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

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.


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][4][5][6][7][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][10][11][12][13][14][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 twocomponent 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.

The Physical Problem and Mathematical Formulation
Let us assume that the upper half of the space ( ≥ 0), which is bounded by an infinite flat plate ( = 0), is filled with a nonhomogeneous mixture of charged particles of electrons and positive ions.An infinite flat plate is fixed at  = 0, and parallel to the -plane.The plate oscillates harmonically in the  direction with frequency ; that is, the velocity of the plate depends on the time  as follows: where the symbol   denotes the real part of complex expression.The quantity  0 is the velocity amplitude, which is assumed to be small when compared with the thermal molecular velocity   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  0 Cos() along the -axis ( 0 and  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 ⃗   and ⃗   acting on each electron and ion respectively be given by [28][29][30] The directions of the considered physical quantities are as follows: where  =  for electrons and  =  for ions.
The   ,   ,   , and   are functions of  and  that satisfie Maxwell's equations.The distribution function (, , ) 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 where and   are the number density, mean drift velocity, and effective temperature obtained by taking moments of   .Some latitude in the definition of   and ⃗   is possible; one choice is in [32] 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  2 (0, ) =  0 Cos() for  > 0, where  2 =   as   > 0 and   is finite as  → ∞ for both ions and electrons.
The model of the cone of influence suggested by Lee's moment method [36][37][38][39][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 where  1 and  2 are four unknown functions of time  and the distance variable .Using Grad's moment method multiplying ( 7) and ( 8) by   ( ⃗ ).Integrating over all values of ⃗ , we can obtain the transfer equations for electrons and ions, respectively, in the form The integrals over the velocity distance are evaluated from the relation where   =   ( ⃗ ),  = 1, 2, and  =       , where   ,   and   are the particles velocities components along -, -, and -axes, respectively.Moreover,   and   may be obtained from Maxwell's equation, for electrons as follows: For ions we obtain where   = ∫   ,     = ∫     , with the initial and boundary conditions (, ) ,   (, ) are finite as  → ∞.
We introduce the dimensionless variables defined by For Ma 2 ≪ 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,   = 1 + (Ma 2 ) and where   is the shear stress [11] that is defined by   =  ∫(  −   )    .
Using the dimensionless variable, if we neglect terms of order (Ma 2 ), (11) for  1 =   and  2 =     is: Similarly, (12) becames: with the initial and boundary conditions In the expressions for the transport coefficients mentioned previously, the fact that the plasma is neutral was used [33], writing  =   =   ,  =   =   , and  = 1, which exploited the fact that the ratio   /  is small.Thus, the ratio between the four distinguished collision frequencies can be rewritten in a form that referred to ]  in formulas ( 9) Therefore, ]  ∼ ]  ≫ ]  ≫ ]  ; the first of these gross inequalities arises because thermal ion speeds are less than thermal electron speeds by the factor √  /  if   ≈   , 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.

Solution of the Initial-Boundary Value Problem
The traveling wave solution method [41,42] is used, considered to make all the dependent variables as functions of .Here  and  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 where  is a positive integer.Substituting from ( 34)-( 35) into (32), we have The boundary and initial conditions became at  = 0, e.g.,  = 1; ,   ,   ,   are finite as  → −∞.
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 with the corresponding boundary and initial conditions 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.

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   ,   , and   , which are the thermodynamic coordinates corresponding to the conjugate intensive quantities   ,   , and   , respectively.The three contributions in the internal energy change in the Gibbs formula we have where    =   is the internal energy change due to the variation of the entropy, which is written in dimensionless form as pol =     is the internal energy change due to variation of polarization, and  para  =     is the internal energy change due to the variation of magnetization, here  is calculated from the equation [44] Introduce the dimensionless variables    =   /  ,    =   (1/      ),    =   (1/      ) in the Gibbs formula to get (after dropping the primes) (b) On the other hand, if the plasma is diamagnetic, the internal energy change due to the extensive variables , , and  represents the thermodynamic coordinates conjugate to the intensive quantities , , and , respectively; therefore, we have three contributions in the internal energy change in the Gibbs formula given by where  dia  = −    is the internal energy change due to the variation of the induced magnetic induction, where   =   (  /  ).Hence, the dimensionless form for  in this case takes the following form: where

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.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  1 and  2 approache to the equilibrium velocity distribution function  0 at  = 0.003, for example.The ions are still departing from equilibrium very slowly; see Figure 1(b).
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 ∼   as the transfer of the energy; hence, ion-electron momentum transfer is small compared with ionion momentum transfer.The transfer of momentum from electrons to ions occurs in a time of the same order ∼   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 (  ∼ 4.35) which will be taken into consideration, henceforth.
In a relative long period, for example,  = 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  ≅ 1380; see Figure 1(c).
All Figures 1(a The   gets a severe decrement with time while   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 √  /  as   ≅   , 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].
The shear stress   is beginning from zero value.It is increasing nonlinearly towards its maximum value at  ∼ 0.5  .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   = −  /(  /) represents the resistance to the motion.It gradually increases nonlinearly as the gas particles move away from the plate for ions   .The   approaches zero for electrons except at a small time interval around  ∼ 0.5  ; this is due to the corresponding maximum value of   in the same time interval.The electrons induced electric field has a sudden increase in the beginning until it reaches its maximum value (  Max ∼ 0.1) due to the sudden oscillation of the plate itself.It decreases nonlinearly until it vanishes at  ∼ 0.5  .After that, it changes its direction until the same maximum value   Max ∼ −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).
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 .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.

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 (  ≅ 4. 35) and ions (  ≅ 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,   /  ≃ (((]  +]  )/2)/]  ).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   :   :  par = 1 : 0.6 × 10 −2 : 0.8 × 10 −4 . ( It is concluded that the effect of the changes of the internal energies for positive ions   and  par due to electric and magnetic fields are small in comparison with   .This happens with the recognition of the fact that these fields are self-induced by the sudden motion of the oscillating plate.

Nomenclature
⃗ : The induced magnetic field vector : The induced magnetic field ⃗ : The induced electric vector : The induced electric field : The velocity distribution function

Figure 1 :
Figure 1: (a) The comparison between the combined perturbed dimensionless velocity distribution functions for electrons   ( 1 (green),  2 (red)) and electrons equilibrium velocity distribution function   (grid) at  = 10 −3 , 2.175, and 4.35 for a fixed  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   ( 1 (green),  2 (red)) and ions equilibrium velocity distribution function   (grid) at  = 10 −3 , 2.175, and 4.35 for a fixed  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   ( 1 (gray),  2 (black)) and ions equilibrium dimensionless velocity distribution function   (grid) at  = 690, 1380 for a fixed  value (0.003) with the Mach number of the plate Ma = 0.065.
) 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.

Figure 6 :
Figure 6: The dimensionless internal energy change versus space  and time : (a)   ; (b)   .

Figure 7 :
Figure 7: The dimensionless internal energy change versus space  and time : (a)   ; (b)   .