Kalempa [20]; Park et al. [21]; Hadjiconstantinou [22]; Emerson et al. [23]; Doi [24]). A brief comparison of the linearized R13 (LR13) equations with DSMC data (Hadjiconstantinou [22]) was carried out by Taheri et al. [25]. Although oscillatory Couette flow is a simple case, it has many related applications in a variety of MEMS devices, for example, the Tang resonator (Tang et al. [26]). A proper understanding of flow phenomena in resonators will therefore help to improve the performance and quality factor (Frangi et al. [27]). In this study, the extended continuum governing equations are employed to compute the details of Stokes’ second problem and oscillatory planar Couette flow in the early transition regime. The purpose of the paper is to analyze and assess the dynamic response capability of the higher-order moment equations for oscillatory flow and compare against the available kinetic data. Gu and Emerson [18] presented results using three different continuum-based models to study oscillatory flow in the transition regime. Data obtained from numerical solutions of the Boltzmann equation and the direct simulation Monte Carlo method are used to assess the ability of the continuum models to capture important rarefaction effects.

The objective of this paper is to seek the unsteady exact solution of the Boltzmann kinetic equation of an inhomogeneous charged gas mixture bounded by an oscillating plate, for the first time. The initial-boundary value problem of the Rayleigh flow problem, applied to the system of the two-component plasma (positive ions + electrons), is solved to determine the macroscopic parameters such as the mean velocity, shear stress, and viscosity coefficient, together with the induced electric and magnetic fields. The investigation of the collisions mutual effects of ions with electrons, in the form of the ion distribution function, is completely operated. The light will be shed upon the effects of collisions of electrons with ions in the form of the electrons distribution function. Using the estimated distribution functions, it is of fundamental physical importance to deliberate the irreversible thermodynamic behavior of the diamagnetic and paramagnetic plasma gas. The results are applied to a typical model of laboratory argon plasma. The agreements of the results with the preceding theoretical studies are clarified.

2. The Physical Problem and Mathematical Formulation

Let us assume that the upper half of the space (𝑦≥0), which is bounded by an infinite flat plate (𝑦=0), is filled with a nonhomogeneous mixture of charged particles of electrons and positive ions. An infinite flat plate is fixed at 𝑦=0, and parallel to the 𝑥𝑧-plane. The plate oscillates harmonically in the 𝑥 direction with frequency 𝛽; that is, the velocity of the plate depends on the time 𝑡 as follows: 𝑉_𝑤=𝑅_𝑒𝑈₀exp−𝑖𝛽𝑡=𝑈₀Cos𝛽𝑡, (1) where the symbol 𝑅_𝑒 denotes the real part of complex expression. The quantity 𝑈₀ is the velocity amplitude, which is assumed to be small when compared with the thermal molecular velocity 𝑉_𝑇𝑒 of the gas. The charged gas is initially in absolute equilibrium and the wall is at rest. Then the plate starts to oscillate suddenly in its own plane with a velocity 𝑈₀Cos(𝛽𝑡) along the 𝑥-axis (𝑈₀ and 𝛽 are constants). Moreover, the plate is considered impermeable, uncharged, and an insulator. The whole system (electrons + ions + plate) is kept at constant temperature. All physical quantities are defined in the Nomenclatures.

Let the forces 𝑓^→_𝑒 and 𝑓^→_𝑖 acting on each electron and ion respectively be given by [28–30] 𝑓^→_𝑒=−𝑒𝐸^→_𝑒−𝑒𝑐_𝑜^→𝑐∧𝐵^→_𝑒,_→𝑓_𝑖=𝑒𝐸^→_𝑖+𝑒𝑐_𝑜^→𝑐∧𝐵^→_𝑖. (2)

The directions of the considered physical quantities are as follows: 𝑉^→_𝛼≡𝑉_𝑥𝛼,0,0,^→𝐽_𝛼≡𝑞_𝛼𝑛𝑉_𝑥𝛼,0,0_→,𝐸_𝛼≡𝐸_𝑥𝛼,0,0,𝐵^→_𝛼≡0,0,𝐵_𝑧𝛼, (3) where 𝛼=𝑒 for electrons and 𝛼=𝑖 for ions.

The 𝑉_𝑥𝛼, 𝐸_𝑥𝛼, 𝐵_𝑧𝛼, and 𝐽_𝑥𝛼 are functions of 𝑦 and 𝑡 that satisfie Maxwell’s equations. The distribution function 𝐹(𝑦,𝑐,𝑡) of the particles for the plasma gas can be obtained from the Bhatnager-Gross-Krook (BGK) model [31] of the kinetic Boltzmann’s equation as 𝜕𝐹_𝑒+𝜕𝑡^→𝑐⋅𝜕𝐹_𝑒𝜕^⇀𝑟+^→𝑓_𝑒𝑚_𝑒⋅𝜕𝐹_𝑒𝜕^⇀𝑐=𝜈_𝑒𝑒𝐹_0𝑒−𝐹_𝑒+𝜈_𝑒𝑖𝐹_0𝑒−𝐹_𝑒forelectrons,𝜕𝐹_𝑖+𝜕𝑡^→𝑐⋅𝜕𝐹_𝑖𝜕^⇀𝑟+^→𝑓_𝑖𝑚_𝑖⋅𝜕𝐹_𝑖𝜕^⇀𝑐=𝜈_𝑖𝑖𝐹_0𝑖−𝐹_𝑖+𝜈_𝑖𝑒𝐹_0𝑖−𝐹_𝑖forions, (4)(5) where 𝐹_0𝛼=𝑛_𝛼(2𝜋𝑅𝑇_𝛼)^−3/2exp(−(^→𝑐−𝑉^→_𝛼)²/2𝑅𝑇_𝛼) and 𝐹_0𝛼=𝑛_𝛼(2𝜋𝑅𝑇_𝛼)^−3/2exp(−(^→𝑐−𝑉^→_𝛼)²/2𝑅𝑇_𝛼).

