Multimoment Hydrodynamics in Problem on Flow around a Sphere : Entropy Interpretation of the Appearance and Development of Instability

Multimoment hydrodynamics equations are applied to investigate the phenomena of appearance and development of instability in problem on a flow around a solid sphere at rest. The simplest solution to the multimoment hydrodynamics equations coincides with the Stokes solution to the classic hydrodynamics equations in the limit of small Reynolds number values, Re 1  . Solution 0 Sol to the multimoment hydrodynamics equations reproduces recirculating zone in the wake behind the sphere having the form of an axisymmetric toroidal vortex ring. The 0 Sol solution remains stable while the entropy production in the system exceeds the entropy outflow through the surface confining the system. The passage of the first critical value 0 Re ∗ is accompanied by the 0 Sol solution stability loss. The 0 Sol solution, when loses its stability, reproduces periodic pulsations of the periphery of the recirculating zone in the wake behind the sphere. The 1 Sol and 2 Sol solutions to the multimoment hydrodynamics equations interpret a vortex shedding. After the second critical value 0 Re ∗∗ is reached, the 0 Sol solution at the periphery of the recirculating zone and in the far wake is replaced by the 2 Sol solution. In accordance with the 2 Sol solution, the periphery of the recirculating zone periodically detached from the core and moves downstream in the form of a vortex ring. After the attainment of the third critical value 0 Re ∗∗∗ , the 2 Sol solution at the periphery of the recirculating zone and in the far wake is replaced by the 1 Sol solution. In accordance with the 1 Sol solution, vortex rings penetrate into each other and form the continuous vortex sheet in the wake behind the sphere. The replacement of one unstable flow regime by another unstable regime is governed the tendency of the system to discover the fastest

Sol solution, vortex rings penetrate into each other and form the continuous vortex sheet in the wake behind the sphere.The replacement of one unstable flow regime by another unstable regime is governed the tendency of the system to discover the fastest

Introduction
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 Re values.The direction of evolution of the ground axisymmetric flow ( ) x [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 Re 1  .The Section 4 is devoted to construction of the 0 Sol solution to the multimoment hydrodynamics equations suitable of interpreting the stationary axisym- metric flow at a moderately high Re values.The Section 5 examines the behavior of the 0 Sol solution that loses its stability after the passage of the first critical Reynolds number value 0 Re * .In Section 6, the characteris- tic 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.

Problem Statement
Consider a space filled with thermodynamically equilibrium gas.Suppose that a solid sphere of radius a moves in gas at constant velocity 0 U along the 0 Z axis of the immobile Cartesian frame of reference 0 0 0 X Y Z .Let us now pass from 0 0 0 X Y Z to the Cartesian frame of reference XYZ with the axes parallel to those of 0 0 0 X Y Z and the origin made coincident with the center of the moving sphere.In the XYZ frame of reference, the sphere is at rest, the inflowing-gas velocity at an infinite distance from the sphere, 0 U , is aligned with the positive direction of the Z axis, and the flow problem is stationary.
The pair distribution function ( )  ( ) corresponding to the equilibrium state is ( ) ( ) where 0 n and 0 T are the local density of the number of particles and the temperature of unperturbed gas, , m is the mass of a gas particle, and 0 G and 0 v are the velocity of the center of mass and relative velocity of the pair of particles.In XYZ , the pair distribution function ( ) can be written as x y , and z be the Cartesian coordinates of point x in the space; , r θ , and ϕ are its spherical coor- The XYZ frame with its origin at the center of the sphere.The r X X X θ ϕ frame with its origin at some point x in the gas.reference r X Y Z θ ϕ into coincidence with point x so that the r X axis is directed along x vector.Mark the projections of vector G onto the XYZ axes by , x y G G , and z G , its projections onto the axes of r X Y Z θ ϕ , by , r G G θ , and G ϕ , and its spherical coordinates, by , G ε , and ω , 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 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 ϕ about the Z axis.Let us compose combinations of trajectory invariants F xy , F zx , and F zy , invariant with respect to this rotation F sin sin sin Ф F F cos sin sin cos sin cos By virtue of the flow geometry, there are two independent regions of integration with respect to G at each point x in the gas.The first region, denote it by 1 W , 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 ( ) W , embraces all possible trajectories of centers of mass of particle pairs converging to point x .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 ( ) ) is controlled by trajectory invariants (2.4), and coefficients klmn c are independent of x .In keeping with definitions given in [5], the expressions for the principal hydrody- namic values-the moments of ( ) 2)-can be written as Here, n is the local density of the number of particles; U is the hydrodynamic velocity; G p , G T , G ij P , and G q are the pressure, temperature, stress tensor, and heat flux corresponding to motion of the centers of mass of pairs of particles; v p , v T , and v q correspond to relative motion of particles within pairs; Nonprincipal hydrodynamic values v ij p and Gv q 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 ϕ about the Z axis) assume the form Here, η  is the coefficient of the dynamic viscosity.From the definitions of gas pressure p , temperature T , viscous stress tensor ij p , and heat flux q [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 v i ∏ and Gv Λ components assume the following form for the cy- lindrically 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 r a = , one has to impose the no-leak and no-slip conditions 0 0 r r a r a nU nU Heat flux r q traversing each element of the sphere surface must be counterbalanced by the heat flux within the sphere cond r q and the thermal radiation E ∆ ( ) Here, γ is the coefficient of the thermal conductivity of the sphere, and σ is the coefficient of the sphere emissivity.

