Modelling of Nonlinear Thermodiffusion for a Spherically Symmetric Case

The paper discusses the properties of the nonlinear thermodiffusion equation corresponding to the heat transfer processes occurring with a finite velocity in gas from a high intensity source. In the previous papers A. J. Janavičius proposed the nonlinear diffusion equation which provided a more exact description of impurities diffusion by fast moving vacancies generated by X-rays in Si crystals. This is similar to the heat transfer in gas with constant pressure by molecules carrying a greater average kinetic energy based on the nonlinear thermodiffusion of gas molecules from hot regions to the coldest ones with a finite velocity by random Brownian motions. Heat transfer in gas must be compatible with the Maxwell distribution function. Heat transfer in gas described by using nonlinear thermodiffusion equation with heat transfer coefficients directly proportional to temperature . The solution of the thermodiffusion equation in gas was obtained by using similarity variables. The equation is solved by separating the linear part of the equation that coincides with Fick's second law. The obtained results coincide with Ya.B. Zeldovich’s previously published solutions of nonlinear equations by changing the respective coefficients.

In the previous papers we have discussed the nonlinear diffusion of impurities in semiconductors [1][2], nonlinear thermodiffusion in gas [2][3] and heat transfer in metals by electrons [4] using mathematical methods of similarity variables [1] for the solution of the nonlinear equation. The obtained results are important for engineering calculations of heat transfer in gas at constant pressure.
It is assumed that the process of thermal transmittance in gas is similar to nonlinear diffusion process of impurities described as Brownian movement of atoms in solids spreading with a finite velocity. Heat transfer can be described by using a modified theory of nonlinear diffusion in solids [1]. In this case the frequency of the jumps of diffusing molecules [2] depends upon the coordinates and changing molecules concentration n and temperature T according law of ideal gas defining pressure  p nkT . The coefficient of thermal conductivity of gas can be expressed in the following way [2], [6] Here λ is the mean value of a free path of diffusing molecules, v -the mean velocities of molecular movement, v c -the molar heat capacity at constant volume, ρ -the density of gas, n -the number of molecules per unit volume, k -Boltzmann constant, T -temperature of gas, μmolar mass, R -gas constant, d -the distance between centers of the diffusing molecules of the gas, ( ) w D Tthe thermodiffusion function of impurities in gas for the isobaric process. The equation of thermodiffusion in gases can be obtained with the coefficient ( ) w D T , which is proportional to the temperature [7] of gas. Similarly, as in the case of nonlinear diffusion in crystal silicon [1] by using (1), the coefficient ( ) The constant pressure ( , ) ( , ) ( , ) ( , )   S S p n r t kT r t n r t kT r t at slow impurities transmission in gas with decreasing temperature ( , ) T r t is compensated by increasing the concentration ( , ) n r t of impurities in gases with heating from the spherical source of temperature S T . Here ws D is the thermodiffusion proportionality constant for the nonlinear thermodiffusion function  which requires that the first Fick law must be improved by introducing the heat flux [1], [2] according the radial direction with the finite length Δr of the jumps with the finite velocity [1] of diffusing particles. It can be assumed that the jump length of the diffusing atoms or molecules from one equilibrium position to another in solids or fluids has the definite value L .
In gases L may be the average free path of diffusing particles. From the conservation the number of diffusing particles and the theorem of Gauss' [6] ( the following nonlinear thermodiffusion equation is obtained By using the nonlinear diffusion equation [5], [6] the following thermodiffusion equation in spherical coordinates can be obtained The equation (6) for temperature ( , ) T r t , 0 0   r r , 0 0   t t can be rewritten in the spherical case which mathematically coincides with the nonlinear diffusion equation [2]. The numerical calculations provided in [2] give the dependence ( , ) n r t as a straight line in the region The temperature ( , ) T r t dependence was obtained in a similar way [7]. The jump of hotter molecules or particles of impurities with a greater kinetic energy in the region Δ  r r with lower temperature is only possible if exists hotter at the point r .
This requirement is equivalent to the approval that thermodiffusion and diffusion must occur with finite value jumps and velocities. This is very important in defining coefficients of thermal conductivity [3], [7] and diffusion [1], [2].
The nonlinear heat conduction equation [8] presented for one dimensional case using energy density E , which cannot be directly measured, is not perspective viable as equation (9) for temperature ( , ) T r t , which can be measured directly for calculation thermal conductivity [9].
The aim of the article is to get the nonlinear equation describing the flow of atoms and molecules in gases by the thermodiffusion for a spherically symmetric case and to find its approximate analytical solution.

THE NONLINEAR THERMODIFFUSION EQUATION FOR A SPHERICALLY SYMMETRIC CASE
The solution of (9) can be obtained by introducing similarity variable [5] ξ and function ( ) ξ Modelling of Nonlinear Thermodiffusion for a Spherically Symmetric Case EEJP. 1 (2021) which depends on the thermodiffusion constant wS D at source with temperature S T . By substituting (11) into (9) the following nonlinear differential equation can be obtained 2 2 The nonlinear equation (12) can be solved approximately by separating first three terms like the linear equation of hot molecules diffusion [2] can occur with the different lengths r  of the some average frequency of jumps as at linear heat transmittance approach The part of nonlinear equation (12) is transformed to linear (13) and obtained the following expression representing thermodiffusion only by nonlinear processes The first [2] and the third terms in (14) represent a nonlinear diffusion or thermodiffusion. The second term in (14) represents the connection with linear thermodiffusion (13) and nonlinear equation (14) by introducing the term 1 f, which will be demonstrated below gets small numerical values in the region 0 2 ξ   .

APPROXIMATE ANALYTICAL SOLUTION
The term 1 ( ) 2 ξ ξ   l P f of nonlinear equation (14) can be excluded. Thus, a simplified equation is obtained The equation (15) is solved for the source point temperature S T and environment temperature e T T T T (16) by satisfying the boundary conditions for the maximum distance e r of heat penetration and by satisfying the boundary condition for temperature S T at the source point The solution of the linear thermodiffusion equation (13) can by expressed [2] by similarity variables where the radius of source is Δξ . The nonlinear thermodiffusion equation (14) 1 The equation (21) can be modified to the following form which is easy to integrate The obtained solution of the equation must satisfy the boundary condition for maximum heat penetration depth The comparison of solutions (Table) ( ) ξ n f of nonlinear thermodiffusion equation (25) and ( ) ξ f of simplified equation (16) shows the coincidence of both numerical solutions which depend on similarity variables ξ .   The presented profiles of ( ) ξ f and ( ) ξ P in Fig. 2 as well as the results presented in Table 1 show that the term 1 ( ) 2 ξ ξ   l P f gets small numerical values in the equation (14), and, consequently, can be removed. The linear effects in equations (12), (13) and nonlinear equations (15) The average value v in (28) must be substituted by the temperature 1 2  S T T . Gas densities n at this temperature collide with the hottest molecules 2 s n when they satisfy the condition 2  s n n , which formats the front of temperature. The dependence of density distribution of hot molecules m c and S T can explain the formation of the barrier of hot molecules with greater kinetic energies