The quantities 𝑛_𝛼, 𝑉^→_𝛼 and 𝑇_𝛼 are the number density, mean drift velocity, and effective temperature obtained by taking moments of 𝐹_𝛼. Some latitude in the definition of 𝑇_𝛼 and 𝑉^→_𝛼 is possible; one choice is in [32] 𝑇_𝑒=𝑇_𝑖,𝑇_𝑖=𝑇_𝑒,𝑉^→_𝑒=𝑉^→_𝑖,𝑉^→_𝑖=𝑉^→_𝑒. (6)

The particles are reflected from the plate with a full velocity accommodation; that is, the plasma particles are reflected with the plate velocity so that the boundary conditions are 𝑉_𝑥2𝛼(0,𝑡)=𝑈₀Cos(𝛽𝑡)for𝑡>0, where 𝑉_𝑥2𝛼=𝑉_𝑥𝛼as𝑐_𝑦>0 and 𝑉_𝑥𝛼 is finite as 𝑦→∞ for both ions and electrons.

Substituting from (2), (3), and (6) into (4) and (5) one obtains 𝜕𝐹_𝑒𝜕𝑡+𝑐_𝑦𝜕𝐹_𝑒−𝜕𝑦𝑒𝐵_𝑒𝑧𝑚_𝑒𝑐₀𝑐_𝑦𝜕𝐹_𝑒𝜕𝑐_𝑥−𝑐_𝑥𝜕𝐹_𝑒𝜕𝑐_𝑦+𝑒𝐸_𝑒𝑥𝑚_𝑒𝜕𝐹_𝑒𝜕𝑐_𝑥=𝜈_𝑒𝑒𝐹_0𝑒−𝐹_𝑒+𝜈_𝑒𝑖𝐹_0𝑖−𝐹_𝑒,forelectrons,𝜕𝐹_𝑖𝜕𝑡+𝑐_𝑦𝜕𝐹_𝑖+𝜕𝑦𝑒𝐵_𝑖𝑧𝑚_𝑖𝑐₀𝑐_𝑦𝜕𝐹_𝑖𝜕𝑐_𝑥−𝑐_𝑥𝜕𝐹_𝑖𝜕𝑐_𝑦−𝑒𝐸_𝑥𝑚_𝑖𝜕𝐹_𝑖𝜕𝑐_𝑥=𝜈_𝑖𝑖𝐹_0𝑖−𝐹_𝑖+𝜈_𝑖𝑒𝐹_0𝑒−𝐹_𝑖,forions, (7)(8) where 𝜈_𝑒𝑒, 𝜈_𝑒𝑖, 𝜈_𝑖𝑖, and 𝜈_𝑖𝑒 are electron-electron, electron-ion, ion-ion, and ion-electron collision frequencies, respectively, which are given by [32–35] 𝜈_𝑒𝑒=4√𝜋𝑛_𝑒𝑒⁴LogΛ_𝑒𝑒3√𝑚_𝑒𝐾_𝐵^3/2𝑇_𝑒^3/2,𝜈_𝑒𝑖=4√2𝜋𝑛_𝑖𝑒⁴𝑍²LogΛ_𝑒𝑖3√𝑚_𝑒𝐾_𝐵^3/2𝑇_𝑒^3/2,𝜈_𝑖𝑖=4√𝜋𝑛_𝑖𝑒⁴𝑍⁴LogΛ_𝑖𝑖3√𝑚_𝑖𝐾_𝐵^3/2𝑇_𝑖^3/2,𝜈_𝑖𝑒=𝑚_𝑒𝑚_𝑖4√2𝜋𝑛_𝑒𝑒⁴𝑍²LogΛ_𝑖𝑒3√𝑚_𝑒𝐾_𝐵^3/2𝑇_𝑒^3/2, (9) where 𝜆_𝐷𝑒=𝜆_𝐷𝑖=𝜆_𝐷, Log[Λ]=Log[4𝜋𝑛𝜆³_𝐷] and Zare the Coulomb Logarithms and the degree of ionization, respectively.

The model of the cone of influence suggested by Lee’s moment method [36–40] for the solution of the Boltzmann’s equation is employed here. Let us write the solution of (4) and (5), as suggested by Kashmarov in the form 𝐹_𝛼=𝐹_1𝛼=𝑛_𝛼2𝜋𝑅𝑇_𝛼^−3/2𝑐1+^𝑥𝑉_𝑥1𝛼𝑅𝑇_𝛼exp−𝑐²2𝑅𝑇_𝛼for𝑐_𝑦𝐹<0,_2𝛼=𝑛_𝛼2𝜋𝑅𝑇_𝛼^−3/2𝑐1+^𝑥𝑉_𝑥2𝛼𝑅𝑇_𝛼exp−𝑐²2𝑅𝑇_𝛼for𝑐_𝑦>0, (10) where 𝑉_𝑥1𝛼 and 𝑉_𝑥2𝛼 are four unknown functions of time 𝑡 and the distance variable 𝑦.

Using Grad’s moment method multiplying (7) and (8) by 𝑄_𝑗(^→𝑐). Integrating over all values of ^→𝑐, we can obtain the transfer equations for electrons and ions, respectively, in the form 𝜕𝑄𝜕𝑡^𝑗𝐹_𝑒𝑑𝑐+𝜕𝑐𝜕𝑦^𝑦𝑄_𝑗𝐹_𝑒𝑑𝑐+𝑒𝐸_𝑥𝑒𝑚_𝑒×𝐹_𝑒𝜕𝑄_𝑗𝜕𝑐_𝑥𝑑𝑐+−𝑒𝐵_𝑧𝑒𝑚_𝑒𝑐₀𝑐_𝑥𝜕𝑄_𝑗𝜕𝑐_𝑦−𝑐_𝑦𝜕𝑄_𝑗𝜕𝑐_𝑥𝑑𝑐=𝜈_𝑒𝑒𝐹_0𝑒−𝐹_𝑒𝑄_𝑗𝑑𝑐+𝜈_𝑒𝑖𝐹_0𝑖−𝐹_𝑒𝑄_𝑗𝑑𝑐,𝜕𝑄𝜕𝑡^𝑗𝐹_𝑖𝑑𝑐+𝜕𝑐𝜕𝑦^𝑦𝑄_𝑗𝐹_𝑖𝑑𝑐−𝑒𝐸_𝑖𝑥𝑚_𝑖×𝐹_𝑖𝜕𝑄_𝑗𝜕𝑐_𝑥𝑑𝑐+𝑒𝐵_𝑖𝑧𝑚_𝑖𝑐₀𝑐_𝑥𝜕𝑄_𝑗𝜕𝑐_𝑦−𝑐_𝑦𝜕𝑄_𝑗𝜕𝑐_𝑥𝑑𝑐=𝜈_𝑖𝑖𝐹_0𝑖−𝐹_𝑖𝑑𝑐+𝜈_𝑖𝑒𝐹_0𝑒−𝐹_𝑖𝑄_𝑗𝑑𝑐. (11)(12)

