Detailed evaluating the results of the direct numerical integration of the Navier-Stokes equations against experiment in problem on flow past a hard sphere at rest is carried out in [1] . Experiment records three stable medium states. Each of these three stable flows begins to develop in its own direction, qualitatively different from other flows when it loses stability. Calculation satisfactory reproduces all three stable medium states observed experimentally. However, the calculation is incapable of producing anything that corresponds to seven unstable regimes observed along the three directions of instability development. The analysis of numerous divergences between the results of numerical integration of the Navier-Stokes equations and the experiment [1] -[3] led to the following conclusion. Solutions to the classic hydrodynamics equations successfully reach the border of the instability field represented by the dashed slanting line in Figure 1 from [3] . As Reynolds number grows, these solutions move along the border of the field. However, the solutions to the classic hydrodynamics equations are unable to cross this border and to pass into the instability field.

The Navier-Stokes equations themselves are called as the most probable reason for discrepancies between calculation and experiment [1] - [4] . It may be likely that the Navier-Stokes equations become inapplicable to unstable phenomena. The responsibility for the failure of the classic hydrodynamics was laid on the approximation made in deriving the Boltzmann equation, namely, the hypothesis of molecular chaos “Stosszahlansatz”. The Boltzmann hypothesis closes the kinetic equation, allowing classic hydrodynamics to be constructed for only three lower principle hydrodynamic values. It turns out that the neglect of higher principle hydrodynamic values does not introduce visually observable changes into the picture of stable flows. This error, however, grows very rapidly after the loss of stability.

In present paper the multimoment hydrodynamics equations [5] are applied to solve a problem on flow past a hard sphere at rest at a wide range of values. The direction of evolution of the ground axisymmetric flow [1] after losing its stability is studied. In Section 2, the problem is formulated. In Section 3, the problem is solved for the Stokes flow around a sphere, i.e., at. The Section 4 is devoted to construction of the solution to the multimoment hydrodynamics equations suitable of interpreting the stationary axisymmetric flow at a moderately high values. The Section 5 examines the behavior of the solution that loses its stability after the passage of the first critical Reynolds number value. In Section 6, the characteristic features of the appearance of instability are interpreted in terms of entropy. The principle of retention and loss of the open system stability is formulated. In Section 7, the characteristic features of the development of instability are interpreted in terms of entropy. The evolution criterion is formulated. The Section 8 is devoted to finding the solutions to the multimoment hydrodynamics equations capable of reproducing a vortex shedding. The Section 9 provides an algorithm to select one unstable solution of many found unstable solutions. The selected solution indicates the direction of system evolution. In Section 10, the results are compared with the experimental data.

Consider a space filled with thermodynamically equilibrium gas. Suppose that a solid sphere of radius moves in gas at constant velocity along the axis of the immobile Cartesian frame of reference. Let us now pass from to the Cartesian frame of reference with the axes parallel to those of and the origin made coincident with the center of the moving sphere. In the frame of reference, the sphere is at rest, the inflowing-gas velocity at an infinite distance from the sphere, , is aligned with the positive direction of the axis, and the flow problem is stationary.

The pair distribution function corresponding to the equilibrium state is

where and are the local density of the number of particles and the temperature of unperturbed gas, , , is the mass of a gas particle, and and are the velocity of the center of mass and relative velocity of the pair of particles. In, the pair distribution function can be written as

Let, and be the Cartesian coordinates of point in the space;, and are its spherical coor- dinates, and, Figure 1. Bring the origin of the Cartesian frame of

reference into coincidence with point so that the axis is directed along vector. Mark the projections of vector onto the axes by, and, its projections onto the axes of, by, and, and its spherical coordinates, by , and, Figure 1.

The basic property of pair distribution functions (17) from [3] is that is conserved with time along the trajectory of the center of mass of a pair of particles. In the stationary case, the function remains unchanged along a straight line parallel to vector

The, , and functions are the first integrals of Equation (2.3). These functions were termed in [6] trajectory invariants. In this study, consideration is restricted to gas flows around a sphere, which are invariant under rotation through an arbitrary angle about the axis. Let us compose combinations of trajectory invariants, , and, invariant with respect to this rotation

By virtue of the flow geometry, there are two independent regions of integration with respect to at each point in the gas. The first region, denote it by, is a spherical cone, Figure 2. This region incorporates the trajectories of centers of mass of pairs of particles that originate and terminate at the surface of the sphere. The integration limits for this group of trajectories are. The second region, , embraces all possible trajectories of centers of mass of particle pairs converging to point. The integration limits for the second group of trajectories are unbounded.

According to the general approach to solving the multimoment hydrodynamic equations in terms of pair functions, outlined in [6] , solution is sought for in the form of an infinite series of the products of trajectory invariants (2.4)

The spatial dependence of (2.5) is controlled by trajectory invariants (2.4), and coefficients are independent of. In keeping with definitions given in [5] , the expressions for the principal hydrodynamic values―the moments of (2.2)―can be written as

,

,

,

, ,

,

,

,

Here, is the local density of the number of particles; is the hydrodynamic velocity;, , , and are the pressure, temperature, stress tensor, and heat flux corresponding to motion of the centers of mass of pairs of particles;, , and correspond to relative motion of particles within pairs;.

