Travelling Waves Solution of the Unsteady Flow Problem of a Collisional Plasma Bounded by a Moving Plate

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.


Introduction
Partially ionized plasmas at low gas pressure have found a wide range of applications in many branches of contemporary technology. Gas discharge lighting, manufacturing of semiconductor chips, plasma treatment of materials [1,2], Astrophysics [3], nanotechnology, plasma chemistry, and bio-medical treatments are the most known applications of low temperature plasma [4]. Plasma (charged gases) based technology underpins some of the world's largest industries producing a significant proportion of the world's global commercial products including; computers, cell phones, automobiles, aero planes, paper and textiles [5]. Foremost among these is the electronics industry, in which plasma-based processes are indispensable for the manufacture of ultra-large-scale integrated microelectronic circuits. The Boltzmann equation represents a well-defined model to describe the motion of plasma, when microscopic effects must be considered. This is the case of an electron flow in MEMS (Micro-Electro-Mechanical Systems).
Abourabiia and Abdel Wahid [6], in the framework of irreversible thermodynamics, examined the characteristics of the Rayleigh flow problem of a rarified electron gas extracted from neutral atoms and proved that, it obeys the entropic behavior for gas system but with an approximate solution and inaccurate formula of the collision frequency. In the present paper, the enhancement and improvement of this study is done. The present study has obvious peculiarities in comparison with the previous one [6]. In principle, the complete and accurate formulas of collision frequencies are used, avoid the discontinuity in the solution using Laplace transformation, used in [6], and introduce the complete value Bounded by a Moving Plate of variables without any cutoff caused from the small parameters method as in [6] and in this study we have no restriction on the non-dimensional parameter like [6].
The objective of this paper is to solve the initial-boundary value problem of the Rayleigh flow problem applied to the system of the plasma to determine the macroscopic parameters such as the mean velocity, shear stress, viscosity coefficient together with the induced electric and magnetic fields. Using the estimated distribution functions, it is of fundamental physical importance to study the irreversible thermodynamic behavior of the diamagnetic and paramagnetic plasma, so that the predictions of the entropic behavior and related thermodynamic functions are investigated. The results are applied to a typical model of laboratory Argon plasma. The results agree with proceeding theoretical studies are illustrated.

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 multi-component plasma. The plasma is initially in absolute equilibrium and the wall is at rest. Then the plate starts to move suddenly in its own plane with a velocity ( -0 t U e α ) along the x-axis ( 0 and U α are constants). Moreover, the plate is considered impermeable, uncharged, and an insulator. The whole system (electrons + ions + neutrals +plate) is kept at a constant temperature. All physical quantities are defined in the nomenclature.
Let the forces e f acting on each electron; be given by [22][23][24]: By taking ( , 0, 0), ( , 0, 0), ( , 0, 0) and (0, 0, ) We assume that V x , E x , B z , and J x are functions of y and t. This choice satisfies Maxwell's equations. The distribution function F (y, c, t) of the particles for the plasma gas can be obtained from the kinetic Boltzmann's equation, which can be written in the BGK (Bhatnager-Gross-Krook) model as: for electrons, Where ( ) The quantities , and n V T α α α are the number density, mean drift velocity, and effective temperature obtained by taking moments of F α .
The particles are reflected from the plate with a full velocity accommodation i. e., the plasma particles are reflected with the plate velocity so that the boundary conditions are: and x V is finite as y → ∞ for electrons [5].
for electrons, where ee ν , ei ν and en ν are electron-electron, electron-ion, and neutral-electron collision frequencies respectively, which are given by [23][24]31 where De Di Log n π λ Λ = and Z are the Coulomb Logarithms and the degree of ionization respectively.
The model of the cone of influence suggested by Lee's moment method [25] for the solution of the Boltzmann's equation is employed here. Let us write the solution of Eq. (4), as suggested by Kashmarov [7] in the form: The integrals over the velocity distance are evaluated from the relation [8][9][10], , and where , , x z We introduce the dimensionless variables defined by: For ≪ 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, i. e., = 1 + ( ) and. = 1 + ( ) Let Where xy P is the shear stress [5,7] with the initial and boundary conditions: and are finite as y , .
x xy 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) [22]: We can reduce the basic Eqs. (17) - (20), after simple algebraic manipulations to a single equation:

