Unsteady Analytical Solution of The Influenced of a Thermal Radiation Force Generated from a Heated Rigid Flat Plate on Non-homogeneous Gas Mixture.

In the present paper, the effect of the non-linear thermal radiation on the neutral gas mixture in the unsteady state is investigated for the first time. The unsteady BGK technique of the Boltzmann kinetic equations for a neutral non-homogenous gas is solved. The solution of the unsteady case makes the problem more general significance than the stationary one. For this purpose, the moments' method, together with the traveling wave method, is applied. The temperature and concentration are calculated for each gas component and mixture for the first time. Furthermore, the study is held for aboard range of temperatures ratio parameter and a wide range of the molar fraction. The distribution functions are calculated for each gas component and the gas mixture. The significant non-equilibrium irreversible thermodynamic characteristics the entire system is acquired analytically. That technic allows us to investigate the consistency of Boltzmann's H-theorem, Le Chatelier principle, and thermodynamics laws. Moreover, the ratios among the different participation of the internal energy alteration are evaluated via the Gibbs formula of total energy. The final results are utilized to the argon-helium non-homogenous gas at different magnitudes of radiation force strength and molar fraction parameters. 3D-graphics are presented to predict the behavior of the calculated variables, and the obtained results are theoretically discussed.


I. Introduction
Thermal radiation is a deep-rooted part of our environment. Also, at high temperatures, the radiative heat transfer is a remarkable incomplete system analysis. Some application where thermal radiation transfer is of primary significance includes cryogenic fuel storage systems, spacecraft cooling systems, boilers and furnaces, and solar collectors [1,2]. Although, the statistical-mechanical study of a non-homogenous gas in a nonequilibrium situation is a fascinating subject from theoretical as well as experimental viewpoint. Few papers are dealing with the gas mixture [3][4][5][6][7][8][9], compared with the significant number of studies in the case of a single neutral gas [10][11][12][13][14][15][16]. It is well known that the particles of a non-homogenous neutral gas radiate and absorb thermal energy. Therefore, in non-homogenous neutral gases, the interaction with thermal radiation is remarkable, unlike solids and liquids, which are capable of undergoing conspicuous volume alteration. Considering this interaction makes the nonhomogeneous neutral gas way more realistic. Further, radiative heat transfer in non-homogenous neutral gases has remarkable applications from combustion processes to atmospheric modeling operations. Abourabia and Abdel Wahid [13] presented a new approach for investigating the effect of the thermal radiation force on the nonhomogenous neutral gases. For steady problem, this idea was applied on a half-space filled off by a nonhomogenous neutral gas bounded by a fixed heated solid plane. The present paper is extended to investigate the non-stationary problem for a neutral non-homogenous gas affected by a thermal radiation force. Therefore, we should solve two systems of non-linear partial differential equations (PDEs) instead of one system of ordinary differential equations (ODEs), as presented in [13]. The Boltzmann kinetic equation is valid for all ranges of Knudsen number [14], while the Navier-Stokes (N.S.) system is suitable to give us acceptable results for the continuum flow only, (where = is the Knudsen number that measures the rarefaction of any gas molecules and represents the ratio between the mean free path λ to a characteristic length L). For this purpose, we utilize coupled systems of non-stationary BGK kinetic equations, one for each component of the neutral gas. The radiation force effect is inserted into the force term of the Boltzmann kinetic equation. These procedures are done by applying the Liu-Lees model for two-side Maxwell non-equilibrium distribution functions using the moment method. Moreover, the manner of the macroscopic characteristics of the non-homogenous gas is estimated for different radiation force strength according to different fixed, rigid plate temperatures. The temperature and concentration are, in turn, substituted into the related nonequilibrium distribution function. This approach permits us to investigate the manner of the equilibrium, nonequilibrium, and non-stationary distribution functions for different magnitudes of the molar fraction parameters. Also, the remarkable non-equilibrium thermodynamic characteristics of the entire system (neutral nonhomogenous gas + heated plane solid plate) are calculated. Especially thermodynamic forces, entropy, entropy generation, entropy flux, and kinetic coefficients are investigated. Furthermore, the consistency of thermodynamics second law, Boltzmann H-theorem, and Onsager's relation are illustrated. The ratios among the different participations of the internal energy alteration are estimated via the Gibbs' formula. The results are applying to the argon-helium neutral non-homogenous gas. Finally, the remarkable conclusions of the paper are indicated.

