_{1}

^{*}

The equations for the pair distribution functions are derived directly from the second equation of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. The derivation is fulfilled within the frameworks of the multiscale method. The equations for the pair distribution functions are the kinetic foundation for the multimoment hydrodynamics equations. Solutions to the equations for the pair distribution functions predetermine the possibility of constructing the hydrodynamics equations with an arbitrary number of principle hydrodynamic values specified beforehand. The tendency to increase the number of principal hydrodynamic values is caused by the necessity of interpreting the behavior of the system after the loss of stability. Solutions to the classic hydrodynamics equations constructed for only three principle hydrodynamic values are unable to predict the direction of instability evolution. Solutions to the multimoment hydrodynamics equations are capable of reproducing correctly the phenomenon of emergence and development of instability.

Possibility to study the unstable phenomena by means of the direct numerical integration of the Navier-Stokes equations became feasible comparatively recently. The direct numerical integration of the Navier-Stokes equations in the problem of a flow around a solid sphere was performed by various numerical methods. Nevertheless, the results of all these numerical experiments were absolutely identical (see, review [

Experiment records two stable medium states presented by the

Non-stationary solutions

The idea of bringing the

As expected, the idea of bringing the

The problems encountered by classic hydrodynamics when interpreting the unstable phenomena were not unexpected. They were predicted in [

The s-particle distribution function

The BBGKY hierarchy is closed by the Liouville equation for

Following common ideology of the multiscale method [

here,

According to Equation (2.2а), the one-particle distribution function

Let us switch from phase coordinates

Let us recast the second equation of the BBGKY hierarchy (2.1) in terms of two-particle distribution functions

In Equation (2.4), the vectors

Express the

here,

Further transformation of relations (2.5) gives:

The

riant along the trajectory of the center of mass of a pair of particles to within

Apply Equation (2.2b) to penetrate the domain

The pair of particles with the parameters

Let us apply the operator

a pair of particles to within

Equation (2.8) is specified in terms of dimensionless time and coordinates on the initial scale. Recast Equation (2.8) in terms of distribution function

Upon substituting the third term on the left hand side of Equation (2.4) into the force term of the first equation of hierarchy (2.1), this equation assumes the form:

Let us bring Equation (2.10) into the dimensionless form and assess the order of magnitudes of its constituent terms. In accordance with Equation (2.2а),

To assess the order of magnitude of the first term on the right hand side of Equation (2.10) we use the Equation (2.9). Generally, there are no reasons to believe that the

Expand the dimensionless one-particle distribution function in a perturbation theory series in terms of the virial parameter

Following Equation (2.11), expand functions

The transition (Equations (2.5) and (2.6)) from the

In accordance with Equation (2.2а),

The spatial integration in Equation (2.14) is performed within the

The

The one-particle distribution function

In Equation (2.16) collision integral assumes the form:

Recast collision integral

The velocity

here

Suppose that

This means that particle collisions cannot influence the formation of hydrodynamic values constructed on the properties of the

is a set of the principal hydrodynamic values. The

The equation for the

It follows [

It follows that, when we pass to the hydrodynamic stage from the phase space of one particle, such hydrodynamics equations cannot be constructed using more than three lower principal hydrodynamic values corresponding to the

The use of the Boltzmann hypothesis (“Stosszahlansatz”) opens up the possibility of approximate passage to hydrodynamics. Following Boltzmann, let us factorize two-particle distribution functions in the

Boltzmann hypothesis (2.23) closes Equation (2.16). The obtained classic kinetic equation for the

So, the physical meaning of the error introduced by the Boltzmann hypothesis (2.23) into hydrodynamics is as follows. It follows that just Boltzmann hypothesis allows us to construct hydrodynamics on only three lower principal hydrodynamic values. It follows that the use of the Boltzmann hypothesis excludes higher principal hydrodynamic values (2.21) from the participation in the formation of classic hydrodynamics equations. To include the higher principal hydrodynamic values, we must find passage to hydrodynamics from the phase space capable of accommodating the whole set of binary particle collision

The second equation of the BBGKY hierarchy (2.1), like the first one, is not closed. The integral term of the second hierarchy equation contains a three-particle distribution function responsible for interaction of particles 1 and 2 with some third particle 3. The absence of closeness of the second hierarchy equation prevents us from the direct transition to the hydrodynamic stage from the phase space of two particles.