Nonprincipal hydrodynamic values and are given by Equation (21) from [3] with the Navier- Stokes accuracy. In terms of spherical coordinates, these values for cylindrically symmetric flows considered in this study (i.e., for flows invariant with respect to rotation through an arbitrary angle about the axis) assume the form

Here, is the coefficient of the dynamic viscosity. From the definitions of gas pressure, temperature, viscous stress tensor, and heat flux [5] it follows that they are linear combinations of moments (2.6) and (2.7)

According to [5] , the overall equations of conservation of the number of particles, momentum, and energy assume the form

The basic property of pair distribution functions (17) from [3] subdivides Equations (2.10) and (2.11). As recast in terms of the spherical coordinates, the and components assume the following form for the cylindrically symmetric flow in question

The boundary conditions must allow for the fact that the contribution of the function given by Equation (2.5) to the hydrodynamic values vanishes at an infinite distance from sphere

At the surface of the sphere, i.e., at, one has to impose the no-leak and no-slip conditions

Heat flux traversing each element of the sphere surface must be counterbalanced by the heat flux within the sphere and the thermal radiation

Here, is the coefficient of the thermal conductivity of the sphere, and is the coefficient of the sphere emissivity.

Let us formulate dimensionless parameters, where denotes the Mach number, and the Reynolds number,. Multimoment hydrodynamics Equations (2.9)-(2.13) will be brought to a dimensionless form in a conventional manner [7] . It turns out that the dimensionless equations contain and. Hydrodynamic values can be represented as parametric series

Here, sequentially assumes values;;, , and;

and, and sequentially assumes values;;;. Symbols and appear

over dimensionless quantities. Multimoment hydrodynamics Equations (2.12), (2.13) are written with the Navier-Stokes accuracy, by which token expansion (3.1) contains no terms with.

Consider the flow regime at small Reynolds number. At the hydrodynamic stage of description characterized by small values of the Knudsen number,. Terms of the order of dominate expansion (3.1) of the second-order moments and in the flow regime in question. In expansions (3.1) of hydrodynamic values, , and, it will suffice to retain only, and, respectively. Expansion (3.1) of the particle density flux is reasonably be re-

stricted to, which provides for the desired order of the terms in Equation (2.7) for

. As noted in [6] , the Navier-Stokes approximation is not accurate enough to calculate temperature components and. Thus, at, the spatial dependence of temperature is neglected.

Let us truncate expansion (2.5) to the three lowest-order terms

Each of three terms of series (3.2) can contribute to each principal hydrodynamic value (, , , , , , and). Considering that the principal hydrodynamic values are linearly independent, coefficients, of expansion (3.2) can be written as linear combinations of six arbitrarily chosen components

To calculate the hydrodynamic values (2.6), the first component of series (3.2) has to be integrated with respect to with the appropriate weight function of velocities and within region, and the second and third components, within region. Why is approximated by Equation (3.2), and its components are integrated in such a way is explained below. The integration yields

Here,

The order of coefficients and appearing in Equations (3.4, 3.5) depends on the only density flux component, , retained in expansion (3.1) at. The order of coefficients, , and de-

pends on components and of expansion (3.1). Using one of six linear combinations of

coefficients, we eliminate the contribution to the particle density, proportional to, which is missed in expansion (3.1). Using two linear combinations of coefficients, we eliminate the contribution

to and that is proportional to and is missed in expansion (3.1). Two other linear combina-

tions of enable us to eliminate the contribution to and that is proportional to. The reason why and contain no terms proportional to will be elucidated in Section 4. The last, sixth linear combination of yields in Equation (3.5). Three linear combinations of enable us

to eliminate the contribution to being proportional to and the contribution to and being proportional to, which are missed in Equation (3.1). The first term of series (3.2) con-

tributes only to the first and the third moments of the function. Two linear combinations of offer means of eliminating the contribution to and being proportional to.

Using boundary conditions (2.15) we obtain,. The components of tensor (2.7) can be determined from Equation (3.4). Substituting and from (3.4) into any of the equations of momentum conservation (2.12), we obtain. When analyzing the multimoment hydrody-

namics equations in the order of [6] , it was inferred that, , hence, ,. Substituting the foregoing expressions for the coefficients, , , , into Equation (3.4) and using definitions (2.8), we obtain

Distributions (3.6), (3.7) follow from the classic hydrodynamics equations [7] for the Stokes flow, which are identical to the multimoment hydrodynamics equations in the limit [6] .

The hydrodynamic values (3.6), (3.7) assume the form of a linear combination of the products of by trigonometric functions. To calculate (3.6), (3.7), the first component of Equation (3.2) was integrated with respect to within the region, and the second and third components, within the region. If, on the contrary, the first component is integrated within, and the others, within, Equation (3.4) contain the

products of irrational and trigonometric functions. However, with such products in Equation

(3.4), it is impossible to satisfy boundary conditions (2.15) and equations of momentum conservation (2.12) simultaneously. When the higher-order terms are being retained in expansion (2.5), the functions will appear in Equation (3.4) which are nonlinear in cosθ and sinθ and the Equations (2.15) and (2.12) cannot, as previously, be satisfied simultaneously.

