Kinetic treatment for the exact solution of theunsteady Rayleigh flow problem of a rarefiedhomogeneous charged gas bounded by anoscillating plate

T.Z. AbdelWahid and S.K. Elagan

Abstract: The exact solution of the unsteady Rayleigh flow problem of a rarefied charged gas bounded by an oscillatingplate has been made. For this purpose, we use the traveling wave solution method. The kinetic and the irreversible thermo-dynamic properties of the charged gas are presented from the molecular viewpoint. Our study is based on the solution ofthe Bhatnager–Gross–Krook (BGK) model of the Boltzmann kinetic equation, with the precision value of the electron–electron collision frequency. The BGK model equation coupled with Maxwell’s equations, for electron gas near anoscillating rigid plane, are solved. The distinction and comparisons between the perturbed and the equilibrium velocitydistribution functions are illustrated. The ratios between the different contributions of the internal energy changes arepredicted via the extended Gibbs equation for both diamagnetic and paramagnetic plasmas. The results are applied to atypical model of laboratory argon plasma.

PACS Nos: 47.45.–n, 05.60.–k, 05.70.–a, 52.25.Kn, 52.25.Fi, 51.10.+y, 52.27.Aj, 52.20.Fs, 47.45.Ab

Résumé : Nous avons obtenu la solution exacte du problème de l’écoulement cavitant de Rayleigh d’un gaz chargé (plasma)de faible densité cantonné par des surfaces planes oscillantes. À cette fin, nous utilisons la méthode de solution en ondesprogressives. Les propriétés cinétiques et thermodynamiquement irréversibles sont traitées d’un point de vue moléculaire.Notre étude est basée sur la solution Bhatnager–Gross–Krook (BGK) de l’équation cinétique de Boltzmann, avec une valeurprécise de la fréquence de collision électron–électron. Nous solutionnons l’équation du modèle BGK couplée à l’équationde Maxwell pour le gaz d’électrons près de la surface oscillante. Nous illustrons la distinction et la comparaison entre lesfonctions de distribution de vitesse perturbée et à l’équilibre. À l’aide des équations généralisées de Gibbs pour des plasmasdiamagnétiques et paramagnétiques, nous prédisons les rapports entre les différentes contributions aux changements de l’é-nergie interne. Ces résultats sont appliqués à un modèle typique de plasma d’argon en laboratoire.

[Traduit par la Rédaction]

Introduction

Plasma (charged gas) based technology underpins some ofthe world’s largest industries producing a significant propor-tion of the world’s global commercial products includingcomputers, cell phones, automobiles, airplanes, paper, andtextiles [1, 2]. Foremost among these is the electronics indus-try, in which plasma-based processes are indispensable forthe manufacture of ultra large-scale integrated microelec-tronic circuits. The Boltzmann equation represents a well-defined model to describe the motion of a rarefied chargedgas (plasma), when microscopic effects must be considered.This is the case of an electron flow in microelectromechan-ical systems (MEMS).The choice of a physical model to adequately describe a

gas flow depends on the flow regime. A rarefied gas may bedivided into several different flow regimes in accordance

with its level of rarefaction as quantified by the Knudsennumber, Kn, defined as the ratio between the mean free pathand a characteristic length scale of the gas flow. A signifi-cantly large number of flows may be classified within thetransition regime (0.1 < Kn < 10). Relevant applicationswithin a rarefied gas include upper atmospheric simulations[3], including the Space Shuttle Orbiter, the Magellan Space-craft, the Stardust Sample Return Capsule, and the Mars Path-finder. Additional relevance outside that of upper-atmosphericstudies includes chemical vapor deposition [4], the microfilter [5], and MEMS [6–9]. Traditionally, the Boltzmannequation, based on kinetic theory, remained the only appro-priate option for the solution of these rarefied gases.The applications of the Boltzmann equation in MEMS

technology are numerous. There are some important factorsthat increase the requirement of the employment of the Boltz-mann kinetic equation in the MEMS applications as:

Received 25 April 2012. Accepted 31 July 2012. Published at www.nrcresearchpress.com/cjp on 7 September 2012.