The integrals over the velocity distance are evaluated from the relation 𝑄_𝑗^→𝑐𝐹_𝛼𝑑𝑐=^∞_−∞⁰_−∞^∞_−∞𝑄_𝑗𝐹_1𝛼𝑑𝑐+^∞_−∞^∞₀^∞_−∞𝑄_𝑗𝐹_2𝛼𝑑𝑐, (13) where 𝑄_𝑗=𝑄_𝑗(^→𝑐),𝑗=1,2, and 𝑑𝑐=𝑑𝑐_𝑥𝑑𝑐_𝑦𝑑𝑐_𝑧, where 𝑐_𝑥, 𝑐_𝑦 and 𝑐_𝑧 are the particles velocities components along 𝑥-, 𝑦-, and 𝑧-axes, respectively. Moreover, 𝐸_𝛼 and 𝐵_𝛼 may be obtained from Maxwell’s equation, for electrons as follows:𝜕𝐸_𝑥𝑒−1𝜕𝑦𝑐₀𝜕𝐵_𝑧𝑒𝜕𝑡=0,𝜕𝐵_𝑧𝑒−1𝜕𝑦𝑐₀𝜕𝐸_𝑥𝑒−𝜕𝑡4𝜋𝑒𝑛_𝑒𝑐₀𝑉_𝑥𝑒=0. (14) For ions we obtain 𝜕𝐸_𝑥𝑖−1𝜕𝑦𝑐₀𝜕𝐵_𝑧𝑖𝜕𝑡=0,𝜕𝐵_𝑧𝑖−1𝜕𝑦𝑐₀𝜕𝐸_𝑥𝑖+𝜕𝑡4𝜋𝑒𝑛_𝑖𝑐₀𝑉_𝑥𝑖=0, (15) where 𝑛_𝛼=∫𝐹_𝛼𝑑𝑐, 𝑛_𝛼𝑉_𝑥𝛼=∫𝑐_𝑥𝐹_𝛼𝑑𝑐, with the initial and boundary conditions 𝐸_𝑥𝛼𝑦,0=𝐵_𝑧𝛼𝐸𝑦,0=0;_𝑥𝛼𝑦,𝑡,𝐵_𝑧𝛼𝑦,𝑡arefiniteas𝑦⟶∞. (16)

We introduce the dimensionless variables defined by 𝑡=𝑡^′𝜏_𝑖𝑒,𝑦=𝑦^′𝜏_𝑖𝑒𝑉_𝑇𝑒√,𝑉2𝜋_𝑥𝛼=𝑉^′_𝑥𝛼𝑉_𝑇𝑒,𝜏_𝑥𝑦𝛼=𝜏^′_𝑥𝑦𝛼𝑉_𝑇𝑒𝑈,Ma=⁰𝑉_𝑇𝑒,𝐵_𝑧𝛼=𝐵^′_𝑧𝛼√2𝜋𝑚_𝑒𝑐₀𝑒𝜏_𝑖𝑒,𝐸^′_𝑥𝛼=𝐸^′_𝑥𝛼𝑚_𝑒𝑉_𝑇𝑒𝑒𝜏_𝑖𝑒,𝜌_𝛼=𝑛_𝛼𝑚_𝛼,𝑉_𝑇𝑒=2𝐾_𝐵𝑇_𝑒𝑚_𝑒,𝑚𝜀=^𝑒𝑚_𝑖,𝑑𝑈_𝛼=𝑑𝑈_𝛼^′𝐾_𝐵𝑇_𝑒,𝐹_𝑗𝛼=𝐹^′_𝑗𝛼𝑛_𝑒2𝜋𝑅𝑇_𝑒^−3/2,𝑗=0,1,2. (17) For Ma²≪1 (low Mach number), we can assume that the density and the temperature variation at each point of the flow and at any time are negligible; that is, 𝑛_𝛼=1+𝑂(Ma²) and 𝑇_𝛼=1+𝑂(Ma²). Put 𝑉_𝑥𝛼=12𝑉_𝑥1𝛼+𝑉_𝑥2𝛼,𝜏_𝑥𝑦𝛼=𝑃_𝑥𝑦𝛼𝜌_𝛼𝑈₀𝑅𝑇_𝑒=𝑉/2𝜋^𝑥2𝛼−𝑉_𝑥1𝛼, (18) where 𝑃_𝑥𝑦𝛼 is the shear stress [11] that is defined by 𝑃_𝑥𝑦𝛼=𝑚∫(𝑐_𝑥−𝑉_𝑥)𝑐_𝑦𝐹_𝛼𝑑𝑐.

