On Kinetic and Irreversible Thermodynamic Treatments of a Rarefied Gaseous Plasma Bounded by a Moving Plate

Received 28 December 2021 Accepted 04 February 2022 The peculiarities of the Rayleigh problem (RP) governing equations of a rarefied gaseous plasma (RGP) are analyzed. They were proven to conform to the entropic performance for the RGP system using the moment method, separation of variables, associated with traveling-wave techniques in irreversible thermodynamics (IT) approach. Maxwell’s equations and the Boltzmann equation (BE) of the BhatnagarGrossKrook (BGK) type were solved. The BE considerable advantage is that it allows us to analyze the depth performance of the equilibrium electrons' velocity distribution function (EVDF) and the perturbed EVDF and their implementation to determine how far the system is from the equilibrium state (ES). As a result, the contrast between the equilibrium EVDF and the perturbed EVDF was conceptually elucidated at various periods. This significant benefit enables us to consider our model's non-equilibrium IT properties. For this purpose, the derived EVDF should be employed in entropy, production, and other critical thermodynamic variables. After analyzing the results, we discovered that H-theorem, thermodynamic principles, and Le Chatelier’s law were consistent with our model. The Gibbs rule was used to express how the various influences of the forces acting on the system's internal energy modification (IEM) are expressed. The findings showed that the proposed model could accurately capture the performance. The suggested type could accurately predict RGP helium and argon gases performance in the upper atmosphere's ionized belts. 3D-Graphics representing the physical parameters were generated using analysis of variance calculations, and the results are thoroughly presented. The importance of this research stemmed from its broad array of utilization in micro-electro-mechanical systems (MEMS), physics, electrical engineering, and nano-electro-mechanical systems (NEMS) technologies in a variety of commercial and industrial utilization.


Introduction
The BE has several uses in MEMS & NEMS technology. One of the most critical considerations driving the use of the BE in MEMS & NEMS implementations is the knowledge that: Since their micron-scale size is generally comparable to the molecule mean free path under normal operating conditions. Thus, the Knudsen flow numbers in MEMS & NEMS are generally far from the continuum regimes. Microflow is a term used to describe flow on a micron scale. The characteristic lengths of the flow gradient in microflows are usually modest and correspond to the molecules' mean free path. Often these MEMS & NEMS typical lengths will be in the micron order or less, resulting in a Knudsen number between 0.001 to 10. Resulting in fluid flows in the slip flow and transition flow conditions like most MEMS & NEMS technology. Microflows in these regimes have characteristics that differ from typical flows with long characteristic lengths. The Navier-Stokes and other traditional continuum models cannot characterize and forecast micro and nano-flows. On the other hand, microflows impact the effectiveness of MEMS & NEMS, such as micro pressure sensors, micro pumps, and valves [1] .
The problem of describing the motion of plasma was developed as a critical one. On the other hand, kinetic forms of the BE and macroscopic templates such as hydrodynamic type are extensively employed in plasma. While hydrodynamic systems are incredibly realistic to represent many observable occurrences, a fluid treatment is insufficient for some, like the RGP. A kinetic type must be applied to get out from the inadequacy of the hydrodynamic types in the RGP system. Nonetheless, kinetic type numerical simulations are prohibitively expensive in CPU time and memory capacity. To precisely explain the complex transfer phenomena in RGP streams circumstances, particle-based RGP dynamics should be used [1-3] . The transport field becomes complicated because nonequilibrium effects occur in the RGP flows [1] . Numerical solutions of the BE, which might be problematic, can characterize these non-equilibrium effects in kinetic gas theory. It is feasible to derive the equivalent macroscopic transport equations from a microscopic equation, such as the BE. The BE's classical approaches for deducing hydrodynamic-like equations are the Chapman-Enskog [4-6] and Grad's moment methods [7-9] .
In laboratory experiments and aerodynamics, the performance of the RGP in the presence of an infinite flat plate unexpectedly shifted in its plane, the RP, is of enormous interest. Particles collisions with rigid surfaces and particle binary collisions are predicted due to RGP rarefaction discontinuities in the surface's macroscopic parameters. Shedlovskii, El-Sakka, et al., Khater, et al. [9][10][11] , among several others, investigated the RP for a great RGP using the collisionless BE, as well as the RGP 's dynamical and electromagnetic field (EMF) characteristics. This investigation aims to apply the precise traveling-wave techniques [12-16] to solve the problem of the RP issue to estimate shear stress, velocity, viscosity coefficient, and induced magnetic and electric fields. IT performance of diamagnetic RGP must be studied by applying estimated EVDF to evaluate entropic predictions performance and associated IT functions.