T.Z. AbdelWahid. Mathematics Department, Faculty of Science and Arts, King Khalid University, Mahail City, Kingdom of SaudiArabia.S.K. Elagan. Mathematics and Statistics Department, Faculty of Science, Taif University, P.O. 888, Kingdom of Saudi Arabia.

Corresponding author: T.Z. AbdelWahid (e-mail: taha_zakaraia@yahoo.com).

987

Can. J. Phys. 90: 987–998 (2012) doi:10.1139/P2012-095 Published by NRC Research Press

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i. The Boltzmann equation is valid for studying theflows in all ranges of the Knudsen number (i.e., thecontinuum (Kn < 0.01), slip (0.01 < Kn < 0.1),transition (0.1 < Kn < 10), and free molecular re-gimes (10 < Kn) [10, 11]), while the Chapman–Enskog method proved the validity of the Navier–Stokes equations as a limiting case correspondingto small Knudsen numbers in the continuum regime.There were some restrictions on macroscopic equa-tions when considering rarefied gas regimes. The at-tempts to construct rarefied gas dynamics similar tothe macroscopic equations of hydrodynamics, with-out using multidimensional phase space, were un-successful. The need to study the Boltzmann kineticequation itself was obvious.

ii. The Knudsen numbers of flow in MEMS are usuallyfar from the continuum regimes because of their micro-scale size, which is often comparable to the mole-cule mean free path under standard working condi-tions. The kind of flow under micro-scale is called amicroflow. In microflows, the characteristic lengthsof the flow gradient are usually very small andcomparable to the mean free path of the molecules.Typically, the characteristic lengths of most MEMSare at or below the order of a micrometre and resultin a Knudsen number between 0.001 and 10, so thatthe flows in most MEMS are in the slip flow andtransition flow regimes. Microflows in these regimeshave different features than normal flows with largecharacteristic lengths. The conventional continuummodels, unlike the Boltzmann equation, fail to de-scribe and predict microflows. On the other hand,microflows play a role in determining the perfor-mances of MEMS, such as micropressure sensors,micropumps, and valves.

iii. The low-speed flows can often be described by theReynolds equation, a simplified form of the Navier–Stokes equations, with negligible convective terms[9]. The Reynolds equation is often used to describefluidic effects in microsystems with gas confined inlong gaps. However, the Reynolds and Navier–Stokes description breaks down when the character-istic size decreases and the flow transitions to therarefied regime. The Boltzmann equation is a gen-eral form of the gas transport equation based on thekinetic theory and can be reduced to Navier–Stokesequations in the near-continuum limit.

For planar flows, the oscillating Couette problem hasmany analogies with Stokes’ second problem, which involvesa flat plate oscillating in an unbounded medium. This prob-lem was first solved by Stokes [12] in 1851 and later by Ray-leigh [13] in 1911. Stokes’ second problem is one of thesimplest configurations to study the behavior of a nonequili-brium gas responding to a plate oscillating in its own plane[14]. A comprehensive study has been carried out with thelinearized Bhatnagar–Gross–Krook (BGK) equation over awide range of Knudsen numbers [15]. The effect of the oscil-lation frequency on the amplitudes and phases of the velocityand shear stress were reported. To improve our understandingof time-periodic, shear driven gas flows, oscillatory Couetteflow provides an ideal test case and has been studied exten-

sively using kinetic theory [16–20]. A brief comparison ofthe linearized R13 (LR13) equations with DSMC data [18]were carried out by Taheri et al. [21]. Although oscillatoryCouette flow is a simple case, it has many related applica-tions in a variety of MEMS devices, for example, the Tangresonator [22]. A proper understanding of flow phenomenain resonators will therefore help to improve the performanceand quality factor [23]. In this study, the extended continuumgoverning equations are employed to compute the details ofStokes’ second problem and oscillatory planar Couette flowin the early transition regime. The purpose of this paper is toanalyze and assess the dynamic response capability of thehigher-order moment equations for oscillatory flow and com-pare against the available kinetic data. Gu and Emerson [14]presented results using three different continuum-based mod-els to study oscillatory flow in the transition regime. Dataobtained from numerical solutions of the Boltzmann equationand the direct simulation Monte Carlo method, are used toassess the ability of the continuum models to capture impor-tant rarefaction effects.The behavior of an ordinary (or rarefied) charged gas in