Using the dimensionless variable, if we neglect terms of order 𝑂(Ma²), (11) for 𝑄₁=𝑐_𝑥 and 𝑄₂=𝑐_𝑥𝑐_𝑦 is: 𝜕𝑉^′_𝑥𝑒𝜕𝑡^′+𝜕𝜏^′_𝑥𝑦𝑒𝜕𝑦^′−𝐸^′_𝑥𝑒=0,𝜕𝜏^′_𝑥𝑦𝑒𝜕𝑡^′+2𝜋𝜕𝑉^′_𝑥𝑒𝜕𝑦^′+𝜈_𝑒𝑒𝜈_𝑖𝑒+𝜈_𝑒𝑖𝜈_𝑖𝑒𝜏^′_𝑥𝑦𝑒=0. (19)

Similarly, (12) becames: 𝜕𝑉^′_𝑥𝑖𝜕𝑡^′+𝜕𝜏^′_𝑥𝑦𝑖𝜕𝑦^′+𝜀𝐸^′_𝑥𝑖=0,𝜕𝜏^′_𝑥𝑦𝑖𝜕𝑡^′+2𝜋𝜕𝑉^′_𝑥𝑖𝜕𝑡^′+𝜈_𝑖𝑖𝜈_𝑖𝑒+𝜈_𝑖𝑒𝜈_𝑖𝑒𝜏^′_𝑥𝑦𝑖=0, (20) with the initial and boundary conditions 𝑉^′_𝑥𝛼𝑦^′,0=𝜏^′_𝑥𝑦𝛼𝑦^′,0=0,2𝑉^′_𝑥𝛼0,𝑡^′+𝜏^′_𝑥𝑦𝛼0,𝑡^′=2MaCos𝛽₁𝑡^′;𝑉^′_𝑥,𝜏^′_𝑥𝑦arefiniteas𝑦⟶∞,𝛽₁=𝛽𝜏_𝑖𝑒. (21)

In the expressions for the transport coefficients mentioned previously, the fact that the plasma is neutral was used [33], writing 𝑛=𝑛_𝑒=𝑍𝑛_𝑖, 𝑇=𝑇_𝑒=𝑇_𝑖, and 𝑍=1, which exploited the fact that the ratio 𝑚_𝑒/𝑚_𝑖 is small. Thus, the ratio between the four distinguished collision frequencies can be rewritten in a form that referred to 𝜈_𝑒𝑒 in formulas (9) 𝜈_𝑒𝑒∶𝜈_𝑒𝑖∶𝜈_𝑖𝑖∶𝜈_𝑖𝑒=1∶√2∶𝑚_𝑒𝑚_𝑖∶𝑚_𝑒𝑚_𝑖=1∶√2∶√𝜀∶𝜀. (22)

Therefore, 𝜈_𝑒𝑒∼𝜈_𝑒𝑖≫𝜈_𝑖𝑖≫𝜈_𝑖𝑒; the first of these gross inequalities arises because thermal ion speeds are less than thermal electron speeds by the factor 𝑚_𝑒/𝑚_𝑖 if 𝑇_𝑒≈𝑇_𝑖, and so the ions take longer period of time to meet each other. The second one reflects the fact that the electrons are not very effective in deflecting the much heavier ions.

For the sake of simplicity, henceforth, we drop the dash over the dimensionless variables. Therefore, we have the following initial-boundary value problem for electrons (neglecting the displacement current) [30]: 𝜕𝑉_𝑥𝑒+𝜕𝑡𝜕𝜏_𝑥𝑦𝑒𝜕𝑦−𝐸_𝑥𝑒=0,𝜕𝜏_𝑥𝑦𝑒𝜕𝑡+2𝜋𝜕𝑉_𝑥𝑒+1𝜕𝑦√𝜀+2𝜀𝜏_𝑥𝑦𝑒=0,𝜕𝐸_𝑥𝑒−𝜕𝑦𝜕𝐵_𝑧𝑒𝜕𝑡=0,𝜕𝐵_𝑧𝑒𝜕𝑦−𝛼_0𝑒𝑉_𝑥𝑒=0,where𝛼_0𝑒=𝑉²_𝑇𝑒𝜏²_𝑖𝑒𝑒²𝑛_𝑒𝑚_𝑒𝑐²₀, (23)(24)(25)(26) with the initial and boundary conditions 𝑉_𝑥𝑒𝑦,0=𝜏_𝑥𝑦𝑒𝑦,0=𝐸_𝑥𝑒𝑦,0=𝐵_𝑧𝑒𝑦,0=0,2𝑉_𝑥𝑒(0,𝑡)+𝜏_𝑥𝑦𝑒(0,𝑡)=2MaCos𝛽₁𝑡𝑉,for𝑡>0;_𝑥𝑒,𝜏_𝑥𝑦𝑒,𝐸_𝑥𝑒,𝐵_𝑧𝑒arefiniteas𝑦⟶∞. (27) In addition, we have the following initial-boundary problem for ions (neglecting the displacement current): 𝜕𝑉_𝑥𝑖+𝜕𝑡𝜕𝜏_𝑥𝑦𝑖𝜕𝑦+𝜀𝐸_𝑥𝑖=0,𝜕𝜏_𝑥𝑦𝑖𝜕𝑡+2𝜋𝜕𝑉_𝑥𝑖+√𝜕𝑦𝜀+1𝜏_𝑥𝑦𝑖=0,𝜕𝐸_𝑥𝑖−𝜕𝑦𝜕𝐵_𝑧𝑖𝜕𝑡=0,𝜕𝐵_𝑧𝑖𝜕𝑦+𝛼_0𝑖𝑉_𝑥𝑖=0,where𝛼_0𝑖=𝑉²_𝑇𝑒𝜏²_𝑖𝑒𝑒²𝑛_𝑖𝑚_𝑒𝑐²₀. (28)(29)(30)(31) Since 𝑛_𝑖=𝑛_𝑒; thus, henceforth, we put 𝛼_0𝑖=𝛼_0𝑒=𝛼₀.

We can reduce our basic (23)–(26), for electrons, after some analytical manipulations to a single equation as follows: 𝜕⁴𝑉_𝑥𝑒𝑦,𝑡𝜕𝑡²𝜕𝑦²𝜕−2𝜋⁴𝑉_𝑥𝑒𝑦,𝑡𝜕𝑡⁴−𝐴_𝑐𝜕³𝑉_𝑥𝑒𝑦,𝑡𝜕𝑡𝜕𝑦²−𝛼₀𝜕²𝑉_𝑥𝑒𝑦,𝑡𝜕𝑡²−𝛼₀𝐴_𝑐𝜕𝑉_𝑥𝑒𝑦,𝑡𝜕𝑡=0, (32) where 𝐴_𝑐=((1/√𝜀)+2/𝜀).