Geometry, Physical, and Mathematical Formulation
Suppose that the top half of space ( ≥ 0), confined by an unlimited flat plate (y = 0), is complete with an RGP of electrons and ions Fig. 1. Because the ratio among electron and ion masses in ionized RGP is too slight ( ≪ 1), the ions will be treated as a motionless neutralizing background. Firstly, the RGP is incomplete ES and the wall rest. The plate then begins to move abruptly in its plane, with velocity 0 − is along the xaxis ( 0 nd are constants). The temperature of the entire system (electrons + ions + surface) is maintained at a constant temperature. The nomenclature defines all physical parameters.
Substituting Eqs. (21)(22) into Eq. (20), we have: Integrating Eq. (23) concerning once, we obtain: } By solving the ordinary differential Eq. (24), we obtain the form of the general solution: (26) ( ) = 2 1 + 3 2 + 4 3 + 1  Substituting from Eqs. (26,35, 37) into the system of Eqs. (38), we obtain three algebraic equations in three unknown constants 2 , 3 and 4 , which can be solved easily by usual methods of algebra to get the values of 2 , 3 and 4 to obtain the complete solution of the problem, as we will do in the applied example. Substituting the calculated velocities 1 and 2 from Eqs. (11, 26, 37) into Eq. (4) for 1 and 2 we may now begin to investigate the system's non-equilibrium thermodynamic characteristics.

The Non-Equilibrium Thermodynamic Descriptions
The IT processes are still representing a hot exciting topic. That is due to the theory's broad theoretical significance and diverse applicability in various fields. We begin by calculating the entropy per unit mass S, going back to the basics of the H-theorem. It has written as In its global form, the principle of entropy generation [28][29][30] is expressed as: We may calculate the thermodynamic force equivalent to the control factors [9] using the grand principle of IT: (42) = .
The connection among entropy generation and thermodynamic forces, but from the other side, takes the form [24][25][26] : In our problem, the condition of the Onsager inequality will be satisfied by the kinetic coefficient 11 and the thermodynamic force as 11 ≥ 0. To investigate the system's IEM. The extended Gibbs relation [9, 27] is introduced, consisting of EMF energy as a part of the total system energy. We distinguish paramagnetic and diamagnetic RGP [9] .
Case1: The RGP is paramagnetic, the IEM is stated in terms of the extended thermodynamics coordinates S, P, and m, which correspond to the intensive thermodynamics' coordinates T, E, and B, correspondingly, the different influences in the IEM in Gibbs methodology: (34) dU = dUS + dUpol + dUpara, where dUS = TdS is the IEM because of a variant of the entropy, dUpol = EdP is the IEM because of a variant of polarization, and dUpara = Bdm is the IEM because of the variant of magnetization. Here m is calculated from the Equation

