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.

A development of the previous paper [

The behavior of charged gases in nonequilibrium states has received considerable attention from the standpoint of understanding the characteristics of nonequilibrium phenomena [

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 [

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.

Let us assume that the upper half of the space (

Let the forces

The directions of the considered physical quantities are as follows:

The

The quantities

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

Substituting from (

The model of the cone of influence suggested by Lee’s moment method [

Using Grad’s moment method multiplying (

The integrals over the velocity distance are evaluated from the relation

We introduce the dimensionless variables defined by

Using the dimensionless variable, if we neglect terms of order

Similarly, (

In the expressions for the transport coefficients mentioned previously, the fact that the plasma is neutral was used [

Therefore,

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) [

We can reduce our basic (

Similarly, the basic (

The traveling wave solution method [

Substituting from (

Now we have an ordinary differential Equation (

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 [

For paramagnetic plasma, the internal energy change is expressed in terms of the extensive quantities

On the other hand, if the plasma is diamagnetic, the internal energy change due to the extensive variables

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:

Consider

The fundamental and the essential inequalities, which we must bear in mind when analyzing the results, are

These inequalities will control the major behavior of both electrons and positive ions in the rest of the discussion. Figure

(a) The comparison between the combined perturbed dimensionless velocity distribution functions for electrons

A comparison between Figures

Figures

In a relative long period, for example,

All Figures

The dimensionless velocity versus space

The shear stress

The dimensionless shear stress versus space

The electrons induced electric field has a sudden increase in the beginning until it reaches its maximum value

The dimensionless induced electric field versus space

The dimensionless induced electric field versus space

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 [

The dimensionless internal energy change versus space

The dimensionless internal energy change versus space

The dimensionless internal energy change versus space

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:

the lighter species (electrons) of the gas mixture reaches equilibrium before the heave one (positive ions), which is in a qualitative agreement with Galkin [

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. [

the ratio between the time that electrons

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

It is concluded that the effect of the changes of the internal energies for positive ions

The same conclusion is applied in the case of electrons such that

The induced magnetic field vector

The induced magnetic field

The induced electric vector

The induced electric field

The velocity distribution function

The local Maxwellian distribution function

Distribution function for going downward particles

Distribution function for going upward particles

The current density

Boltzmann’s constant

The plate Mach number

Specific magnetization

Polarization

The gas constant

Entropy per unit mass

The temperature

The internal energy of the gas

Plate initial velocity

The mean velocity

The mean velocity related to

The mean velocity related to

Gas volume

Thermal velocity of electrons

Thermal velocity of ions

The speed of light

The velocity of the particles

Particle diameter

The electron charge

Lorantz’s force vector

Electron mass

Ion mass

The mean density

Electrons concentration

Ions concentration

Pressure

The position vector of the particle

Time variable

The mean velocity of the particle

The internal energy change due to the variation of entropy

The internal energy change due to the variation of polarization

The internal energy change due to the variation of magnetization

The internal energy change due to the variation of the induced magnetic field

Displacement variable

Ionization.

Dimensionless variable.

Related to electrons

Related to ions

Equilibrium

The relaxation time

The shear stress

Viscosity coefficient

The mean free path

Dimensionless parameter

Frequency

Collision frequency

Mass ratio

Mean free path

Debye radius.