the vicinity of an infinite flat plate oscillating or moving inits own plane — the Rayleigh flow problem — which is con-sidered in the present study, is of great interest in laboratoryexperiments and aerodynamics [24]. It is important to deter-mine the role governing the motion due to the collision ofthe molecules with solid surfaces and the binary collision be-tween the molecules themselves due to the rarefaction of thegas. Therefore, discontinuities in the macroscopic parametersat the surfaces are expected. Shidlovskiy, El-Sakka et al., andKhater and El-Sharif [25–27], among other investigators,studied the Rayleigh flow problem for a highly rarefied gasof a homogeneous system of charged particles in the frame-work of the kinetic theory of gases using the collisionlessBoltzmann equation, and they examined the dynamical andelectromagnetic behavior of the gas.Abourabia and AbdelWahid [24], in the framework of irre-

versible thermodynamics, examined the characteristics of theRayleigh flow problem of a rarified electron gas extractedfrom neutral atoms and proved that it obeys the entropic be-havior for gas systems but with an approximate solution andinaccurate formula of the collision frequency. The enhance-ment and improvement of this study is done in ref. 2, wherethe exact solution and accurate formula of electron–electroncollision frequency are successfully introduced.Kinetic theory has contributed not only to the understand-

ing of nonequilibrium transport phenomena in gases, but alsoto the development of general nonequilibrium statisticalphysics. It is well accepted that the Boltzmann equation [28–35] is one of the most reliable kinetic models for describingnonequilibrium phenomena in gas phase. Following its suc-cess and usefulness, the Boltzmann equation is widely usedto describe various gas-phase transport phenomena, such asplasma gases, granular gases, polyatomic gases, relativisticgases, and chemically reacting gases [36–39]. The kineticequation of gas flow based on the Boltzmann equation, hasobvious peculiarities in comparison with the macroscopic de-scription found using the Navier–Stokes equations [24, 28].Because the full Maxwell–Boltzmann equation requires moreeffort to solve, various approximations have been suggested[40], such as the Chapman–Enskog procedure, Krook’s

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model, and Lee’s moment method [41–45] for the solution ofBoltzmann’s equation. Lees et al. [44] applied the two-sidedmaxwellian distribution to the problem of a conductive heattransfer. Khater and El-Sharif [45] used a two-stream max-wellian distribution function with two unknown parameterscorresponding to the mean velocity and the shear stress toobtain an approximate analytical solution of the Rayleighflow problem for a rarefied gas of an inhomogeneous systemof charged particles.The goals of this study are the following. (i) Obtaining an

analytical exact solution of the unsteady Boltzmann equationof a rarefied electron gas filling the upper half of the space.The rarefied electron gas is bounded by an infinite flat platethat suddenly oscillates on its own plane, producing self-generated magnetic and electric fields. (ii) Illustrating theeffects of the collisions of electrons–electrons on the formof the distribution function. In addition, (iii) indicating thenonequilibrium thermodynamic behavior of the system. Forthis propose, we solve the initial-boundary value problem ofthe Rayleigh flow problem applied to the system of thecharged gas (electrons) to determine macroscopic parame-ters, such as the mean velocity, shear stress, viscosity coef-ficient, together with the induced electric and magneticfields. Using the estimated distribution functions, it is of fun-damental physical importance to study the irreversible ther-modynamic behavior of the diamagnetic and paramagneticelectron gas, so that the predictions of the entropic behaviorand related thermodynamic functions are investigated. Theresults are applied to a typical model of laboratory argonplasma.