Consider a flow at a moderately high and. In this case, will be retained together with

two other components of expansion (3.1) of the flow density: and, In ad-

dition to already obtained components of expansions (3.1) of the zero- and second-order moments of the pair distribution function, , , and with и, will be taken into account. Expansions (3.1) of the energy fluxes and will be limited to and with и

Expressions for hydrodynamic values (2.6) will be constructed by the same principle as in Section 3―as linear combinations of the products of and trigonometric functions. This principle governs both the structure of the retained terms of expansion (2.5) and the region of integration with respect to for each term of expansion (2.5). In addition to three terms given by Equation (3.2), the following components of expansion (2.5) will be included

Generally, each component of expansions (4.1, 4.2) contributes to each principle hydrodynamic value, irrespective of the order of terms retained in expansion (3.1). Considering that the principle hydrodynamic values are linearly independent, each coefficient, should be represented as series (3.3) of six terms for any

considered order of expansion (3.1). Function given by (4.1) is used to calculate the zero- and the second-order moments. The contribution of is eliminated from the first- and third-order moments

through the use of Equation (3.3), as in Section 3. Function (4.2) is used exclusively to calculate the first- and the third-order moments. The terms of Equation (4.1) proportional to, , and the terms of Equation (4.2) proportional to, are integrated with respect to within region. The terms of Equation (4.1) proportional to and the terms of Equation (4.2) proportional to are integrated with respect to within region.

Distributions of the principle hydrodynamics values (2.6) are given in [8] by Equations (A.1)-(A.7). The expressions (A.1)-(A.7) are written in terms of coefficients. Coefficients are expressed in terms of which appear in Equations (4.1), (4.2). Each coefficient is expanded into series (3.3) of six terms for each considered order of expansion (3.1). In what follows, the relationships between and required to derive explicit expressions for and, will be omitted. Each coefficient, contains dimensionless multiplier.

In order to calculate, one needs, as in Section 3, to satisfy boundary conditions (2.14)-(2.16) and equations of conservation (2.12), (2.13). Imposing boundary conditions (2.15) upon the particle density flux (A.5) from [8] , we express 16 coefficients, in terms of 34 remained coefficients [8] . As revealed in [6] , the -order approximation to the total energy-flux vector is

Here is the coefficient of the thermal conductivity. Substituting the expressions for hydrodynamic values (A.1)-(A.4) from [8] into Equation (4.3), it is possible to express 8 coefficients, , in terms of 26 remained coefficients. Substitute Equations (A.2), (A.5), and (A.6) from [8] into equation of mo-

mentum conservation (2.12) and equate the coefficients of the products for either of the

orders, and. Then 6 coefficients, , can be expressed in terms of 20 remained coefficients, , [8] . Eventually, the expressions for the principle hydrodynamic values (2.6) can be written as

In (4.4)-(4.10), for, for.

The terms proportional to and, were dropped from Equations (4.4)-(4.7) for the zero-and the second-order moments, respectively, because for Equation (2.13) to be satisfied in the order of, all the coefficients of these terms must be zero. For Equation (2.13) to be good in the order of

, one has to put expression (4.9) for From the conditions for no heat flux at the surface of the sphere (at) and at infinite distance from the sphere the expression follows (4.10) for Let us next substitute Equations (4.4)-(4.10), and (3.6), (3.7) into Equation (2.13) and equate the coefficients of products of the order of. It turns out that all but the

and products satisfy the law of energy conservation. These two products satisfy Equation (2.13) at

As previously, Equation (2.13) must be satisfied for each product of the order of individually. Eventually we obtained a nonlinear set of eighteen algebraic equations (A.8-A.10) from [8] . In [8] , of eighteen equations we retained sixteen ones.

The total energy flux equals the heat flux at the surface of the sphere,. Using the solutions of

the internal boundary problem for the Laplace equation, the temperature distribution inside the sphere can be expanded in terms of Legendre polynomials [9] . The coefficients of this expansion can be calculated by matching the temperature distribution (4.4)-(4.7) at the surface of the sphere with the distribution derived from the solution of the internal boundary problem. The resulting balance of heat fluxes at the surface of the sphere (2.16) in order of becomes

Here,.

Sixteen equations (П.8, П.9, П.13) from [8] supplemented with three Equation (4.12), and Equation (4.11) form closed nonlinear set of twenty algebraic equations for twenty coefficients,. Four coefficients, namely, , , and, appear in the distribution of particle density (4.4). Seven coefficients, , , , , , and, are responsible for the distributions of pressure and stress (4.6), (4.7). Seven coefficients, , , , , , , and, appear in the distributions of energy flux (4.9), (4.10), two coefficients, and , govern the distribution of particle-density flux (4.8).

