Gaseous Plasma's Time-behavior on a Plate Motion Damping with Time Influenced by a Non-Uniform Non-Stationary Electric Field

The research subject is the kinetic and irreversible non-equilibrium thermodynamics ( INTD ) behaviour of gaseous plasma ( GP ) flow limited by a moving rigid flat plate ( RFP ), its motion damping with time. The effects of non-stationary nonlinear applied electric field ( NLAEF ) were examined on the GP . To explore GP with the electron velocity distribution function ( EVDF ), researchers have concentrated on the Bhatnagar-Gross-Krook ( BGK ) – model of the kinetic Boltzmann equation ( BE ). An analytical solution was found using the moment ( MM ), travelling wave, and shooting method. An interesting comparison between the non-equilibrium EVDF and the equilibrium EVDF is made carefully in the present and three previous critical studies. We discovered that the NLAEF has a significant impact on GP . Compared to the influence of the nonlinear applied magnetic field ( NLAMF ), it caused it to vary and disturb substantially. To save the equilibrium state ( ES ) for a GP , we need to apply NLAMF rather than NLAEF in the plasma controlling procedure. Moreover, we found that the oscillating boundaries keep ES rather than the moving boundaries in both NLAEF and NLAMF . We also found that the system goes to ES vastly in the case of NLAMF . The relations between the variables that participated in internal energy modifications ( IEM ) are examined. The importance of this research stems from its wide applications in domains such as physics, electrical engineering, micro-electro-mechanical systems ( MEMS ), and nano-electro-mechanical systems ( NEMS ) technologies in industrial and commercial sectors.


Introduction
In the MEMS industries, BE provides numerous advantages. Because their micron-scale size is ordinarily relative to the molecule mean free path under typical operating settings, one of the numerous essential issues leading to the use of BE in MEMS and NEMS performances is the awareness that: As a result, in MEMS and NEMS, the Knudsen flow values are frequently distant from the continuum rules. The word "microflow" describes flow on a micron size. In microflows, flow gradient lengths are typically modest and resemble the molecules' mean free path. MEMS and NEMS typically have lengths in the micron range or more minor, resulting in Knudsen numbers ranging from 0.001 to 10. Low-temperature GP or vacuum GP technologies have been widely used in semiconducting and circuit boards for over five decades. They are widely used in magnetic media, metal processing, and cleaning, including carbon removal after laser cutting. These systems achieve new industrial implementations in microelectronics and photonics by reengineering the surface characteristics of polypropylene. That permits such manufacturers to modify the physicochemical characteristics of the material's surface without influencing the properties of the entire material [1][2][3][4] . The temperature, Tα, of low-temperature plasmas, may be characterized as Ti ≈ Tn ≈Te, where Ti is ions temperature, Te is electrons temperature, and Tn is neutral atom temperature. These characteristics help create GP-based technology that employs various electron and ion MVDFs. They characterize the kinetic behaviour of the GP by conveying the energy accessible to a large variety over multiple collisional phenomena. As a result, it is critical to compare the electrons' EVDF with the ES and modify it to follow the electron behaviour. It is generally known that the BE is often used to characterize GP mobility in compressional GP when microscopic factors are considered. Many analytical and experimental contributions [6], [7] are concerned with creating approaches for obtaining realistic solutions to BE. BGK-type [8][9][10] is the most impressive approximation of the difficult collision factor in BE. As a result, various trials employing MM [11]- [12] have been established to handle the BE depending on the BGK-type.
Greater mean free path values sufficiently explained the rarefied gases for ionized flows throughout various Knudsen numbers (Kn ~ 0.1-10). Modified amounts of the recognized size in MEMS and NEMS technologies are widely utilized in various commercial activities (see Refs. [11][12][13][14][15] ). In GP technology, particularly in lowtemperature GP, the contact between flowing GP and rigid surfaces is critical. The ionization degree of charged particles is modest, with a single charged particle having between 100,000 and 1,000,000 neutrals. As a result, instability, discharge management, and the creation of excited atoms are all handled by ions.
For example, Ref. [15] investigates the flow dynamics of GP over RFPs using the BE. Furthermore, [16]- [29] have investigated GP's various flows. They used a numerical solution of the collision frequencies to characterize the behaviour of the charged particle. Yan [30] presented a Hybrid methodology combining Monte Carlo, Particles in Cell, and Macro-Micro decomposed techniques. Pan et al. [31] examined the possibility of gas EVDF in discontinuous speed space utilizing the finite volume approach, using GP transportation in the accelerating electric term of the BGK-model. Juno et al. [32] used finite elements and Runge-Kutta algorithms to examine a novel method for 6D GP to discretize the Vlasov-Maxwell system to find new EVDF alternatives. Using a preservation kinetic gas method, Liu et al. [33] devised a new approach for GP flow problems with large Kn and Debye length quantities. Now we are looking into the influence of electron-ion-neutral collisions in the BGK-kind and then calculating the exact solutions for the problem. Furthermore, we handle the entire BGK-kind collision frequency to prevent model discontinuities without any cut-off based on the total values of the independent variable [16]- [29] . The structure of equations was explained with several assumptions in these studies.
The BE's improvement allows us to examine the concepts of INTD. According to research findings, the displacement current associated with the NLAEF could not be negated in the problem therapy. The non-stationary oscillating flow is studied by Wahid et al. [34] in an INTD environment utilizing the Stokes problem. Under the NLAEF effect, we wanted to employ GP simulation next to a limitless oscillating RFP. The BGK-kind model was combined with the ME model. Furthermore, two-sided MVDFs were used with MM and travelling wave solutions.
Finally, theoretical descriptions of thermodynamic and electromagnetic variables were provided. The difference in variation amongst the MVDFs and the perturbed MVDFs is shown. Because of the BE's characteristics, we investigate INTD concepts. The findings are in good accord with those found in [34] .