In accordance with Equation (3.1), under absence of collisions with particle 3, particles 1 and 2, which at the time

In accordance with Equation (3.2), the distribution function

Let us evaluate the typical number of particles

In accordance with this estimation, the order of magnitude of

In accordance with ideas of the kinetic theory of gases [

Let particles 1 and 2 are located so that at time

sion cylinder

Equation (3.3) has a clear physical meaning. It asserts that a pair of flying-apart particles, that is, of particles that have already left the domain of their interaction

Let particles 1 and 2 are located so that at time

Monotonic increase (3.2) of the second term on the right hand side of Equation (3.1) contradicts the ideas of the kinetic theory of gases on a free path of particle. Indeed, particle 1 will experience collision with particle 2 at the time ^{−} and the number of collisions N_{4} within the volume V. Thus, the assumption that a particle may be present at all phase space locations with equal probabilities sets too high estimation of the second term on the right hand side of Equation (3.1). Evaluation of the second term on the right side of Equation (3.1) requires a more accurate estimation of the order of magnitude of the

Equation (3.4) has a clear physical meaning. It asserts that a pair of drawing together particles, that is, of particles that have not yet reached the domain of their interaction

Let particles 1 and 2 are located so that at time

here,

Generally, any medium particle forms a pair with every other particle. A medium therefore contains

Expand the dimensionless two-particle distribution function in a perturbation theory series in terms of the virial parameter

Let us substitute Equation (3.7) into the dimensionless Equation (3.5). Equating the multipliers at equal degrees of

Integrating Equation (3.8) with respect to

where

Equation (3.8) allows putting on the trajectory

Let us recast Equation (3.11) in the dimensionless form. It turned out that

erate with

along the trajectory of a center of mass of a pair of the

В (3.12)

The first term in the right hand side of Equation (3.12) contains information, which is excessive for the kinetic stage of gas description. This is information about mutual disposition of particles in the pair contained in the

The replacement (3.14), being substituted in Equation (3.12), makes this equation irreversible.

Expanding the left hand side of Equation (3.12) into series in terms of

When writing the right hand side of Equation (3.15), we took into account that, according to Equation (3.13), function

The spatial integration in Equation (3.16) is implemented near point

Multiplying

Let us average Equation (3.15) within the

Let

Equation for

where

Without performing the irreversible replacement in Equation (3.21), we integrate this equation with respect to

The solution to the closed set of Equations (3.19) and (3.22) discovers the basic property of the pair distribution functions. It shows that the distribution functions

This means that particle collisions cannot influence the formation of hydrodynamic values constructed on the pair properties of

is a set of principle hydrodynamic values. The relationship between the functions

The Navier-Stokes equations have been used with much success for more than one and a half century as a tool for very accurately describing stable incompressible flows, both stationary and non-stationary. However, in recent decades, the Navier-Stokes equations have been faced with insurmountable difficulties in interpreting flows, losing its stability. In accordance with the interpretation of [

Classic hydrodynamics equations are constructed for only three lower principle hydrodynamic values: the density of particles number, the hydrodynamic velocity, and the pressure. Higher principle hydrodynamic values are not used in the formation of classic hydrodynamics equations. The possibility of the formation of hydrodynamics equations with an arbitrary number of principle hydrodynamic values specified beforehand was found in [

The distribution functions for pairs of drawing together and flying apart particles referred to as pair distribution functions are constructed in [

In [

The common ideology of the multiscale method is described in [