When constructing the distributions of hydrodynamic values in Section 3, the series (2.5) was truncated to the terms that contribute linearly in and to these distributions. Going beyond the limits of the case, we retained seven terms in Equation (4.1) and eleven terms in Equation (4.2). The expansion (4.1) is used to calculate the zero- and the second-order moments and makes the contributions to Equations (4.4)-(4.7) that are proportional to 1, , and. The expansion (4.2) is used to calculate the first- and the third- order moments and gives rise to the -components of Equations (4.8)-(4.10) proportional to and. With higher-order terms of expansions (4.1), (4.2) taken into account, the contributions to Equations (4.4)-(4.10) are proportional to more high powers of. We were compelled to make one exception associated with retention of the trajectory invariants proportional to, , and in Equation (4.2). Owing to these invariants, the distribution (4.8) involves the component proportional to, lacking in the expansions of the other hydrodynamic values (2.6). The terms of expansion (4.2) proportional to, , and enabled us to allow for the terms of the order of, , in the distribution of particle-density flux (4.8). The terms of this order dominate at.

Numerical integration of the nonlinear set of 20 algebraic equations was carried out at, and. Calculations have revealed a great many roots. However, at, the set has only one invariably stable root, , displayed in Figure 4 in [8] . According to the, , solution, an axisymmetric recirculating zone is formed in the wake behind the sphere at Re~20. This recirculating zone has the shape of an axisymmetric toroidal ring. It expands as grows but its shape remains unchanged (Figure 3(а)). At, the system becomes unstable. At, the Barnett corrections [7] are known to be commensurate with the Navier-Stokes terms. For this reason the calculations [8] were interrupted at.

As the flow around sphere becomes unstable, the problem becomes nonstationary. As in the stationary case, the pair distribution function is represented as

Here, is given by Equation (2.2), and is given by Equation (2.5) with time-de- pendent coefficients. In going to a stationary flow, pair distribution functions lose their main property (17) from [3] . Retaining, as previously, 21 trajectory invariants in expansion (2.5), we arrive at distributions (A.1)-(A.7) from [8] of hydrodynamic values with time-dependent coefficients,. Coefficients are derived from the condition that hydrodynamic values satisfy boundary conditions (2.14)-(2.16)

and nonstationary Equations (2.9)-(2.11). Executing the sequence of transformations, which in Section 4 led us to the closed algebraic set of twenty equations, we obtain a closed nonstationary set of twenty equations denoted hereafter by. Sixteen algebraic Equations (A.8), (A.9), and (A.13) from [8] supplemented with three algebraic Equation (4.12), remain unchanged when passing from the stationary problem to the nonstationary one. However, Equation (4.11) changes to

where.

Expressions (4.4)-(4.10) with time-dependent coefficients, , define distributions of principle hydrodynamic values. The set is applied to study the time history of coefficients

. The stationary solution is described in Section 4. Functions give deviations from the stationary solution.

As revealed by numerical integration of the set, solution, , remains asymptotically stable up to certain critical value,. This means that the small deviations from the

stationary solution, , are damped at. The passage of is accompanied by the

, , solution stability loss. Starting with, the, , small axisymmetric deviations begin to increase exponentially at. The solid curve in Figure 4 is the time history of at. The, , deviations grow up to time. At, the solution is cut off. Why the solution to the set terminates at is best explained in terms of trajectories on the phase plane, (Figure 6 in [8] ). It turned out that at, the set becomes unsuitable for modeling evolution of the physical system.

Time history of the coefficient at creates time dependence at the distribution of particle- density flux (4.8). The distribution (4.8) corresponds to observed evolution of the periphery of the recirculating zone in the wake behind a sphere [10] [11] . In accordance with experiment, starting with, and up to, the periphery of the recirculating zone in the wake behind a sphere moves translationally, receding from the sphere. Starting with the periphery of the recirculating zone moves back towards the sphere [10] [11] .

Suppose that at time, velocities of all gas particles flowing around the sphere reversed their direction, and the boundary conditions were also reversed. Then the periphery of recirculating zone in the wake behind the sphere starts moving back towards the sphere at. As proven in [6] , macroscopic motion of the gas with reversed velocities of its constituent particles is governed by the reverse equations for pair functions. So, observed reverse motion of the periphery of recirculating zone should be described by means of the reverse equations.

However, upon reversing velocities of all gas particles and the boundary conditions, certain details of macroscopic motion of the gas disagree with experiment [10] [11] . For example, upon reversal of gas-particle velocities, the gas involved in vortex motion starts circulating in the opposite direction. However, all gas particles cannot be set in reverse motion at. Nevertheless macroscopic reversal of all gas-particle velocities and boundary conditions is by no means the only way of initiating reverse motion in the wake as a whole. At, reverse motion in the physical system starts spontaneously [10] [11] , i.e., without any interference from outside. Insofar as the reverse equations apply to a vortex propagating backward upon reversal of all gas-particle velocities, it is reasonably assumed that the same equations apply to a vortex set in reverse motion in some other way.

Both the direct and the reverse multimoment hydrodynamics equations are specified by Equations (54)-(56) in [5] . However, expressions for reverse nonprinciple hydrodynamic values and are specified by Equation (2.7), in which each term of the right-hand side has opposite sign [6] .