GEOMETRY AND GOVERNING EQUATIONS OF THE PROBLEM
Analyzing a GP helium system with an unsteady NLAEF, parallel to flow path positioned near an infinite horizontally RFP = 0, bordered by the upper half-space at ≥ 0, Fig. . Initially, GP is contained in an ES. It gets hypothesized that the GP is isothermal ≈ and quasi-neutral. We indicate the NLAEF direction on the x-axis and the induced magnetic field on the z-axis. GP flows to an unstable NLAEF, and the surface RFP moves with a velocity equal to 0 − along the x-axis. Furthermore, the reflected particles acquired RFP velocity in every case.
The applied non-uniform unstable electric field and the induced one are represented by and , respectively, and the 0 , 1 , and 2 constants have restricted values to prevent the electric field from diverging.
, and collision frequencies take the form [37,38] : Here, are Debye lengths, Coulomb logarithm, and ionization degree. Lee's approach to inherently unknowable functions of time and space, 1 and 2 [38,39], can be used to solve Eq. (4): Utilizing Grad MM with Eq. (4), we acquire the transfer equations [6]: The corresponding formula of the functions ( ⃗) integrals over = , The flat plate The ME can be stated as: Observe that the current displacement component in the ME is factored into the equation, although in prior research, it was omitted; see [17,20,26,28] . The electromagnetic fields can be computed using the standard forms.

METHOD OF SOLUTION FOR THE NONDIMENSIONALIZED SYSTEM
The boundary value issue for electrons will be simplified by removing the star from the nondimensional parameters.
where 0 = ( ℎ 2 2 2 2 ). Solving BE by applying the travelling wave approach, the electron kinetics be able to be explained in total, and the new item can be represented as follows: [40] - [42] : Here is a positive integral number. Replacing Eqs. (21)-(22) in Eqs. (17)-(20), we have the next differential equation In its dimensionless form, initial and boundary conditions become: In addition, given Eq. (1), it is observed that, for limited > 0 and restricted time intervals, t of applicable concern, the utilized electric field, , stays finite. Solving the governing equations defining the research lab GP helium using symbolic software (Mathematica).

THERMODYNAMIC TREATMENT OF HELIUM-PLASMA MODEL
Examining the INTD features of the GP helium while considering the problem's INTD phenomena. The thermodynamic characteristics of the GP components will be determined by using EVDF in Eq. (6) and ions, electrons, and neutrals collision frequencies. In addition, we will see if our approach is coherent with several thermodynamic principles and the H-theorem for the GP helium. In physics, entropy is derived from INTD. It is one of the state parameters of matter used to characterize how energy is degraded. It refers to a measurement of the status of some material systems in general. The essence of entropy, the degree of internal chaos of a system, is gradually described with the advent of statistical physics and information theory. Cybernetics, probability theory, life science, and astrophysics are just a few of the domains where it is functional.

GIBBS FORMULA AND ITS CONSEQUENCES
Gibbs's formula can calculate the overall system's internal energy modification (IEM) [44] . The two significant types of GP helium magnetization are paramagnetic and diamagnetic. As a result, for both kinds, the total energy alterations can be stated as follows: IEM in Gibbs' equation is described by the extensive factors S, P, and B related to intensive factors T, E, and MB. As a result, the Gibbs law's IEM can be expressed as [29], [44]