II. The Physical and Mathematical Formulation of the Problem
Consider a neutral non-homogenous gas consisting of two species, for example, the A-Kind and B-Kind. The gas fills in the upper half of the space in the system ( 0 y  ), and bounded by an illimitable immobile plane solid plate (y=0), in a constant pressure S P [2,13]. The solid plate is suddenly heated. That is generating a thermal radiation force. The flow is non-stationary and compressible. In a frame co-moving with the fluid the manner of the neutral non-homogenous gas is studied under the following conditions: (I) The two velocities of the particles (incident and reflected) are equals at the fixed plane solid plate but in opposite signs. That result happens as a result of Maxwell's formula of momentum defuses reflection. Furthermore, the exchange will be generated by the temperature difference among the particles and the heated plane solid plate. That is taking the form of full energy accommodation [17]. (II) A thermal radiation force is acting from the heated plane solid plate on the neutral non-homogenous gas, written in vector form [17][18]  the velocity of light, the temperature, and the thermal radiation force component along y-axis direction, respectively. For a non-stationary motion, the operation in the entire system subject to y f (the thermal radiation force) can be formulated in terms of the BGK kinetic equations in the following form [4,7]: Here , , and Lee's moment method [19][20] is employed here to gain the solution of the BGK kinetic equation. By adding heat to a non-homogeneous neutral gas, allowing it to expand, it is made rarer than the non-homogeneous gas neighbor sections. It continues to create an upward stream of the heated non-homogenous gas, which is usually followed by a flow in the opposite direction by the more distant parts of the non-homogenous gas. The fresh portions of non-homogenous gas are carried into the heat source neighborhood, taking their weather along with them to other regions [4,7]. We assume that the temperature of the non-homogenous gas particles rising upward is while the temperature of the non-homogenous gas particles going downward is 2v T [13,21]. The correspondent concentrations are 1v n and 2v n . We utilize the Liu-Lees model of the two-sided Maxwell non-equilibrium distribution function near the solid plane plate [17] for particles of the v species that can be expressed as: The obtained equations (5) are known as the general equations of transfer [22]. Applying the non-dimensional forms by assuming: 4 ( )  gg  , and 2 g  are calculated, then the macroscopic quantities (velocity, density, and temperature, etc.) can be evaluated as; The bulk velocity of the gas:

N T T T n n n T T T n n n dU dU K T n cm R
The number density: The temperature: The static pressure normal to the heated solid plate: The y-component heat flux: The functions 1 species. The energy conservation equation has the form: The y-direction component of the heat flux has the form: ( ) (  )   3  33  22  1  1  2  2  22  1  1  2  2  1  1  2  2  1  1  2  2  12   3  3  3  3  1  1  2  2  2  2  2  2  1  1  2  2  1  1  2  2 The equations of state complement the above equations: 5 Thus, we acquire the fourth equation in the form: Traveling wave solution method [23] is utilized to solve the problem, by assuming where n is a positive integer. Substituting from Eqs. (11)(12) into Eqs. (7-8) we acquire: Know, we intend to solve Eqs. (14)- (19) with the boundary value problem to evaluate 1 2 1 2 , , and T T n n     , as follows; Substituting from Eqs. (14) and (20) into Eq. (18), taking into consideration Eq. (10), we get: Integrating Eq. (21), for  , we get after factorization Moreover, 2 C  is the integration constants. We can easily show that 12 ,   they are constants. To make our calculation simpler, and make better use of the Eq. (20), let us introduce the function () H   in the following form: From Eqs. (23 and 24) we can obtain: Now, we can integrate equation (19) w.r.t.  after performing some necessary algebraic manipulations, taking into consideration Eqs. (14) and (22) To solve this equation, we obtain eight roots for () H   , using any symbolic software. We take root into account, which preserves the positive signs of both concentration and temperature. The magnitudes of the integrated constants can be evaluated under the initial and boundary condition ( , ) (0, 0) 0 yt is the molar fraction.
Also, the temperature of the incident particles is T2. The heat of the reflected particles from the solid plane plate is represented by temperature T1, which can be expressed, as mentioned in [4,7]:  is the ratio among the solid plane plate and the temperature of the non-homogenous neutral gas. The parameter v  takes an arbitrary positive value less than unity to ensure that the solid plane plate remains hotter than the non-homogenous neutral gas. By solving the system of algebraic Eqs.
The obtained formulae represent the initial and boundary conditions of the problem. Substituting the calculated quantities 1 2 1 2 , , and T T n n into the two-side Maxwell non-equilibrium distribution function in the form , we can acquire the sought non-equilibrium distribution functions, which enable us to investigate the manner of the non-homogenous neutral gas particles. That was not possible if we utilized the N.S. equations system instead of the BGK kinetic equations.