Executing the sequence of transformations, which led us to the closed nonstationary set, we obtain a closed nonstationary set of twenty equations for coefficients, , denoted hereafter by. Sixteen algebraic Equations (A.8), (A.9), and (A.13) from [8] supplemented with three algebraic equations (4.12) remain unchanged in going from the direct to the reverse nonstationary problem. However, Equation (5.2) changes to

It turned out that the, , solution to the set existed in the neighborhood of the cut-off point. According to [12] [13] , the law of large numbers is violated near singular points (bifurcations, regions of the coexistence of several stable solutions, etc.), and large spontaneous fluctuations may appear in the system. In conformity with [12] [13] , a large spontaneous fluctuation causes the transfer of the system from the cut-off point to the, , solution at time. Starting with, axi-

symmetric deviations, , begin to decay exponentially,.

At time, the solution reaches the neighborhood of the, , stationary solution. At time further evolution of the, , solution finishes. The, , solution exists at, the, , solution exists at,. At any time instant:

A large spontaneous fluctuation causes the transfer of the system from the, , solution

to the, , solution at time. The, , solution is unstable. It follows that

small axisymmetric deviations, , begin to grow starting with time. This pro-

cess is repeated periodically. So, we obtained the solution, , , referred hereinaf-

ter as.

Numerical integration of the, set draws the time history of coefficients,. The

time history of and shown in Figure 4 by a solid line corresponds to initial deviation

. Deviations are dependent quantities and can be calculated from the set, in which equation (5.2) is substituted by equation. Calculation gives,. The equations of the, set were integrated with the accuracy of the order of. Accuracy is a measure of fluctuations of hydrodynamic values at thermodynamic equilibrium. The time history of the coefficients and from Figure 4 corresponds to axisymmetric pulsations of the periphery of recirculating zone in the wake behind a sphere. The half-period of pulsations was estimated at approximately one minute at and.

The dashed curve in Figure 4 is the, dependence calculated at

with accuracy. Thus, the, nonlinear set “draws apart” initially close trajectories. This sensitivity to initial conditions was called the Loretz “butterfly effect” [14] .

The multimoment hydrodynamics equations [5] , as well as the classic hydrodynamic equations, govern space and time evolution of the whole ensemble of systems (Gibbs ensemble) rather than of some individual system. All the microscopic parameters of each individual system are compatible with the initial macroscopic parameters which are present not in the form of particular values but in the form of intervals allowing for their possible fluctuations [15] . Thus, each statistical coefficient, , , is a linear combination of a great many dynamic coefficients, denote their number by K which can be infinitely large

Dynamic coefficients are calculated within the classic mechanics. Fluctuation at any

time is defined as a difference between the dynamic and statistical coefficients,

, or which is the same, as a difference between two arbitrary dynamic coefficients,.

Consider the range of the system parameters within which the equations for statistical coeffi-

cients have stable solution,. Within this range, the overwhelming majority of dynamic coefficients passing in the immediate vicinity of at will remain in the vicinity of indefinitely long. Statistical solution, , describes most of the dynamic trajectories, , with the accuracy.

The situation in the unstable range is radically different. Consider two ensembles of systems. The first -ensemble at incorporates systems with coefficients within

small vicinity near -solution, here,. The se- cond -ensemble at incorporates systems with coefficients within small vicinity near -solution, here,. Let, at time. Time history of the coefficients and is

drawn in Figure 4 (solid and dashed curves). Initially close trajectories diverge. It immediately follows that each dynamic -coefficient, , strictly speaking, behaves in its own way. There is no unique, , which describes any dynamic coefficient from the set, , for the whole ensemble with the accuracy. The Gibbs ensemble disintegrates.

Disintegration of the Gibbs ensemble in the unstable region suggests the follows. The multimoment hydrodynamics Equations (2.9)-(2.11) governing the Gibbs ensemble as a whole are invalid in the region where solutions to these equations become unstable. The and dissipation moments (2.7) are also true for the ensem-

ble as a whole. Deviations, , are nothing but fluctuations of stationary coefficients,

.

Fluctuation of some coefficient, namely, , is preset at, for example,

. Fluctuations, , of other coefficients are calculated from the set, in which Equation (5.2) is substituted by equation. Time evolution of the so for-

mulated fluctuation is approximated by solution of the, set. Fluctuation of any hydrodynamic value at any instant and any point is calculated from Equations (4.4)-(4.10). The fluctuations are interrelated in time and space. In [8] , these interrelated fluctuations have been termed regular. However, there is a factor always present in real physical systems. It is the spontaneous chaotic fluctuation [16] . Spontaneous fluctuations are random independent events. Thus, fluctuation of the statistical coefficient is

After the attainment of the first critical value, the solution to the multimoment hydrodynamics equations loses its stability. The conservation laws (2.9)-(2.11) governing the Gibbs ensemble as a whole become invalid. Regular fluctuations alone cannot provide for fulfillment of Equations (2.9)-(2.11). Strictly speaking, to solve the unstable problem accurately, one needs to switch from the statistical to the dynamic level of description and apply the equations of classical mechanics modeling the dynamics of each individual gas particle. However, numerical integration of the classical mechanics equations for a tremendous number of particles (which is sometimes infinite) is an extremely arduous problem. This line of attack seems ill advised.