Solution of the Initial-Boundary Value Problem
We will use the traveling wave solution method [28][29] considering ly mt Such that 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 [28][29]. From Eq. (23) where a is a positive integer. Substituting from Eqs. (23)(24) into Eqs. (22) to get: The boundary and initial conditions become: Now we have an ordinary differential Eq. (25) with the boundary and initial conditions (26).
Exactly the help of symbolic computer software, with their boundary and initial conditions (26), can solve the ordinary fourth order homogeneous differential Eq. (25). The sought solutions will be applied to a typical model of laboratory Argon plasma.

The Non-Equilibrium Thermodynamic Predictions of the Problem
The application of Onsager's principle to a thermodynamic system as a whole or to an individual part of it (locally) is based on a representation of the entropy production as a finite sum of products of fluxes and forces. In addition, on linearity of the description of the nonequilibrium state of the system or its parts, this gives a linear connection between the fluxes and the forces. The symmetry of the coefficients of this connection constitutes Onsager's principle [11][12][13][14][15]. Local application of the principle to a fluid particle of the plasma gas requires its state to be near local equilibrium and its description by a finite set of macroscopic parameters. The problems of the thermodynamics of irreversible processes continue to present great importance. This is associated both with the general theoretical importance of this theory and its numerous applications in various branches of science. Starting from the essentials of the H-theorem, we begin with the evaluation of the entropy per unit mass S . It is written in dimensionless form as [11][12][13][14][15] The law of entropy production [8,28] is written in the local form as: Following the general theory of irreversible thermodynamics we could estimate the thermodynamic force [6,8,10] corresponding to the plate Mach number, presented as a time dependent controlling parameter On the other hand, the relationship between the entropy production, thermodynamic fluxes and forces has the form [8,10]: Near the thermodynamic equilibrium, the following linear relation between the fluxes and the forces holds: In this study, this will be satisfied by the coefficient To study the internal energy change for the system we introduce the extended Gibbs relation [32]. It includes the electromagnetic field energy as a part of the whole energy balance, which distinguish the charged gas into paramagnetic and diamagnetic ones. According to whether if there are unpaired electrons in the molecular orbital diagram, the gas is paramagnetic, or if all electrons are paired, the gas is considered as a diamagnetic one. To get the work term in the first law of thermodynamics, we should write the internal energy balance including the electromagnetic field energy as follows: A) For paramagnetic plasma; the internal energy change is expressed in terms of the extensive quantities S, P and M , which are the thermodynamic coordinates corresponding to the conjugate intensive quantities T, E and B respectively. The three contributions in the internal energy change in the Gibbs formula: where dU S =TdS is the internal energy change due to variation of the entropy, dU pol =E dP is the internal energy change due to variation of polarization, and dU para =B dM is the internal energy change due to the variation of magnetization, here m is calculated from the equation [3,29] Introducing the dimensionless variables in the Gibbs formula to get (after dropping the primes) 1 1 B) On the other hand, if the plasma is diamagnetic; the internal energy change due to the extensive variables S, P and B represent 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: where dU dia = -M dB is the internal energy change due to the variation of the induced magnetic induction, where Hence, the dimensionless form for dU in this case takes the form: 1 1