Similarly, the basic (28)–(31) can be reduced for ions to obtain 𝜕⁴𝑉_𝑥𝑖𝑦,𝑡𝜕𝑡²𝜕𝑦²𝜕−2𝜋⁴𝑉_𝑥𝑖𝑦,𝑡𝜕𝑡⁴−𝐵_𝑐𝜕³𝑉_𝑥𝑖𝑦,𝑡𝜕𝑡𝜕𝑦²−𝛼₀𝜀𝜕²𝑉_𝑥𝑖𝑦,𝑡𝜕𝑡²−𝛼₀𝜀𝐵_𝑐𝜕𝑉_𝑥𝑖𝑦,𝑡𝜕𝑡=0, (33) where 𝐵_𝑐=(√𝜀+1).

3. Solution of the Initial-Boundary Value Problem

The traveling wave solution method [41, 42] is used, considered 𝜉=𝑙𝑦−𝑚𝑡, (34) to make all the dependent variables as functions of 𝜉. Here 𝑙 and 𝑚 are transformation constants, which do not depend on the properties of the fluid but as parameters to be determined by the boundary and initial conditions [41]. Using (34) we obtain the derivatives 𝜕𝜕𝜕𝑡=−𝑚,𝜕𝜕𝜉𝜕𝜕𝑦=𝑙,𝜕𝜕𝜉𝜕𝑡^𝑎𝑎=(−1)^𝑎𝑚^𝑎𝜕^𝑎𝜕𝜉^𝑎,𝜕^𝑎𝜕𝑦^𝑎=𝑙^𝑎𝜕^𝑎𝜕𝜉^𝑎, (35) where 𝑎 is a positive integer.

Substituting from (34)-(35) into (32), we have 𝑚²𝑙²−2𝜋𝑚⁴𝑑⁴𝑉_𝑥𝑒(𝜉)𝑑𝜉⁴+𝐴_𝑐𝑚𝑙²𝑑³𝑉_𝑥𝑒(𝜉)𝑑𝜉³−𝛼₀𝑚²𝑑²𝑉_𝑥𝑒(𝜉)𝑑𝜉²+𝛼₀𝐴_𝑐𝑚𝑑𝑉_𝑥𝑒(𝜉)𝑑𝜉=0. (36) The boundary and initial conditions became 𝐸_𝑥𝑒(𝜉=0)=𝐵_𝑧𝑒(𝜉=0)=𝜏_𝑥𝑦𝑒(𝜉=0)=0,2𝑉_𝑥𝑒(𝜉=−𝑚)+𝜏_𝑥𝑦𝑒(𝜉=−𝑚)=2MaCos𝛽₁,𝑉at𝑦=0,e.g.,𝑡=1;_𝑥𝑒,𝜏_𝑥𝑦𝑒,𝐸_𝑥𝑒,𝐵_𝑧𝑒arefiniteas𝜉⟶−∞. (37)

Now we have an ordinary differential Equation (36) with the boundary and initial conditions (37). Next, let us solve (33) by using the same tackling. Substituting from expressions (34)-(35) into (33), we get 𝑚²𝑙²−2𝜋𝑚⁴𝑑⁴𝑉_𝑥𝑖(𝜉)𝑑𝜉⁴+𝐵_𝑐𝑚𝑙²𝑑³𝑉_𝑥𝑖(𝜉)𝑑𝜉³−𝛼₀𝜀𝑚²𝑑²𝑉_𝑥𝑖(𝜉)𝑑𝜉²+𝛼₀𝜀𝐵_𝑐𝑚𝑑𝑉_𝑥𝑖(𝜉)𝑑𝜉=0, (38) with the corresponding boundary and initial conditions 𝐸_𝑥𝑖(𝜉=0)=𝐵_𝑧𝑖(𝜉=0)=𝜏_𝑥𝑦𝑖(𝜉=0)=0,2𝑉_𝑥𝑖(𝜉=−𝑚)+𝜏_𝑥𝑦𝑖(𝜉=−𝑚)=2MaCos𝛽₁;𝑉_𝑥𝑖,𝜏_𝑥𝑦𝑖,𝐸_𝑥𝑖,𝐵_𝑧𝑖arefiniteas𝜉⟶−∞. (39) The two ordinary, fourth order homogeneous differential equations (36) and (38) can be solved exactly by the help of symbolic computer software, with their boundary and initial conditions (37) and (39). The sought solutions will be applied to a typical model of laboratory argon plasma.