In [17] , when modeling an individual system, each hydrodynamic value in the equations of conservation was supplemented with its spontaneous fluctuation component. As was done in [17] , let us attract spontaneous fluctuations. With spontaneous fluctuations taken into account, equations of conservation (2.9)-(2.11) are satisfied in the case of both direct and reverse multimoment hydrodynamics equations. To satisfy the Equations (2.9)-(2.11), the contribution of the time derivative of regular fluctuations must be counterbalanced by the con-

tribution of time and space derivatives of spontaneous fluctuations. Namely

The functions incorporate space derivatives of spontaneous fluctuations. In the case of reverse equations of conservation, besides of Equation (5.7), the change in third equation of set (4.12) (namely, the replacement of by) must be counterbalanced by the contribution of spontaneous fluctuations.

Let us now transform Equation (5.7) to the dimensional form and assess the order of spontaneous fluctuations varying on time and space scales and. Possible large-scale spontaneous fluctuations with and are of the order of large-scale regular fluctuations. Realistic small scale spontaneous fluctuations with and contribute very little, if at all, to the distributions of hydrodynamic values. However, their time and space gradients are of the basic order of magnitude.

Let us mentally circumscribe a sphere of radius around the sphere of radius. Let us term the gas confined between the surfaces of the coaxial spheres the physical system or simply the system. Let us now turn the radius of the larger sphere to infinity. In pursuance of the condition, the number of particles clinging to the surface of the sphere of radius at any instant is negligibly small compared to the total number of particles in the system. The condition makes it possible to form quasi-isolated system. The processes occurring in a quasi-isolated system are studied without regard for interaction of this system with the ambient medium [16] . According to [16] , let us neglect fluctuations of hydrodynamic values characterizing the system as a whole, which arise due to permeability of the external sphere of radius. In pursuance of the, these fluctuations cannot alter the general physical pattern of the processes.

When deriving equations of entropy conservation (А.6), we reasoned from the concept of a Gibbs ensemble of systems. When modeling an individual system, each hydrodynamic value in the equations of conservation should be supplemented with its fluctuation component [17] . Let us correct Equation (А.9) for the

fluctuation of the pair entropy,. Here, superscript (0) corres-

ponds to the 0-solution to the multimoment hydrodynamics equations. Integrating the resulting equations with respect to over the volume of the system yields

In accordance with (6.1), evolution of the pair entropy is defined by two factors, by the

entropy production in the system and the entropy outflow through the surface confining the system.

Balance Equation (6.1) is valid for and individually. The function characterizes the ensemble of systems as a whole. The second term of the left-hand side of Equation (6.1) is integrated over surface confining volume (over the surface of the spheres of radii and). Note that the physical meaning of the Boltzmann entropy was established without resorting to the concept of a Gibbs ensemble [18] . Inequalities (А.3) were also derived without invoking this concept. Thus, inequalities (А.3) are also valid for, , and, and entropies, , and also convey the meaning of volume occupied by the system in the Г-space.

In terms of Equation (5.6), the fluctuation can be written as

Here, superscripts “R” and “S” mark the contribution of regular and spontaneous fluctuations to the entropy.

Explicit analytical expressions for space distributions of principle hydrodynamic values (4.4)-(4.10) contain dimensionless parameters (and). The -dependence is contained within the, , coefficients. In terms of Equations (4.4-4.10), the components of Equation (6.1) can be written in the form of an infinite series in

Here, , ,. Substituting distributions of hydrodynamic values (3.6), (3.7), (4.4)-(4.10) into (А.7), (А.10) and integrating the resulting expressions with respect to over the volume yield

Here, , , ,. Coefficients

, are given in Section 4. The terms of distributions (4.4)-(4.10) proportional to give rise to terms proportional to and in Equation (6.6) upon integration with respect to over volume. That is why, in deriving (6.6), the integration limit for indefinitely increasing terms is changed by putting.

The function (6.5) incorporates the total number of particles in the system and the total thermal energy. Owing to aforesaid condition and the boundary conditions (2.16) adapted to an individual system, fluctuations of these quantities (and) can be excluded from consideration. Thus,. So, pair entropy accurate to the order of does not change at time. Hence, the system stability is independent of terms of this order. The study of Equation (6.1) undertaken in [19] has revealed that entropy

production in the system (6.7) compensates the entropy outflow through the surface confining the system, then, because the term is

omitted for the reason to be explained later in this section.

In problem with time independent boundary conditions, entropy balance Equation (6.1) accurate to the order of assume the form

As is noted in Section 5, spontaneous fluctuations in large systems tend to be as small as possible. Further, the nearest vicinities of the point at which the solution breaks will be omitted from consideration. In this case, in accordance with the law of large numbers [12] [13] , large spontaneous fluctuations can be disregarded. The contempt for large spontaneous fluctuations makes it possible to omit the, , terms allowing for small spontaneous fluctuations. However, varying with space and time on the scales and, small spontaneous fluctuations keep equations of conservation (6.9) valid. Thus, space and time gradients of small spontaneous fluctuations must be retained on the left hand side of Equation (6.9).