2. The physical problem and mathematicalformulationConsider a rarefied gas of charged particles of electrons

and ions occupying a semi-infinite space (y ≥ 0). An infiniteflat plate fixed at (y = 0), and parallel to the xz-plane oscil-lates harmonically in the x-direction with frequency a, that is,the velocity of the plate depends on the time t as

Vw ¼ Re½U0 expð�iatÞ� ¼ U0 cosðatÞwhere the symbol Re denotes the real part of complex expres-sion. The quantity U0 is the velocity amplitude, which is as-sumed to be small when compared with the thermalmolecular velocity VTe of the gas. Because the ratio betweenelectron and ion masses and electron and neutral atom massis small (me/mi ≪ 1, me/mN ≪ 1) in ionized gases, the ionsand neutrals will be considered an immobile neutralizingbackground. The charged gas is initially in absolute equili-brium and the wall is at rest. Then the plate starts to oscillatesuddenly in its own plane with a velocity (U0 cos(at)) alongthe x-axis (U0 and a are constants). Moreover, the plate isconsidered impermeable, uncharged, and an insulator. Thewhole system (electrons + ions + plate) is kept at a constanttemperature. All physical quantities are defined in the no-menclature.Let the force fe acting on each electron; be given by [46–49]

fe ¼ �eE� e

coðc^BÞ ð1Þ

By taking

V � ðVx; 0; 0Þ J � ðqnVx; 0; 0Þ E � ðEx; 0; 0Þand B � ð0; 0; BzÞ ð2ÞWe assume that Vx, Ex, Bz, and Jx are functions of y and t.

This choice satisfies Maxwell’s equations. The distributionfunction Fe(y, c, t) of the particles for the electron gas canbe obtained from the kinetic Boltzmann’s equation, whichcan be written in the BGK model [50] as

@Fe

@tþ c � @Fe

@rþ feme

� @Fe

@c¼ neeðF0e � FeÞ ð3Þ

where

F0e ¼ nð2pRTÞ�3=2 exp�ðc� VÞ2

2RT

� �

The quantities n, V, and T are the number density, meandrift velocity, and effective temperature obtained by takingmoments of Fe.In addition, we assume that the particles are reflected from

the plate with a full velocity accommodation, that is, thecharged gas particles are reflected with the plate velocity sothat the boundary conditions are taken as:

Vx2ð0; tÞ ¼ U0 cosðatÞ for t > 0

where Vx2 = Vx as cy > 0 and Vx is finite as y → ∞.Substituting from (1) and (2), into (3) one obtains

@Fe

@tþ cy

@Fe

@y� eBz

mec0cy

@Fe

@cx� cx

@Fe

@cy

� �

þ eEx

me

@Fe

@cx¼ neeðF0e � FeÞ ð4Þ

where nee is the electron–electron collision frequency, whichis given by [51–54]

nee ¼4ffiffiffip

pne4 log½Lee�

3ffiffiffiffiffiffime

pk3=2B T3=2

ð5Þ

where log½Lee� ¼ log½4pnl3De�, Z, and lDe are the Coulomblogarithms, the degree of ionization, and Debye radius, re-spectively.The moment method [55] for the solution of Boltzmann’s

equation is employed here. One of the most important advan-tages of this method is that the surface boundary conditionsare easily satisfied. Maxwell converted the Maxwell–Boltzmann equation into an integral equation of transfer, ormoment equation, for any quantity Qj(c) that is a functiononly of the molecular velocity. The distribution functionused there should be considered a suitable weighting func-tion, which is not the exact solution of the Maxwell–Boltzmann equation in general. Lees found that the distribu-tion function employed in Maxwell’s moment equation mustsatisfy the following basic requirements: (i) it must have the“two-sided” character that is an essential feature of highlyrarefied gas flows; (ii) it must be capable of providing asmooth transition from free molecule flows to the contin-uum regime; and (iii) it should lead to the simplest possibleset of differential equations and boundary conditions consis-tent with conditions i and ii.