4. The Investigation of the Behavior of the Internal Energy Change

The studying of the behavior of the internal energy change for the physical systems presents a great importance in science. The extended Gibbs relation for electrons and ions is introduced to study the internal energy change for the system, based on the solution of the nonstationary Boltzmann equation [43]. It includes the electromagnetic field energy as a part of the complete energy balance. This procedure distinguishes the charged gas into paramagnetic and diamagnetic ones. If there are unpaired electrons in the molecular orbital diagram, the gas is paramagnetic. If all electrons are paired, the gas is considered as a diamagnetic one. We should write the internal energy balance, including the electromagnetic field energy, to get the work term in the first law of thermodynamics as follows.(a)For paramagnetic plasma, the internal energy change is expressed in terms of the extensive quantities 𝑆_𝛼, 𝑃_𝛼, and 𝑀_𝛼, which are the thermodynamic coordinates corresponding to the conjugate intensive quantities 𝑇_𝛼, 𝐸_𝛼, and 𝐵_𝛼, respectively. The three contributions in the internal energy change in the Gibbs formula we have 𝑑𝑈_𝛼=𝑑𝑈_𝑆𝛼+𝑑𝑈_pol𝛼+𝑑𝑈_par𝛼, (40) where 𝑑𝑈_𝑆 𝛼=𝑇𝑑𝑆_𝛼 is the internal energy change due to the variation of the entropy, which is written in dimensionless form as 𝑆_𝛼=−𝜋^3/2𝑉²_𝑥1𝛼+𝑉²_𝑥2𝛼−32, (41)𝑑𝑈_pol𝛼=𝐸_𝛼 𝑑𝑃_𝛼 is the internal energy change due to variation of polarization, and 𝑑𝑈_para𝛼=𝐵_𝛼 𝑑𝑀_𝛼 is the internal energy change due to the variation of magnetization, here 𝑀 is calculated from the equation [44] 𝜕𝑆_𝛼𝜕𝑀_𝛼𝐵=−^𝛼𝑇_𝛼⟹𝑀_𝛼𝑇=−^𝛼𝐵_𝛼𝜕𝑆_𝛼𝜕𝑦_𝑡𝑑𝑦. (42) Introduce the dimensionless variables 𝑈^′_𝛼=𝑈_𝛼/𝐾𝑇_𝛼, 𝑀^′_𝛼=𝑀_𝛼(1/𝑒_𝛼𝜏_𝑒𝑖𝑉_𝑇𝑒), 𝑝^′_𝛼=𝑝_𝛼(1/𝑒_𝛼𝜏_𝑒𝑖𝑉_𝑇𝑒) in the Gibbs formula to get (after dropping the primes) 𝑑𝑈_𝛼=𝑑𝑆_𝛼+𝑓_𝛼1𝐸_𝛼𝑑𝑝_𝛼+𝑓_1𝛼𝐵_𝛼𝑑𝑀_𝛼. (43)(b)On the other hand, if the plasma is diamagnetic, the internal energy change due to the extensive variables 𝑆, 𝑃, and 𝐵 represents the thermodynamic coordinates conjugate to the intensive quantities 𝑇, 𝐸, and 𝑀, respectively; therefore, we have three contributions in the internal energy change in the Gibbs formula given by 𝑑𝑈_𝛼=𝑑𝑈_𝑆𝛼+𝑑𝑈_pol𝛼+𝑑𝑈_dia𝛼, (44) where 𝑑𝑈_dia𝛼=−𝑀_𝛼 𝑑𝐵_𝛼 is the internal energy change due to the variation of the induced magnetic induction, where 𝑀_𝛼=𝑇_𝛼(𝜕𝑆_𝛼/𝜕𝐵_𝛼). Hence, the dimensionless form for 𝑑𝑈 in this case takes the following form: 𝑑𝑈_𝛼=𝑑𝑆_𝛼+𝑓_𝛼1𝐸_𝛼𝑑𝑝_𝛼−𝑓_𝛼1𝑀_𝛼𝑑𝐵_𝛼, (45)where 𝑓₁=𝑚_𝑒𝑉²_𝑇𝑒𝐾𝑇₀,𝑑𝑆_𝛼=𝜕𝑆_𝛼𝜕𝑟𝛿𝑦+𝜕𝑆_𝛼𝜕𝑡𝛿𝑡;𝛿𝑦=1,𝛿𝑡=4.35≃𝑡_𝑒𝑞𝑒. (46)

5. Discussion

In this problem, the unsteady behavior of an inhomogeneous mixture of charged gas, bounded by an oscillating plate, is investigated. This study is based on the kinetic theory via the BGK model of the Boltzmann equation. Our computations are performed according to typical data for argon plasma as a paramagnetic medium in the case of the argon gas losing single electrons subjected to the following conditions and parameters: 𝐾_𝐵=1.380710⁻¹⁶erg/K,  𝑇₀=600K, 𝑛_𝑒=7 10¹⁶cm⁻³, 𝑑=3.84 10⁻⁸cm (diameter of the argon atom); the electron rest mass, mass of ions, and charge are 𝑚_𝑒=9.093×10⁻²⁸gm, 𝑚_𝑖=6.633×10⁻²³gm, 𝑒=4.810⁻¹⁰ esu, which are used to calculate electron-ion relaxation time 𝜏_𝑖𝑒=1.4859×10⁻¹⁰ sec. The dimensionless parameter is 𝛼₀=7.9×10⁻² and the mean free path of the electrons 𝜆_𝑖𝑒=2.110⁻³cm compared to the electron Debye length.

Consider 𝜆_𝐷𝑒=(𝐾_𝐵𝑇₀/4𝜋𝑛_𝑒𝑒²)^1/2=6.39×10⁻⁷cm, 𝑓₁=2. Using the idea of the shooting numerical calculation method, the transformation constants are evaluated to obtain 𝑚=2.8,𝑙=−1.2, and the plate Mach number Ma=6.510⁻².

The fundamental and the essential inequalities, which we must bear in mind when analyzing the results, are 𝑚_𝑖≫𝑚_𝑒makes𝜀²=𝑚_𝑒𝑚_𝑖=1.3710⁻⁵,𝜏_𝑒𝑒∶𝜏_𝑒𝑖∶𝜏_𝑖𝑖∶𝜏_𝑖𝑒=1∶7.07×10⁻¹∶2.69×10²∶5.15×10⁴⟹𝜏_𝑒𝑒∼𝜏_𝑒𝑖≪𝜏_𝑖𝑖≪𝜏_𝑖𝑒. (I-a)(I-b)

These inequalities will control the major behavior of both electrons and positive ions in the rest of the discussion. Figure 1(a) clarifies that in the course of time the perturbed velocity distribution functions 𝐹₁ and 𝐹₂ approache to the equilibrium velocity distribution function 𝐹₀ at 𝑦=0.003, for example. The ions are still departing from equilibrium very slowly; see Figure 1(b).

(a)

(b)

(c)

Figure 1 

(a) The comparison between the combined perturbed dimensionless velocity distribution functions for electrons 𝐹𝑒 (𝐹1𝑒 (green), 𝐹2𝑒 (red)) and electrons equilibrium velocity distribution function 𝐹𝑜𝑒 (grid) at 𝑡=10−3, 2.175, and 4.35 for a fixed 𝑦 value (0.003) with the Mach number of the plate  Ma=0.065. (b) The comparison between the combined perturbed dimensionless velocity distribution functions for ions 𝐹𝑖 (𝐹1𝑖 (green), 𝐹2𝑖 (red)) and ions equilibrium velocity distribution function 𝐹𝑜𝑖 (grid) at 𝑡=10−3, 2.175, and 4.35 for a fixed 𝑦 value (0.003) with the Mach number of the plate Ma = 0.065. (c) The comparison between the combined perturbed dimensionless velocity distribution functions for ions 𝐹𝑖 (𝐹1𝑖 (gray), 𝐹2𝑖 (black)) and ions equilibrium dimensionless velocity distribution function 𝐹𝑜𝑖 (grid) at 𝑡=690, 1380 for a fixed 𝑦 value (0.003) with the Mach number of the plate Ma=0.065.