Discussion
The unsteady performance of RGP is investigated by applying the traveling-wave techniques and the kinetic theory of IT processes. Our calculations are based on detailed data for RGP in Argon RGP [28] as a diamagnetic medium in which the Argon RGP loses electron pairs as a function of the voltage used to ionize the Argon atoms, exposed to the following parameters:  = 0.05. It shows that the distance from ES is small. The system, over time, works to restore the ES, which proves that the system complies with the 2 nd law of thermodynamics and conforms to the Le Chatelier ES principle. Fig. 3 Illustrates (a) the equilibria EVDF 0 (b) the combination perturbation EVDF , and (c) a combination of equilibrium and non-equilibrium EVDF. All of them show that the system almost approaches ES as t=100. The advantage of the EVDF representation and calculation is that it illustrates how the system is far from ES and when it may reach an ES. We can do that after many evolutions of both perturbed and When the equilibrium EVDF matches the unbalanced EVDF thus, we conclude that the system reaches the ES. That advantage cannot be reached by solving macroscopic types like Navier-Stocks and other macroscopic magnetohydrodynamic models.
The mean velocity is seen from Fig. 4, at the neighborhood of the sudden movement of the plate, it is a maximum (≈ ) and then declines time exponentially. It decreases non-linearly with the distance y. Fig. 4 The velocity Vx vs. y and t A similar performance holds for the shear stress except that equals zero at t=0 and y=0 also, it has a negative sign owing to its direction from bottom to upper, see Fig. 5. The mean velocity is seen from Fig. 4, at the neighborhood of the sudden movement of the plate, it is a maximum (≈ ) and then declines time exponentially. It decreases nonlinearly with the distance y. A similar performance holds for the shear stress except that equals zero at t=0 and y=0 also, it has a negative sign owing to its direction from bottom to upper, see Fig. 5. We considered the RGP is taken as Newtonian [9] . The viscosity impairs the motion, which is non-linear incrementally increasing as the RGP deviates from the surface. It increases with increasing time because of the decrement of the velocity of the electrons, see Figs. 6, 7.
The amplitude of the induced electric field increases exponentially with time t and distance y; see Fig. 7. The same observations hold for the induced magnetic; see Fig. 8.
The thermodynamic performance is illustrated in Figs. (9-15). The performance of the entropy S has a good agreement with the 2 ed law of thermodynamic and the H-theorem that it increases with the increment of time t, see Fig. 9. Also, Fig. 10 verifies how the performance of the system coincides with Htheorem since the entropy generation is positive ≥ 0 [24][25][26] . Indeed, the coefficient LMp satisfies the thermodynamics restrictions as it has a non-negative value for all the interval time and all y; see Fig. 11, 12.   Fig. 6 Viscosity coefficient vs. y and t Fig. 7 The electric field Ex vs. y and t Fig. 8 The magnetic field Bz vs. y and t Fig. 9 Entropy S vs. y and t

Fig. 10
Entropy production σ vs. y and t Fig. 11 The kinetic coefficient vs. y and t The electron loses (or gains) energy as it passes through RGP due to interactions with the particles of the surroundings caused by RGP polarization and collisions. The forces acting on electrons in the RGP from the EMF generated by the particles affect an electron's energy loss (or gain). The abruptly shifting surface produces work on the RGP, altering the IEM of the RGP [29] .
As shown in Fig. 13, IEM is smoothly damping with time away from the plate due to entropy variant . It also has a linear rise in moving plate proximity owing to the energy wasted and received from the particles and surface. Fig. 14 demonstrates that the internal energy owing to the polarization variant grows as both the time t and the distance y increase, which is consistent with the performance of EMFs (see Figs. 7,8,14).
While the IEM varies smoothly between two maximum negative values for the diamagnetic RGP, we found variants in the intensity of the induced magnetic field, as seen in Fig. 15.
Because of variations in the strength of the EMF, the IEM ranges non-linearly between two maximum negative values for the Para-magnetic RGP, dU par , As shown in Fig. 16.  Fig. 16 dUdia vs. y and t

Conclusions
The finding of the investigation of the unsteady BGK type of the BE in the case of an RGP applying the moments, traveling wave, and separation of variables methods of the two-sided EVDF and Maxwell's equations is developed within the restrictions of slight deviation from ES, RGP, and low Mach numbers. This technique allows us to calculate the flow velocity and other plasma parameters and variables. We evaluated the entropy, its production, and other important thermodynamic important variables by utilizing them in the EVDF and applying the H-theorem. The results correlated well with the H-theorem, thermodynamics rules, and Le Chatelier's concept. Via Gibbs' formula, the ratios between the various influences of the IEM are evaluated. We found that the maximum numerical values of the three IEM influences in the Dia-magnetic case are ordered in magnitude as: : : = 1: 1.67 × 10: 2.5 × 10 −4 . While In the Para-magnetic scenario, the maximum quantities of the three IEM impacts are ordered in magnitude as: : : = 1: 1.67 × 10: 1.67.
It is concluded: various IEM influences, such as , , , owing to EMF, are dominated compared to . In recognition that the flow had not been influenced by variation in temperature. In contrast, it was influenced by the EMF that was self-produced by the sudden motion of the rigid plate. 3D-Graphics illustrate the calculated variables' performances are shown, and their behavior is profoundly examined. Our finding concluded that: Our model and its solution and all calculated variables are compatible with IT laws.

Data availability
The data used to support the findings of this study are included in the article.

Declaration of Competing Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments
This study is supported by the Egyptian Academy of Scientific Research and Technology by the associated grant number (No. 6508) under the ScienceUP Faculties of Science program.
Many thanks to the reviewers for their outstanding efforts in the review process. Many thanks to the editor-in-chief and all the journal editorial family members.