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Let us write the solution of (4), in the form [25]

Fe ¼F1 ¼ nð2pRTÞ�3=2 1þ cxVx1

RT

� �exp

�c2

2RT

� �for cy < 0

F2 ¼ nð2pRTÞ�3=2 1þ cxVx2

RT

� �exp

�c2

2RT

� �for cy > 0

8>>>><>>>>:

ð6Þ

where Vx1 and Vx2 are two unknown functions of time t andthe single distance variable y.The solution of the Boltzmann equation is difficult, and

the velocity distribution function F is not directly of interestto us at this stage, but the moments of the distribution func-tion are of interest. Therefore, we derive Maxwell’s momentequations by multiplying the Boltzmann equation by a func-tion of velocity Qj(c) and integrating over the velocity space.How many and what forms of Qj(c) will be used is dependenton how many unknown variables need to be determined andhow many equations need to be solved.Using Grad’s moment method [41] multiplying (4) by Qj(c)

and integrating over all values of c, we obtain the transferequations in the form

@

@t

ZQjFedc þ @

@y

ZcyQjFedc þ eEx

me

ZFe

@Qj

@cxdc

� eBz

mec0

Zcx

@Qj

@cy� cy

@Qj

@cx

� �dc ¼ nee

ZðF0e � FeÞQjdc ð7Þ

The integrals over the velocity distance are evaluated fromthe relation [25]

ZQjðcÞFdc ¼

Z1�1

Z0�1

Z1�1

QjF1dcþZ1�1

Z10

Z1�1

QjF2dc ð8Þ

where Qj = Qj(c), j = 1, 2, and dc = dcxdcydcz, where cx, cy,and cz are the particles’ velocity components along x-, y-, andz-axes, respectively. Moreover, Ex and Bz may be obtainedfrom Maxwell’s equation

@Ex

@y� 1

c0

@Bz

@t¼ 0 ð9Þ

@Bz

@y� 1

c0

@Ex

@t� 4pen

c0Vx ¼ 0 ð10Þ

where

n ¼ZFdc nVx ¼

ZcxFdc

with the initial and boundary conditions

Exðy; 0Þ ¼ Bzðy; 0Þ ¼ 0

Exðy; tÞ and Bzðy; tÞ are finite as y ! 1 ð11Þ

We introduce the dimensionless variables defined by

t ¼ t0tee y ¼ y0teeVTeffiffiffiffiffiffi

2pp

!Vx ¼ V 0

xVTe

txy ¼ t 0xyVTe Ma ¼ U0

VTe

Bz ¼ B0z

ffiffiffiffiffiffi2p

pmec0

etee

!

Ex ¼ E0x

meVTe

etee

� �r ¼ nm VTe ¼

ffiffiffiffiffiffiffiffiffiffiffiffi2kBTe

me

r

3 ¼ me

midU ¼ dU 0ðkBTeÞ and

Fj ¼ F 0j neð2pRTeÞ�ð3=2Þ j ¼ 0; 1; 2 ð12Þ

For Ma2 ≪ 1 (low Mach number), we can assume that thedensity and the temperature variation at each point of theflow and at any time are negligible, that is, n = 1 + O(Ma2)and T = 1 + O(Ma2). Let

Vx ¼ 1

2ðVx1 þ Vx2Þ

txy ¼ Pxy

rU0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiRTe=2p

p ¼ ðVx2 � Vx1Þ ð13Þ

where Pxy is the shear stress [42] defined by

Pxy ¼ m

Zðcx � VxÞcyFdc

We take Q1 = cx and Q2 = cxcy, and substitute (6) into (7),using the dimensionless variable defined in (12). After drop-ping the bars, we get the following equations (neglecting thedisplacement current) [56]:

@Vx

@tþ @txy

@y� Ex ¼ 0 ð14Þ

@txy

@tþ 2p

@Vx

@yþ txy ¼ 0 ð15Þ

@Ex

@y� @Bz

@t¼ 0 ð16Þ

@Bz

@y� a0Vx ¼ 0 ð17Þ

where

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a0 ¼ V2Tet

2eee

2n

mec20

With the initial and boundary conditions

Vxðy; 0Þ ¼ txyðy; 0Þ ¼ Exðy; 0Þ ¼ Bzðy; 0Þ ¼ 0

2Vxð0; tÞ þ txyð0; tÞ ¼ 2Ma cosðb1tÞ for t > 0

Vx; txy; Ex; andBz are finite as y ! 1ð18Þ

We can reduce basic equations (14)–(17), after simple al-gebraic manipulations to a single equation