III.
The Non-Equilibrium Thermodynamic Characteristics of the System: while the Boltzmann's entropy generation in the non-stationary state S  is formulated as mentioned in: According to the theory of thermodynamics [4, 7, and 13], the thermodynamic forces can be evaluated: The first is n X that related to the modification in the concentration (  is the non-dimensional radiation force energy that affected the non-homogenous neutral gas particles and y  is the thickness of the layer near to the solid plane plate that has units as the mean free path. After calculating the thermodynamic forces and the entropy generation, the kinetic coefficients Lij can be obtained from the following relation [24][25][26]: ( ) The Gibbs relation for the modification of the internal energy ( , ) dU y t utilized to the entire system (neutral nonhomogenous gas + heated plane solid plate) is ( , )

IV. Results and Discussion
The manner of the neutral non-homogenous gas was examined under the influence of a thermal radiation force in the non-stationary state of a plane heat transfer problem in the entire system (neutral non-homogenous gas + heated plane solid plate). Also, the thermal radiation is presented in the force term of the Boltzmann kinetic equation for a nonhomogenous neutral gas subject to the inequality d    , where is the average spacing among the molecules, both  and d are the mean free path and the molecular diameter, respectively. In all calculations and figures, we take the magnitudes of the following parameters from [28][29] for the argon-helium neutral non-homogenous gas, where the particles of the A and B components are the argon and helium non-homogenous neutral gases, respectively:  Now, we will discuss the manner of the non-homogenous neutral gas particles far from the equilibrium state: Figures (1-a, Figures (3a, 3b, 3c) shed light upon an increment in 1 f  and a decrement in 2 f  compensating for each other, compared with 0 f for each gas and gas mixture, too. That gives a greet qualitative agreement in the behavior of the distribution functions with [21]. That is explained by a departure from the equilibrium state, where the gas particles, having temperature 1 T and density 1 n , after being heated, are replaced by a counterchange by the non-homogenous neutral gas particles having temperature 2 T and density 2 n . This behavior agrees with Le-Chatelier's principle.

9
The entropy S always growing with time, and the entropy production σ has non-negative magnitudes for all values of t  and the molar fraction A C . That gives a complete satisfaction of the second law of thermodynamics and the Boltzmann Htheorem. This manner agrees with Le Chatelier's principle, as shown in Figs. (4-5, 21-22, 38-39). The manner of the different participation of the internal energy modification can be clarified in Figs. (7-9, 24-26, 41-43). The Gibbs formula estimates the numerical ratios among the diverse involvement of the internal energy alteration based upon total derivatives of the extensive parameters. Taking into consideration their different tendencies, the maximum numerical magnitudes of the three participations at different radiation force strengths (correspondent to different plane rigid plate temperatures), are ordered in quantity as follows:  Fig. (14). The T X (according to the gradient of temperature) will have the opposite direction to the n X (according to the gradient of the density), as shown in Figs. (10)(11). That gives a qualitative agreement in the manner with the thermodynamic forces determined in [28], comparing with the same neutral non-homogenous gas (Ar-He) at the same molar fraction 0.5 A C = .
(ii) For a fixed value ( 0.5 ) that represents the concentrations of the heavier non-homogenous neutral gas (argon) indirectly, and the lighter non-homogenous neutral gas (helium) respectively. That means that, the more, the lighter mass of the nonhomogenous neutral gas species, the more the efficiency of the thermal radiation energy participation in the entire system total energy modification, as shown in Fig. (26). That is also according to the manner of the correspondent value of the R X itself related to each component in the neutral non-homogenous gas, as shown in Fig. (29). The T X (according to the gradient of temperature) will have the opposite direction to the n X (according to the slope of the density), as shown in Figs. (27)(28). That gives a qualitative agreement in the manner with the thermodynamic forces determined in [28], comparing with the same neutral non-homogenous gas (Ar-He) at the same molar fraction 0.5   = that means that the more the temperature difference (among the fixed plate surface temperature and the neutral non-homogenous gas temperature), the more will be the active participation of the thermal radiation energy in the entire system total energy modification. That is attributed to the manner of the correspondent magnitudes of the R X itself, as shown in Fig. (44).

) and variable magnitudes of
According to our calculations, the constraints imposed on the kinetic coefficients ij L are fulfilled. 11

V. Conclusions
The calculated results were demonstrated to conclude the following: a) The predominant vital factors that are directly proportional to the efficiency of the radiation energy in the total energy modifications of the entire system are: i) The ratio among the temperature of the plane solid plate surface and the neutral non-homogenous gas particles.
ii) The ratio among the mass of each component of the neutral non-homogenous gas.
iii) The molar fraction of each component of the non-homogenous gas. b) At a temperature ( 1200K T  ), the radiation energy participation in the modified total internal energy becomes the predominant one. c) The lighter non-homogenous neutral gas component, helium, is influenced by a thermal radiation force more than the more massive gas component, argon. d) The first laws of thermodynamics (notably, the second law), the Le-Chatelier's rule, the Boltzmann H-theorem, and the Onsager's reciprocal relations, are all fulfilled for the investigated non-homogenous gas entire system. e) The negative sign at specific kinetic coefficients, related to cross effects, implies in such cases that according to the imposed thermodynamic force (gradient), there is a heat flux in the opposite direction to the main flux. For instance, the negative sign in front of implies that there is a flow caused by the temperature gradient, from a lower to a higher temperature, known as thermal diffusion (or effect of Soret), which gives a qualitative agreement with the investigation results [23,28].

Data Availability
• No data, models, or code were generated or used during the study.

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