Derivative is calculated from Equation (6.6). One of ways of allowing for the contribution of spontaneous fluctuations is to introduce a random source [17] . However, numerical integration of the, set is as much in error as the random source method. Thus, the contribution of spontaneous fluctuations is taken into account involuntary, owing to the error in numerical calculation.

The study of the spatial variation of the pair entropy [19] has revealed that the

term models entropy removal from the system exclusively through the surface of the

solid sphere of radius. By virtue of the condition, the term models entropy removal by spontaneous fluctuations also through the surface of the sphere. The entropy balance in the solid sphere is maintained by thermal radiation.

Suppose that at time, the system produces an entropy fluctuation. The

values at close to are plotted at Figure 5. As revealed by calculations, at and at. Thus, at, any fluctuation with generated by the system tends to fade out, whereas at, any fluctuation with tends to grow. Time history of is illustrated in Figure 6. From Figure 6 follows that at and. Hence, entropy fluctuations generated by the system die

down monotonically, and the entropy of the system tends to its stationary value. At and,. Thus, entropy fluctuations produced by the system build up

monotonically up to point where the solution breaks, and the entropy of the system deviates increasingly farther from its stationary value with increasing. As revealed by calculations, fluctuations always fade out at, and build up at. The function is given by Equation (6.6).

From balance Equation (6.1) follows that at, the pair entropy formed in the system per unit time due to binary collisions,

exceeds the pair entropy removed through the surface confining the system per unit time,

with the result that fluctuations produced by the system die out. At, conversely, the pair entropy outflow exceeds its production, and fluctuations build up. Thus, the case of instability onset in a flow around a sphere is a prevalence of pair entropy outflow over its production at the instant the Reynolds number reaches its critical value.

As revealed by calculations [19] , the Boltzmann entropy behaves exactly as the pair en-

tropy does, when the system passes through. Hence, the cause of instability onset in the system, regardless of whether it is formulated in terms of pair entropy or in terms of Boltzmann entropy, remains the same.

Earlier in the analysis we restricted consideration exclusively to entropy fluctuations with. The reason for this “asymmetry” is as follows. Under the second principle of thermodynamics, the probability of

fluctuations in an isolated system is so much greater than that of

fluctuations that the latter rarely, if ever, occur in nature [16] . As revealed by study [19] , entro-

py fluctuations with direct the system along extremely unlikely, i.e., impractic-

able path. Nevertheless, the probability of fluctuations in an open system is not to be ruled out [19] .

Evolution of fluctuations generated by system depends on two factors, on entropy production and removal through the surface confining the system. The fact that these factors were analyzed without resorting to any kinds of approximations encourages us to believe in the universal nature of the established cause of instability onset. Therefore, the principle according to which an open system retains (or loses) its stability can be formulated as follows.

An open system with time-independent boundary conditions has a stable stationary a-state with entropy while entropy production in it exceeds entropy outflow through the surface confining the system for and does not exceed entropy outflow for

As soon as the parameters characterizing the system reach the values, at which at least one of inequalities (6.10а) and (6.10b) fails, the stationary a-state of the open system becomes unstable.

The principle originally formulated for open system with time-independent boundary conditions can be expected to the case of open systems with time-dependent boundary conditions. Generally entropy corresponding to an ensemble of systems may not be reckoned as stationary value. If so, Equation (6.1) no longer implies that the entropy production equals its outflow. That is why, generally, the stability principle is formulated in terms of excess of the entropy production and excess of the entropy outflow.

The a-state with entropy of an open system remains stable while the excess of entropy production generated in the system exceeds its excess of outflow through the surface confining the system for and does not exceed the excess of outflow for

As soon as the parameters characterizing the system reach the values, at which at least one of inequalities (6.11а) and (6.11b) fails, the a-state of the open system becomes unstable.

Inequalities (6.10) for systems with time-independent boundary conditions are reduced to inequalities (6.11). However, the stability principle for stationary states (6.10) seems to be more “transparent”. The above formulated stability principle remains invariant in going from pair to Boltzmann entropy [19] .

In accordance with the principle of retention and loss of stability (6.11), in an open unstable system, any entropy fluctuation begins to grow. In particular, for a system with time-independent boundary conditions

Based on the expression (7.1), we can formulate the criterion of evolution of an open system with lost stability.

An open unstable system with time-independent boundary conditions, takes a direction of evolution that provides the most rapid decrease in entropy. Namely, of the two directions of development of the instability, having the same values of the entropy and entropy derivative at the time, fluctuations find such a direction that is characterized by lower value of the second derivative of entropy:

wherein

That is, at the time, the system takes the α-direction, for which the second derivative of the entropy with respect to time has a lower value compared to the respectively derivative for the β-direction.

In constructing the approximate solution to the equation for the pair function, only a limited number of terms is retain in the series of products of trajectory invariants (2.5). Different approximate solutions compatible with the boundary conditions of the problem differ in the number of terms retained in expression (2.5). To select the optimal approximate solutions, it is necessary to introduce an additional criterion. The logic of selecting one of the set of approximate solutions can be seen in the formulation of the criterion of evolution (7.2).