@4Vxðy; tÞ@t2@y2

� 2p@4Vxðy; tÞ

@t4� @3Vxðy; tÞ

@t@y2

� a0@2Vxðy; tÞ

@t2� a0

@Vxðy; tÞ@t

¼ 0 ð19Þ

3. Solution of the initial-boundary valueproblemWe will use the traveling wave solution method [57, 58]

considering

x ¼ ly� mt ð20Þsuch that all the dependent variables are functions of x. Herel and m are transformation constants that do not depend onthe properties of the fluid but are parameters to be deter-mined by the boundary and initial conditions [58]. From(20) we get the derivatives

@

@t¼ �m

@

@x

@

@y¼ l

@

@xand

@a

@ta¼ ð�1Þama @a

@xa@a

@ya¼ la

@a

@xað21Þ

where a is a positive integer.

Fig. 1. The comparison between the combined perturbed velocity distribution functions Fe [F1 (gray), F2 (black)] and electrons equilibriumvelocity distribution function Fo (grid) at (t = 0.001, 12.5, and 25) for a fixed y value (2.5) with the Mach number of the plate Ma = 0.033.

Fig. 2. The velocity Vx versus space y and time t.

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Substituting from (20) and (21) into (19) to get

m2l2 � 2pm4� � d4VxðxÞ

dx4þ ml2

d3VxðxÞdx3

� a0m2 d

2VxðxÞdx2

þ a0mdVxðxÞdx

¼ 0 ð22Þ

The boundary and initial conditions become

Exðx ¼ 0Þ ¼ Bzðx ¼ 0Þ ¼ txyðx ¼ 0Þ ¼ 0

2Vxðx ¼ �mÞ þ txyðx ¼ �mÞ ¼ 2Mp cosðb1Þat y ¼ 0 e:g:; t ¼ 1

Vx; txy; Ex; andBz are finite as x ! �1 ð23Þ

Now, we have an ordinary differential (22) with the boun-dary and initial conditions (23).The ordinary fourth-order homogeneous differential (22),

can be successfully solved exactly, with the help of symboliccomputer software, with their boundary and initial conditions(23). The sought solutions will be applied to a typical modelof laboratory argon plasma.

4. The investigation of the behavior of theinternal energy change

The studying of the behavior of the internal energy changefor the physical systems, presents a great importance in sci-ence. To study the internal energy change for the system,

Fig. 3. The shear stress txy versus space y and time t.

Fig. 4. The viscosity coefficient m versus space y and time t.

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based on the solution of the nonstationary Boltzmann equa-tion [59], we introduce the extended Gibbs relation [2, 24,60]. It includes the electromagnetic field energy as a part ofthe whole energy balance, which distinguishes the chargedgas into paramagnetic and diamagnetic ones. If there are un-paired electrons in the molecular orbital diagram, the gas isparamagnetic, and if all electrons are paired, the gas is con-sidered a diamagnetic one. To get the work term in the firstlaw of thermodynamics, we should write the internal energybalance including the electromagnetic field energy as follows.

4.1 Paramagnetic plasmaThe internal energy change is expressed in terms of the ex-

tensive quantities S, P, and M, which are the thermodynamic

coordinates corresponding to the conjugate intensive quanti-ties T, E, and B, respectively. The three contributions in theinternal energy change in the Gibbs formula

dU ¼ dUS þ dUpol þ dUpar ð24Þwhere dUS = TdS is the internal energy change due to variationof the entropy S, where the entropy per unit mass S [15, 23, 52]

S ¼ �p3=2 V2x1 þ V2

x2

� �� 3

2

� �ð25Þ

where dUpol = EdP is the internal energy change due to var-iation of polarization P, and dUpar = BdM is the internal en-ergy change due to the variation of magnetization, here M iscalculated from the equation [24, 60]

Fig. 5. The induced electric fiel Ex versus space y and time t.

Fig. 6. The induced magnetic field Bz versus space y and time t.