Results and Discussions
In this problem, the unsteady behavior of a rarefied electron gas is studied based on the kinetic theory of irreversible processes using the exact traveling wave analytical solution method via the BGK model of the Boltzmann equation with the exact value of electro-electron collision frequency. Those computations are performed according to typical data for electron gas in Argon plasma [32,33,34] as a paramagnetic medium in the case where the argon gas loses single electrons or as a diamagnetic medium in the case where the argon gas loses electron pairs, depending on the ionizing potential applied to the argon atoms. The following conditions and parameters apply: All the variables of the problem satisfy the initial and the boundary conditions, Eq.(21), of the problem, see Figures (2,3,5,6). This cannot be exactly examined in the previous study [6], because of the discontinuity in the. Figure (1) shows that the deviation from equilibrium is small and in a course of time the perturbed velocity distribution functions 1 2 and F F approach to equilibrium velocity distribution function 0 F as (y=0.001), i. e. in the region near the moving plate, which represents the region that we are interested for studying. This gives a good agreement with the famous Le Chatelier principle. Figure (2) illustrated that, the mean velocity of electrons, near the moving plate, has a maximum value equal the Mach number (=Ma) of the plate, which satisfies the condition of the problem. The behavior of the shear stress compatibles with the behavior of the velocity itself, see Figure (3).
The viscosity coefficient [30] for a rarefied gas in a slaw fellow is xy x V y τ µ = ∂ ∂ . Its behavior is represented in Figure (4), which shed light upon the resistant to the motion, which increases with time. This is because if any change is imposed on a system that is in equilibrium then the system tends to adjust to a new equilibrium counteracting the change. Near the moving plate, Figures (5, 6) indicate the behavior of the self-generated fields. The induced electric field decreases with time while the induced magnetic field increases with it adjacent to the plate. The induced electric field increases with time while the induced magnetic field decreases with it far from the plate. Clearly, they satisfy the conditions of the problem.
The entropy S increases with time, which gives a good agreement with the second law of thermodynamics, see Figure  ( Upon passing through a plasma, a charged particle (electron) looses (or gains) part of its energy because of the interaction with the surroundings because of plasma polarization and collisions [35]. 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 particle itself, since the suddenly moving plate causes work to be done on the gas, changing the internal energy of the gas U. As seen in Figures (10-13), the change in the internal energy due to the variation of entropy and paramagnetic is smoothly dampened with time by energy lost to and gained from the ions and plate, respectively. While the change in internal energy increases gradually with time because of the intensive variables, corresponding to either polarization or diamagnetic plasma.

Conclusions
The solution of the unsteady BGK Boltzmann kinetic equation in the case of a rarefied electron gas using the method of the moments of the two-sided distribution function together with Maxwell's equations is developed within the travilling wave exact solution method and the exact value of electronelectron, electron-ion and electron-neutral collision frequencies. This solution allows for the calculation of the components of the velocity of the flow. By inserting them into the suggested twosided distribution functions and applying the Boltzmann Htheorem, we can evaluate the entropy, entropy production, thermodynamic force, and kinetic coefficient. Via Gibbs' equations, the ratios between the different contributions of the internal energy change are evaluated based upon the total derivatives of the extensive parameters. The predictions, estimated using Gibbs' equations, reveal 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:

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 F 0 The local Maxwellian distribution unction F 1 Distribution function for going downward particles c y < 0 F 2 Distribution function for going upward particles c y >0 J The current density J y (S) The entropy flux component along y-axis direction K B Boltzmann constant (Erg/K°) 1.38 10 -16 L 11 The kinetic coefficient 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 U 0 Plate initial Velocity V x The mean velocity V x1 The mean velocity related to F 1 V x2 The mean velocity related to F 2 V Gas volume Te V Thermal velocity of electrons Ti V Thermal velocity of ions X 11 Thermodynamic force c 0 The speed of light c The velocity of the particles d particle diameter e The electron charge The mean velocity of the particle dU S The internal energy change due to the variation of entropy dU Pol The internal energy change due to the variation of polarization dU par The internal energy change due to the variation of magnetization dU dia 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 eq Equilibrium Greek letters τ The relaxation time τ xy The shear stress µ Viscosity coefficient λ The mean free path α 0 Dimensionless parameter β Damping constant ν Collision frequency ε Mass ratio λ mean free path λ D Debye radius