A comparison between Figures 1(a) and 1(b) shows that the collisions of ions with electrons have a very little effect on the form of the ions nonequilibrium distribution function. On the other hand, collisions of electrons with ions have an important effect on the form of the electrons nonequilibrium distribution function, which is in a good agreement with the well-known study of Braginskii [33]. This is because the transfer of momentum from ions to electrons occurs in about the same time ∼𝜏_𝑒𝑖 as the transfer of the energy; hence, ion-electron momentum transfer is small compared with ion-ion momentum transfer. The transfer of momentum from electrons to ions occurs in a time of the same order ∼𝜏_𝑒𝑒 as electron-electron transfer time, so that collisions of electrons with the ions affect the electrons nonequilibrium distribution function.

Figures 1(a) and 1(b) indicate that the lighter species (electrons) in the mixture of gases reach equilibrium before the heavier one (positive ions) that is in a qualitative agreement with the investigation made by Galkin [45]. Figure 1(a) makes it clear that the time needed for the electrons to reach equilibrium is (𝑡_𝑒𝑞∼4.35) which will be taken into consideration, henceforth.

In a relative long period, for example, 𝑡=690, the departure of positive ions from equilibrium becomes obvious; the distinction between ions equilibrium distribution function and their nonequilibrium distribution function becomes substantial. After a relatively long time the heavy species (positive ions) reach equilibrium at 𝑡≅1380; see Figure 1(c).

All Figures 1(a) and 1(b) to 8(a) and 8(b) shed light upon the boundary and initial conditions for both electrons and ions hold. As revealed in Figures 2(a) and 2(b) at the vicinity of the suddenly oscillating plate the mean velocities for both electrons and positive ions have a value = Ma of the oscillating plate which satisfies the conditions of the problem. The 𝑉_𝑥𝑒 gets a severe decrement with time while 𝑉_𝑥𝑖 shows a gradually linear decrement; this is due to two reasons, firstly, because thermal ion speeds are less than the thermal electron speeds, by the amount 𝑚_𝑒/𝑚_𝑖 as 𝑇_𝑒≅𝑇_𝑖, and so ions take longer time to encounter each other while electrons do not. Secondly, this behavior reflects the fact that the electrons are not very effective in deflecting the much heavier ions, while ions have a very successful effect on electrons motion, referring to inequalities (I-a) and (I-b). Therefore, the change of all kinetic and thermodynamic variables belongs to the positive ions that has unnoticed nonlinear gradually slow changes; see Figures 1(b) to 8(b). This is in a qualitative agreement with the previous paper [15].

(a)

(b)

Figure 2 

The dimensionless velocity versus space 𝑦 and time 𝑡: (a) 𝑉𝑥𝑒; (b) 𝑉𝑥𝑖.

The shear stress 𝜏_𝑥𝑦𝑒 is beginning from zero value. It is increasing nonlinearly towards its maximum value at 𝑡∼0.5𝑡_𝑒𝑞𝑒. After that, it decreases with the same behavior until it vanishes; see Figure 3(a). This is according to the behavior of the electrons velocity itself. Since the deviation from equilibrium is small, the electron gas is rarefied and the flow is slow. Thus, the gas is Newtonian [46]. It follows that the viscosity 𝜇_𝛼=−𝜏_𝑥𝑦𝛼/(𝜕𝑉_𝑥𝛼/𝜕𝑦) represents the resistance to the motion. It gradually increases nonlinearly as the gas particles move away from the plate for ions 𝜇_𝑖. The 𝜇_𝑒 approaches zero for electrons except at a small time interval around 𝑡∼0.5𝑡_𝑒𝑞𝑒; this is due to the corresponding maximum value of 𝜏_𝑥𝑦𝑒 in the same time interval.

(a)

(b)

Figure 3 

The dimensionless shear stress versus space 𝑦 and time 𝑡: (a) 𝜏𝑥𝑦𝑒; (b) 𝜏𝑥𝑦𝑖.

The electrons induced electric field has a sudden increase in the beginning until it reaches its maximum value (𝐸_𝑥𝑒Max∼0.1) due to the sudden oscillation of the plate itself. It decreases nonlinearly until it vanishes at 𝑡∼0.5𝑡_𝑒𝑞𝑒. After that, it changes its direction until the same maximum value 𝐸_𝑥𝑒Max∼−0.1 in the opposite direction. It is pushing electrons towards equilibrium. This is because of the famous le Chatelier principle that states, “If the system going away from equilibrium, its particles take the behavior that decreases the departure from equilibrium” that is pushing the system towards equilibrium again; see Figure 4(a). The same behavior holds for electrons induced magnetic field; see Figure 5(a).

(a)

(b)

Figure 4 

The dimensionless induced electric field versus space 𝑦 and time 𝑡: (a) 𝐸𝑥𝑒; (b) 𝐸𝑥𝑖.

(a)

(b)

Figure 5 

The dimensionless induced electric field versus space 𝑦 and time 𝑡: (a) 𝐵𝑧𝑒; (b) 𝐵𝑧𝑖.

Upon passing through a plasma, a charged particle (electron) losses (or gains) part of its energy because of the interaction with the surroundings (positive ions) due to plasma polarization and collisions. The energy loss (or gain) of an electron is determined by the work of the forces acting on the electrons in the plasma by the electromagnetic field generated by the moving particles themselves [47], since the suddenly oscillating plate causes work to be done on the gas, changing the internal energy of the gas 𝑈. As shown in Figures 6–8, the change in the internal energy due to the variation of entropy and polarization varies smoothly with time by the energy lost to and gained from the ions and plate, respectively. The change in internal energy varies chaotically because of the intensive variables, corresponding to paramagnetic plasma; at the end each tends to zero.