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@S

@M¼ �B

T) M ¼ �

ZT

B

@S

@y

� �t

dy ð26Þ

Introducing the dimensionless variables

U 0 ¼ U

KTM 0 ¼ M

1

eteeVTe

� �P0 ¼ P

1

eteeVTe

� �

in the Gibbs formula to get (after dropping the primes)

dU ¼ dSþ f1EdPþ f1BdM ð27Þ

4.2 Diamagnetic plasmaOn the other hand, if the plasma is diamagnetic; the inter-

nal energy change due to the extensive variables S, P, and Brepresent the thermodynamic coordinates conjugate to the in-tensive quantities T, E, and M, respectively, therefore, wehave three contributions in the internal energy change in theGibbs formula given by

dU ¼ dUS þ dUpol þ dUdia ð28Þ

Fig. 7. Internal energy change dUS versus space y and time t.

Fig. 8. Internal energy change dUpol versus space y and time t.

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where dUdia = –Mdb is the internal energy change due to thevariation of the induced magnetic induction, where M =T(@S/@B) [24, 60].Hence, the dimensionless form for dU in this case takes

the form

dU ¼ dSþ f1EdP� f1MdB ð29Þwhere

f1 ¼ meV2Te

KT

� �dS ¼ @S

@r

� �dyþ @S

@t

� �dt dy ¼ 5

dt ¼ 5

5. Discussion

In this problem, the unsteady behavior of a rarefied elec-tron gas is studied based on the kinetic theory of irreversibleprocesses using the exact traveling wave analytical solutionmethod via the BGK model of the Boltzmann equation withthe exact value of electron–electron collision frequency. Ourcomputations are performed according to typical data forelectron gas in argon plasma [61] as a paramagnetic mediumin the case where the argon gas loses single electrons or as adiamagnetic medium in the case where the argon gas loseselectron pairs, depending on the ionizing potential applied tothe argon atoms. The following conditions and parameters

Fig. 9. Internal energy change dUdia versus space y and time t.

Fig. 10. Internal energy change dUpar versus space y and time t.

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are applied: kB = 1.3807 × 10–22 J/K, T0 = 1200 K, ne = 7 ×1013 cm–3, d = 3.84 × 10–8 cm (diameter of the argon atom),the electron rest mass and charge me = 9.093 × 10–28 g, e =4.8 × 10–10 esu (1 esu = 0.333 564 1 nC) are used to calculatethe dimensionless parameter a0 = 1.22 × 10–6, the electron–electron collision relaxation time tee = 1.306 39 × 10–11,and the mean free path of the electron gas

l ¼ 1ffiffiffi2

ppned2

¼ 2:180 cm

compared to the electron Debye length

lDe ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiKBT0

4pnee2

r¼ 2:85� 10�5 cm f1 ¼ 2

Using the idea of the shooting numerical calculationmethod, we evaluate the transformation constants to obtainm = 1.3, l = 0.1, and the plate Mach number Ma = 3.310–2.All the variable of the problem satisfies the initial and the

boundary conditions, (18), of the problem (see Figs. 2, 3, 5,and 6).Figure 1 shows that the deviation from equilibrium is

small and over the course of time the perturbed velocity dis-tribution functions F1 and F2 approach equilibrium velocitydistribution function F0e. This is in good agreement with thefamous Le Chatelier principle. Figure 2 illustrated that themean velocity of electrons, near the oscillating plate, has amaximum value equal to the Mach number (= Ma) of theplate, which satisfies the condition of the problem. The neg-ative sign of the shear stress represents its direction, fromdown to up, and the shear stress decreases with time due tothe behavior of the velocity itself, see Fig. 3.The viscosity coefficient [62] for a rarefied gas in a slaw

flow is m = txy/(@Vx/@y). Its behavior is represented inFig. 4, which shed light upon the resistance to the motion,which gradually increases with time. This is due to the factthat, if any change is imposed on a system that is in equili-brium, then the system tends to adjust to a new equilibriumcounteracting the change. This gives an agreement with theLe Chatelier principle. Near the oscillating plate, Figs. 5 and6 indicate the behavior of the self-generated fields. The in-duced electric field increases with time while the inducedmagnetic field decreases. Clearly, they satisfy the conditionsof the problem.Upon passing through a plasma, a charged particle (elec-