CALCULATIONS AND RESULTS
This part provides an example of flow in GP [45] . Solving the BGK-kind for GP helium predicated on the INTD utilizing the travelling wave methodology for the , , and colliding frequencies, the GP kinetics can be characterized in depth. The findings are displayed in figures and 3D-graph s utilizing a wide variety of standard data for electron gas in GP helium. In the experimental context, the findings in the figures reflect the normal verification consistency between BE and research lab GP estimates [20,37] . The GP helium loses one electron in the paramagnetic condition. However, the helium-GP loses two electrons due to the atoms' ionizing potential in the diamagnetic scenario. We investigate the accurate values of 1 , 2 , 3 using the idea of the numerical shooting technique obtained 1.3807 10 erg/K, B k =  0 = 300 K, = 10 11 cm -3 . The atom diameter of the helium-GP is = 1.4 × 10 −8 cm, and 0 = 2.9 × 10 −6 . The mean free path is = 11.483 × 10 3 cm, which is large compared with the Debye length = 3.779 × 10 −4 cm. Lastly, , , and collision frequencies are = 1.32 × 10 9 −1 , = 9.34 × 10 8 −1 ,and = 2.399 × 10 −3 −1 , correspondingly.
It is clear from the behaviours of the EVDF that the ascending particles and their turbulence after colliding with the plate are much more affected than the descending particles that have not yet collided with the plate, see Figs. 2A, 2B. With the passage of time and the deviation of the system to ES, we find that this disturbance in the behaviours of the particles diminishes little by little; see the gradation and evolution of these behaviours from t = 0.1, 5, 20, 25.9, see Figs. 2A, 2B. The collision of the ascending turbulent particles with the unperturbed descending particles works to decay these behaviours, seen gradually in Figs. 2A, 2B. Comparing the EVDF of turbulent electrons and their ES ones shows that this turbulent behaviour is more for the ascending particles than for the descending particles. That is very clear; see Figs. 3A, 3B.  The first component of the two-branched EVDF of descending particles deduces the tendency to ES much faster than the ascending particles. That is directly affected by the movement of the plate and is also affected by the electromagnetic fields created, which of course, are as strong as possible near the plate forming it, see Figs. 3A, 3B. We explore the characteristics of GP in the presence of an applied non-uniform non-stationary electric field that varies in space and time and the impact of identical restrictions. We can study a comparison between different previous studies in Refs. [7,12,35] . First comparison: Comparing our Figs. 3-A, B with the case study in Ref. [12] , where NLAMF affected a GP bound by a moving plate. We found that the NLAEF in the case of moving plate has a more substantial influence on EVDF related to GP than the NLAMF compared with Figs. 1-A, B in Ref. [12] , that swiftly drives the GP out of its ES as (t=2.5). The EVDF of the GP achieve ES over time, consistent with the laws of INTD as (t=10), the GP is a little out of ES owing to NLAEF; see the last 3D-graph of Fig. 1-A, whereas it has achieved ES owing to NLAEF; check its last 3D-graph of Fig. 1 in Ref. [12] . We find that the NLAEF has a strong effect on GP. When contrasted to the action of the NLAMF, it causes it to vary and perturb aggressively. To keep the ES, we should use the impact of NLAMF rather than the impact of NLAEF in the GP controlling process, which agrees with the same results concluded from the study in Ref. [7] .
Second comparison: Comparing our Figs. 3-A, B with the case study in Ref. [35] , where NLAMF affected a GP bounded by an oscillating plate. We found that the NLAEF in the case of moving plate has a more substantial influence on EVDF related to GP than the NLAMF in the case of moving plate compared with Figs. 1-A, B in Ref. [35] , that swiftly drives the GP out of its ES as (t=5). The EVDF of the GP achieves ES over time, consistent with the laws of INTD, as (t=10), the GP are a little out of ES owing to NLAEF; see the last 3D-graph of Fig. 1-A, whereas it has achieved ES owing to NLAEF; check its last 3D-graph of Fig. 1 in Ref. [35] . We find that the NLAEF has a strong effect on GP. When contrasted to the action of the NLAMF, it causes it to vary and perturb aggressively.
To keep the ES, we should use the impact of NLAMF rather than the impact of NLAEF in the GP controlling process, which agrees with the same results concluded from the study in Ref. [7] . Third comparison: Comparing our Figs. 3-A, B with the case study in Ref. [35] , where NLAEF affected a GP bounded by an oscillating plate. We found that the NLAEF in the case of a moving plate has a more substantial influence on EVDF related to GP than the NLAEF in a case of an oscillating plate compared with Figs. 1-A, B in Ref. [35] . Figures 4-A, B show that the NLAEF impacts dramatically raising GP velocity in a short period. That is in harmony with the perturbed EVDF's manner. Away from RFP, the shear stress increases nonlinearly. The shear stress linearly behaves next to the RFP; see Figs 5-A, B. The NLAMF, on the other hand, behaves nonlinearly, neighbouring to the RFP [35] . The viscosity parameter performs in a linear manner neighbouring the surface and in a nonlinear manner far from it; see Figs. 6-A, B. The NLAMF, on the other hand, behaves in a nonlinear fashion [34] . Figs. 7-A, B show how the NLAEF's nonlinear acts influence GP  The behaviour of the trustworthy induced magnetic field linked to the NLAEF compared to the NLAMF in [34] is depicted in Figs. 8-A, B. The fluctuation and perturbation of GP can be explained in this way. When contrasted, a high induced magnetic field is related to the NLAEF, but a small electric field is connected to the NLAMF.
Boltzmann's H-theorem, thermodynamic principles, and Le Chatelier's concept agree that entropy and how it is excessive (2 × 10 3 times) relative to the earlier study's comparable figure; review Fig. 5 in Ref. [35] , since it is merely a generated magnetic field formed by RFP motion. It was found to be linked to NLAMF [35] . increases with time. It is, however, slow-growing, which justifies the latency of GP electrons reaching an ES in this situation compared to the earlier work [35] ; review Figs. 7-A, B. Compared to the NLAMF [34] , we conclude that the pace of reaching an ES in the NLAEF instance is plodding. That supports the hypothesis that employing an NLAMF rather than an NLAEF is optimal for managing electron GP and transferring them to the ES.   Figs. (11-A, B), (12-A, B),  and (13, A, B ): ( ) = 1 : 2 × 10 5 . As shown in [35] , the IEM dominates in the NLAEF instance compared to the matching NLAMF cases.
In the case of a research project: : : = 1 : 0.4 × 10 2 : 1.33 × 10 5 . Figs. 14-A, B show that with NLAEF, the gyrofrequency , decreases nonlinearly with space and time, whereas it decreases uniformly with space and time in the Ref. [35] . That is because of the powerful electromagnetic fields occurring in this scenario. Gyro-radius near the solid horizontal RFP shows a substantial perturbation in Figs. 15-A, B, whereas the NLAMF shows a smooth and consistent variation [35] .