(a)

(b)

Figure 6 

The dimensionless internal energy change versus space 𝑦 and time 𝑡: (a) 𝑑𝑈𝑠𝑒; (b) 𝑑𝑈𝑠𝑖.

(a)

(b)

Figure 7 

The dimensionless internal energy change versus space 𝑦 and time 𝑡: (a) 𝑑𝑈𝑝𝑒; (b) 𝑑𝑈𝑝𝑖.

(a)

(b)

Figure 8 

The dimensionless internal energy change versus space 𝑦 and time 𝑡: (a) 𝑑𝑈par𝑒; (b) 𝑑𝑈par𝑖.

6. Conclusions

The solution of the unsteady BGK equation in the case of an inhomogeneous rarefied charged gas, bounded by an oscillating plate, is investigated. We use the method of moments of the two-sided distribution function together with Maxwell’s equations. This is developed within the restrictions of small deviation from equilibrium, rarified gas mixture, and slow flow. This solution allows for the calculation of the components of the velocity of the flow for both electrons and positive ions. Inserting them into the suggested two-sided distribution functions and analyzing the results, it is found that:(a)the lighter species (electrons) of the gas mixture reaches equilibrium before the heave one (positive ions), which is in a qualitative agreement with Galkin [37];(b)definitely, the equilibrium times for electrons and for ions are calculated. The relation between those times and the relaxation time for both species of the mixture is deduced. We proved that the collision of ions with electrons has very little effect on the form of the ions nonequilibrium distribution function. On the other hand, we found that the collisions of electrons with ions have an important effect on the form of the electrons nonequilibrium distribution function, which is in a qualitative agreement with the study done by Park et al. [21];(c)the ratio between the time that electrons (𝑡_𝑒𝑞𝑒≅4.35) and ions (𝑡_𝑒𝑞𝑖≅1380) need to reach equilibrium is approximately equal to the same order of the reverse ratio of the ion-ion collision frequency to the mean value of the electron-electron and electron-ion collision frequencies; that is, 𝑡_𝑒𝑞𝑖/𝑡_𝑒𝑞𝑒≃(((𝜈_𝑒𝑒+𝜈_𝑒𝑖)/2)/𝜈_𝑖𝑖). This conclusion is different from the situation when the plate is moving with damping velocity and not oscillating; see the previous paper [15].

The predictions, estimated using Gibbs’ equation, reveal that the following order of maximum magnitude ratios between the different contributions to the internal energy change based on the total derivatives of the extensive parameters is, for ions 𝑑𝑈_𝑆𝑖∶𝑑𝑈_𝑝𝑖∶𝑑𝑈_par𝑖=1∶0.6×10⁻²∶0.8×10⁻⁴. (47)

It is concluded that the effect of the changes of the internal energies for positive ions 𝑑𝑈_𝑝𝑖 and 𝑑𝑈_par𝑖 due to electric and magnetic fields are small in comparison with 𝑑𝑈_𝑆𝑒. This happens with the recognition of the fact that these fields are self-induced by the sudden motion of the oscillating plate.

The same conclusion is applied in the case of electrons such that 𝑑𝑈_𝑆𝑒∶𝑑𝑈_𝑝𝑒∶𝑑𝑈_par𝑒=1∶10⁻⁶∶10⁻². (48)

Nomenclature

𝐵→:	The induced magnetic field vector
𝐵:	The induced magnetic field
𝐸→:	The induced electric vector
𝐸:	The induced electric field
𝐹:	The velocity distribution function
𝐹0:	The local Maxwellian distribution function
𝐹1:	Distribution function for going downward particles 𝑐𝑦<0
𝐹2:	Distribution function for going upward particles 𝑐𝑦>0
𝐽:	The current density
𝐾𝐵:	Boltzmann’s constant (Erg/K0)1.3810−16
Ma:	The plate Mach number
𝑀:	Specific magnetization
𝑃:	Polarization
𝑅:	The gas constant
𝑆:	Entropy per unit mass
𝑇:	The temperature
𝑈:	The internal energy of the gas
𝑈0:	Plate initial velocity
𝑉𝑥:	The mean velocity
𝑉𝑥1:	The mean velocity related to 𝐹1
𝑉𝑥2:	The mean velocity related to 𝐹2
𝑉:	Gas volume
𝑉𝑇𝑒:	Thermal velocity of electrons
𝑉𝑇𝑖:	Thermal velocity of ions
𝑐0:	The speed of light
𝑐:	The velocity of the particles
𝑑:	Particle diameter
𝑒:	The electron charge
𝑓→:	Lorantz’s force vector
𝑚𝑒:	Electron mass
𝑚𝑖:	Ion mass
𝑛:	The mean density
𝑛𝑒:	Electrons concentration
𝑛𝑖:	Ions concentration
𝑝:	Pressure
→𝑟:	The position vector of the particle
𝑡:	Time variable
→𝑢:	The mean velocity of the particle
𝑑𝑈S:	The internal energy change due to the variation of entropy
𝑑𝑈𝑃:	The internal energy change due to the variation of polarization
𝑑𝑈par:	The internal energy change due to the variation of magnetization
𝑑𝑈dia:	The internal energy change due to the variation of the induced magnetic field
𝑦:	Displacement variable
𝑍:	Ionization.

Superscripts

′∶	Dimensionless variable.

Subscripts

𝑒∶	Related to electrons
𝑖∶	Related to ions
𝑞∶	Equilibrium
𝛼∶	=𝑒 for electrons or =𝑖 for ions.

Greek Letters

𝜏∶	The relaxation time
𝜏𝑥𝑦:	The shear stress
𝜇∶	Viscosity coefficient
𝜆∶	The mean free path
𝛼0:	Dimensionless parameter
𝐵∶	Frequency
𝜈∶	Collision frequency
𝜀∶	Mass ratio
𝜆∶	Mean free path
𝜆𝐷:	Debye radius.