Let be the set of parameters characterizing the jth stable stationary solution, , , to the multimoment hydrodynamic equations, and let an increase in be accompanied by the de-

parture of the system state from the state of statistical equilibrium. In the simplest case, the entropy is

calculated in the entire space from one jth solution. Suppose that, at some value of the parameter, the

pair entropy, calculated from the solution, , , occurs in a small vicinity

of the pair entropy. The entropy is calculated based on the solution to the clas-

sic hydrodynamics equations, which are valid in the limit. Based on expression (7.2), let us formulate a criterion for selecting the approximate stable solution for open system.

In interpreting the behavior of open system with time-independent boundary conditions, a solution that provides the fastest drop in entropy should be chosen from the set of stable approximate solutions to the multimoment hydrodynamics equations. Namely, at a certain value of the parameter, an approximate solution with the lowest value of the entropy derivative should be chosen among a few approximate solutions with the same entropy values in a small vicinity of the pair entropy:

This means that, at the given, the priority lies with the kth approximate stable solution to the multimoment hydrodynamics equations for which the derivative of the pair entropy with respect to has the lowest value among the respective derivatives provided by other similar solutions. In more complex cases, it may be necessary to compare the second derivatives of the entropy, as is done in (7.2).

Criteria (7.2), (7.3) are suitable for interpreting open systems with time-dependent boundary conditions.

According to the solution, the recirculating zone is formed in the wake behind a sphere. After the attainment of, the periphery of the recirculating zone begins to pulsate periodically. Pulsating periphery demonstrates the absence of slightest indications of detachment from the core of the recirculating zone. Consequently, there is no vortex street in the far wake behind the sphere. Therefore, the solution does not describe the vortex shedding.

Let the distribution of the particle-density flux be:

To obtain distribution (8.1), the second term on the right side of Equation (4.8) is written in -variables. Distribution (8.1) defines a vortex ring at distance from the sphere center, Figure 3(b). The vortex ring center is located at the axis, which forms the -angle with the -axis, Figure 7. The inflowing gas velocity is aligned with the positive direction of the axis of reference frame XYZ. The XYZ frame has its origin at the center of the sphere.

Let us retain all the terms proportional to in the distributions of principal hydrodynamic values (4.4-4.10). These distributions have the form of linear combinations of the products of the inverse power functions of and trigonometric functions of the polar angle. To bring distribution (8.1) to this form, we change in Equation (8.1) from the variables to the variables, expand these expressions in a Taylor series in powers of, and retain in this series only the terms independent of and linear in:

Then, we change from the spherical coordinates to the spherical coordinates. The variables and are, respectively, the polar and azimuthal angles of the vector in the Cartesian reference frame with the axis defining the direction of the flow incoming on the sphere and in the Cartesian reference frame with the axis forming an angle with the axis, Figure 7. The transformations performed give rise to azimuthal angle dependent terms in the distributions of the particle-density flux:

Here,

Along with the coefficient, the transformed expression (8.3) for the distribution of the particle-density

flux contains two additional coefficients: and. The coefficient characteriz-

es the distance from the vortex ring to the sphere surface, whereas the coefficient characterizes the deviation of the vortex ring center from the axis. Trajectory invariants for expansion (4.2), which lead to the distribution (8.3), are not written in the present Section.

The distributions of the principal hydrodynamic values (4.4)-(4.10) associated with the coefficients should be supplemented with distribution (8.3). The presence of -dependent terms in Equation (8.3) for the distribution of the particle-density flux leads to the appearance of these terms in the hydrodynamics equations. Taking into account the dependence on is beyond the accuracy limits of the approximation under consideration.

Expand expression (8.3) for the distribution of the particle-density flux in a Fourier series in the azimuthal angle and substitute the zeroth-order Fourier expansion term in the multimoment hydrodynamics Equations (2.9)-(2.11). Eighteen nonlinear algebraic equations are presented by relationships (A8)-(A10) in [8] . Right- hand sides of Equations (A8)-(A10) from [8] contain twenty coefficients,. In accordance with the algorithm presented in [8] , we supplement these equations by terms containing the coefficients and. Let us supplement eighteen algebraic Equations (A8-A10) from [8] with three algebraic Equation (4.12) and differential Equation (4.2). As a result, we obtain a closed set of nonlinear equations of twenty-second order S22 for the coefficients,. It turned out that, in the investigated range of, of great many solutions to the system S22, only two solutions correspond to such an entropy value that allows these solutions to compete with the solution. We denote these solutions as and.

In the rearrangement of the distribution (8.1), the terms nonlinear in were omitted. Calculations showed that the omitted terms have no significant influence on the solutions and. Substitute the full Fourier expansion of expression (8.3) for the distribution of the particle-density flux into the multimoment hydrodynamics Equations (2.9)-(2.11) and integrate the resulting nonlinear system of differential equations over. In this case, we arrive at system S22 supplemented by several terms. Calculations have shown that these supplements produce no significant influence on the solutions and.