CONCLUSIONS
We found that the system goes to an equilibrium state (ES) compatible with Le Chatelier's principles. The relations between the various macroscopic variables of the GP are studied. INTD properties of the system are presented. Entropy and entropy generation are derived, and their behaviour is investigated. The essence of entropy, the degree of internal chaos of a system, is gradually described with the advent of statistical physics and information theory. Cybernetics, probability theory, life science, and astrophysics are just a few of the domains where it is functional. According to our findings, NLAEF has a strong effect on GP. Compared to the effect of the nonlinear applied magnetic field (NLAMF), it causes it to vary and disturb substantially. To save the ES for a GP, we should employ NLAMF rather than NLAEF in the GP management procedure. The importance of this research stems from its wide applications in domains such as physics, electrical engineering, micro-electromechanical systems (MEMS), and nano-electromechanical systems (NEMS) technologies in industrial and commercial sectors.
Plasmas are being applied in a wide range of applications. The ability to modify the GP's behaviour via the NLAEF effect is key to these applications. For this, we contrasted the current work, which looks at GP performance with the NLAEF influence, in conjunction with the former work, Ref. [35] , which looked at GP behaviour under the NLAMF effect. That was accomplished in the instance of an RFP moving to a GP. A rigorous analytical examination of the impact of timely and spatially variable NLAMF on GP helium was carried out. In addition, the current displacement term in the ME equations is regarded, which has hitherto been overlooked [17][18][19][20] . We developed a model to assess the GP's EVDF, velocity, and electromagnetic field. Obtained results correlate well with Boltzmann's H-theorem, thermodynamic principles, and Le Chatelier's rule. We determine that: 1-NLAEF has a strong influence on GP. In comparison to the impact of the NLAMF on it, it caused it to fluctuate and perturb aggressively; see [7], [12], [35]. To keep the ES, we should use the action of NLAMF rather than the effect of NLAEF in the GP process monitoring.
2-Moreover, we found that the oscillating boundaries keep ES rather than the moving boundaries in both cases of NLAEF and NLAMF; see [7], [12], [35]. 3-We also found that the system goes to ES vastly in the case of NLAMF; see [7], [12], [35]. 4-The NLAEF is related to a large induced magnetic field, whereas the NLAMF is connected to a mild induced electric field. 5-Compared to the NLAMF [34], the rate of reaching an ES in the NLAEF example is prolonged. That supports the idea that when controlling GP electrons to achieve an ES state, it is better to use an NLAMF rather than an NLAEF to get them there. 6-When compared with equivalent values in the situation of the NLAMF in [35], the IEM helps develop the NLAEF. In the case of a research project: : : = 1 : 0.4 × 10 2 : 1.33 × 10 5 . 7-All thermodynamic considerations are congruent with our model and all estimated variables.

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

Conflict of Interest
The author declares that there is no conflict of interest regarding the publication of this paper.

Acknowledgements
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.

Author Contribution Statement
All authors contributed equally to the paper. Kinetic and thermodynamic treatments of a neutral binary gas mixture affected by a nonlinear thermal radiation field. Canadian Journal of Physics, 90(2): 137 -149