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In the article correct method for the kinetic Boltzmann equation asymptotic solution is formulated, the Hilbert’s and Enskog’s methods are discussed. The equations system of multicomponent non- equilibrium gas dynamics is derived, that corresponds to the first order in the approximate (asym- ptotic) method for solution of the system of kinetic Boltzmann equations.

In 1912 Hilbert considered the kinetic Boltzmann equation for one-component gas as an example of integral equation and proposed a “recipe” for its approximate (asymptotic) solution (see [

The approach of Struminskii, who had proposed in 1974 in [

In section 2 below will be proposed the correct method of asymptotic solution of the kinetic Boltzmann equations system for multicomponent gas mixture for the approach, that combines Enskog’s and Struminskii’s approaches; in particular, it will be shown, how one has to modify Enskog’s method: in addition to asymptotic expansion of the velocity distribution function i-component particles of gas mixture it is necessary to determine and to use the expansion of the particle number density

Further, in the Section 3 the system of infinitesimal first order equations of multicomponent non-equilibrium gas dynamics, appearing during the process of the solution of the system of Boltzmann equations by successive approximations method in the Section 2 as necessary condition of the existence of approximate (asymptotic) solution of the integral equations system, is considered in more detail.

This article is condensed version of our article arXiv:1303.6275. Notations, used below, are close to notations in [

The Boltzmann equations system, that describes change of dependent on t and spatial coordinates, prescribed by radius-vector

in (2) N is a set of indexes, that are numbering components of the mixture;

by prime in (20) and below the velocities and the functions of velocities after the collision are denoted.

Let us introduce following notations:

to differ velocities of colliding molecules of the same kind in (22) the one velocity is denoted by

In Enskog’s approach the differential parts of the Boltzmann Equations (2), that are denoted by

In Struminskii’s approach to the asymptotic solution of the Boltzmann equations system the differential parts of the Boltzmann Equations (2) and the collision integrals of the particles of i-component with the particles of the other components are considered to be small as compared with the collision integral of the particles of i-component between each other, therefore the indicator of infinitesimality

It is possible to combine Enskog’s approach with Struminskii’s approach. For this purpose we divide the set of mixture components N into two subsets: the subset of components, that we call formally inner components (we could consider the case, when there are some subsets of inner components, but this case does not fundamentally differ from the one, considered below, the only difference is that the notation become more complicated) and the subset of components, that we call external components. To differ the two groups of mixture components we

denote the subset of indexes of inner components

intersection of the sets

Let us write the asymptotic expansion of the velocity distribution function

The differential parts of the Equations (3) are written as:

where

―cf. with [

Substituting (10) and (11) in (8) and equating coefficients at the same powers of

Similarly substituting (10) and (11) in (9) and equating coefficients at the same powers of

functions of particles of external components of gas mixture

Speaking about an order of approximation below, we assume the order to be equal to the value of index r in (14), (16). According to (5), (13), in zero order approximation we have the following system of integral equations to find the velocity distribution functions of particles of inner components of gas mixture

The general solution of the equations system (17) can be written as a set of the Maxwell functions:

where k is the Boltzmann constant.

Particle number density

in (21)

that is convenient to use below instead of definition (21).

According to definitions (19), (20), (21), in addition to the asymptotic expansion (10) it is necessary to determine asymptotic expansions for particle number density

mean mass velocity

and temperature

Substituting (10) and (23)-(25) in (19), (20), (22) and equating terms of the same infinitesimal order we obtain

In (27), (28) the notations are introduced

In particular, for

density, the mean mass velocity and the temperature of inner components of the mixture:

According to (4), (15), zero order integral equations, from which the velocity distribution functions

―are simpler than Equations (17) and differ actually from (17) only by lack of summation over components. Therefore, similarly (18), the general solution of the equations system (34) can be written as a set of the Maxwell functions:

where

Let’s add to the definition of the number density of particles of i-component definitions of mean velocity

from (19), (36), (37) the equality is obtained:

that is convenient to use below instead of definition (37).

Let’s enter similar (24)-(25) asymptotic expansions of outer

and outer

Substituting (10), (23), (39), (40) in (19), (36), (38) and equating terms of the same infinitesimal order we obtain for each

cf. with (26)-(28). In (42), (43) the notation is used

For

velocity and the temperature of outer

For

into account, can be rewritten in the form

in (49) functions

The left-hand sides of Equations (49) involves functions, that are known from the previous step of the successive approximations method. Unknown functions

Multiplying Equations (50) by

From (51) we conclude, that

where

To simplify further evaluations according to the expression for

where

where

Multiplying Equations (55) by

Among (infinitesimal) set of particular solutions of the system of Equations (55), different from each other on some solution of the system of homogeneous Equations (50), unique solution

Having substituted expression for

in (26)-(28), taking (18), (29)-(33) and (58)-(59) into account, we obtain a system of

from which we find expressions for functions

fficients of asymptotic expansions of the particle number density of

Then the fulfillment of equalities (56)-(57) can be considered as the differential equations, the r-order equations of gas dynamics, for finding

The partial solution of the system of inhomogeneous Equations (55)

coefficients, depending on

For

where

The fulfillment of analogous (56)-(57) equalities

can be considered as the differential equations, the r-order equations of gas dynamics, for finding

Let us consider in more detail the system of infinitesimal first order Equations (56)-(57), (71)

To simplify transformations, according to the expressions for velocity distribution functions of particles of infinitesimal zero order (18), (35), functions

for inner components

At transformation of differential parts of the Equations (56)-(57) and (71) we use equalities:

In (75)-(77) the bar above symbol with index

depending on external forces

After simple transformations from (56)-(57) and (71)

In accordance with the general definition of pressure tensor of i-component of gas mixture

and with the general definition of i-component heat flux vector

(cf. with [

is inner components pressure tensor of zero order,

is inner components heat flux vector of zero order,

is zero order internal energy of particles of inner components per unit volume, which is equal, in this case, to energy of their translational chaotic motion, however, the energy transfer equations, written in form (81) and (84) can be used in more general cases as well (cf. with [

is

is

is zero order internal energy of particles of

General analytic expressions for integrals

System of infinitesimal first order equations of multicomponent non-equilibrium gas dynamics (79)-(84) is proposed to use for describing turbulent flows

S. A. Serov,S. S. Serova, (2016) Asymptotic Solutions of the Kinetic Boltzmann Equation and Multicomponent Non-Equilibrium Gas Dynamics. Journal of Applied Mathematics and Physics,04,1687-1697. doi: 10.4236/jamp.2016.48177