tron) loses (or gains) part of its energy because of the inter-action with the surroundings because of plasma polarizationand collisions [63]. The energy loss (or gain) of an electronis determined by the work of the forces acting on the elec-trons in the plasma, by the electromagnetic field generatedby the moving particle itself (because the suddenly oscillatingplate causes work to be done on the gas), and this produces achange in the internal energy of the gas U. As seen inFigs. 7–10, the change in the internal energy due to the var-iation of entropy and paramagnetism is smoothly dampenedwith time by energy lost to and gained from the ions andplate, respectively. While the change in internal energy in-creases gradually with time because of the intensive varia-bles, corresponding to either polarization or diamagneticplasma.

6. Conclusion

The solution of the unsteady BGK Boltzmann kineticequation in the case of a rarefied electron gas using themethod of the moments of the two-sided distribution functiontogether with Maxwell’s equations is developed within thetraveling wave exact solution method and the exact value ofelectron–electron collision frequency. Tackling this allowsone to calculate the components of the flow velocity. Insert-ing the velocity components into the suggested two-sided dis-tribution functions and applying the Boltzmann H-theorem,we can evaluate the entropy, entropy production, thermody-namic force, and kinetic coefficient. Via Gibbs’ equations,the ratios between the different contributions of the internalenergy change are evaluated based upon the total derivativesof the extensive parameters. The predictions, estimated usingGibbs’ equations, reveal the following order of maximum nu-merical magnitude ratios between the different contributionsto the internal energy change, based on the total derivativesof the extensive parameters:

dUS : dUpol : dUdia ¼ 1 : 2:5� 10�1 : 3:1� 10�4

dUS : dUpol : dUpar ¼ 1 : 2:5� 10�1 : 3:7� 10�4

It is concluded that the effect of the changes in the internalenergies dUpol, dUdia, and dUpar due to electric and magneticfields are very small in comparison with dUS, in recognitionof the fact that these fields are self-induced by the suddenmotion of the plate.

AcknowledgmentThe authors would like to thank F.M. Sharipov for com-

ments and criticism.

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List of symbols

B induced magnetic field vectorB induced magnetic fieldBz induced magnetic field in the z-directionBz′ dimensionless induced magnetic field in the z-directionc0 speed of lightcx speed of the particles in the x-directioncy speed of the particles in the y-directiond particle diameter

dUdia internal energy change due to the variation of the in-duced magnetic field

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dUpar internal energy change due to the variation of magneti-zation

dUPol internal energy change due to the variation of polariza-tion

dUS internal energy change due to the variation of entropye electron chargeE induced electric vectorE induced electric fieldEx induced electric field in the x-directionEx′ dimensionless induced electric field in the x-directionF velocity distribution function

F0e local maxwellian distribution function for electronsF1 distribution function for descending particles cy < 0F2 distribution function for ascending particles cy > 0Fe velocity ditribution function for electronsFj velocity ditribution functionFj′ dimensionless velocity ditribution functionfe Lorentz’s force vectorJ current density vectorJx current density in the x-directionKB Boltzmann constant 1.3807 × 10–22 J/KKn Knudsen numberM specific magnetizationM′ dimensionless specific magnetizationMa plate Mach numberme electron massmi ion massmN neutral massn mean densityne electron concentrationP polarizationP′ dimensionless polarizationPxy shear stress

Qj(c) a function only of the molecular velocityq charge

R gas constantRe gas constant for electronsr position vector of the particleS entropy per unit massT temperatureTe electron temperaturet time variablet′ dimensionless time variableU′ dimensionless internal energy of the gasU0 Plate initial VelocityVTe electron thermal velocityVx mean velocityVx′ Dimensionless mean velocityVx1 mean velocity related to F1Vx2 mean velocity related to F2y displacement variabley′ dimensionless displacement variableZ degree of ionizationa frequencya0 dimensionless parameterb1 dimensionless frequency3 mass ratio

Lee Coulomb logarithml mean free path

lDe Debye radius of electronsm viscosity coefficientn electron–electron collision frequencyx wave parameterr densityt relaxation time

tee electron–electron relaxation timetxy shear stresstxy′ dimensionless shear stress

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