Flow around a Sphere at Small Re
Let us formulate dimensionless parameters . 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 2 Ma and Re .Hydrodynamic values can be represented as parametric series Here, ( ) ( ) , respectively.Expansion (3.1) of the particle density flux nU is reasonably be re-

nU
, which provides for the p .As noted in [6], the Navier-Stokes approximation is not accurate enough to calculate temperature components . Thus, at Re 1  , the spatial dependence of temperature is neglected.
Let us truncate expansion (2.5) to the three lowest-order terms s , v q , and G q ).Considering that the principal hydrodynamic values are linearly independent, coefficients , 1, 2, 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 G with the appropriate weight function of velocities G and v within region 2 W , and the second and third components, within region 1 W .Why ( ) 2), and its components are integrated in such a way is explained below.The integration yields 0,0 2 0 0 2 The order of coefficients 1 A and 2 A appearing in Equations (3.4, 3. enable us to eliminate the contribution to v q and G q that is proportional to 0 0 0 n kT U .The reason why v q and G q contain no terms proportional to 0 0 0 n kT U will be elucidated in Section 4. The last, sixth linear combination of ( ) ( ) A in Equation (3.5).Three linear combinations of ( ) ( )  (3.4) into any of the equations of momentum conservation (2.12), we obtain ( ) . When analyzing the multimoment hydrodynamics equations in the order of 2 0 0 Ma Re n kT [6], it was inferred that . Substituting the foregoing expressions for the coefficients ( ) 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 Re 1  [6].
The hydrodynamic values (3.6), (3.7) assume the form of a linear combination of the products of ( ) k a r by trigonometric functions.To calculate (3.6), (3.7), the first component of Equation (3.2) was integrated with respect to G within the 2 W region, and the second and third components, within the 1 W region. If, on the contrary, the first component is integrated within 1 W , and the others, within 2 W , 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.

Stationary Flow around a Sphere at Moderately High Re
Consider a flow at a moderately high Re and  , will be taken into account.
Expansions (3.1) of the energy fluxes G Q and v Q will be limited to Expressions for hydrodynamic values (2.6) will be constructed by the same principle as in Section 3-as linear combinations of the products of ( ) k a r 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 G 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 , , , , , , , 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 nU (A.5) from [8], we express 16 coefficients ˆi C , 34, , 44, 46, ,50 i =   in terms of 34 remained coefficients [8].As revealed in [6], the 2 1 0 0 0 Ma Re l n kT U − -order approximation to the total energy-flux vector Q ( ) 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 ˆi C , 26, ,33 i =  , in terms of 26 remained coefficients.Substitute Equations (A.2), (A.5), and (A.6) from [8] into equation of momentum conservation (2.12) and equate the coefficients of the ( ) cos sin , can be expressed in terms of 20 remained coefficients ˆi , [8].Eventually, the expressions for the principle hydrodynamic values (2.6) can be written as The terms proportional to 0 Re l n and 0 0 Re , 1, 2, l n kT l =  , 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 0 0 0 n kT U a , all the coefficients of these terms must be zero.For Equation (2.13) to be good in the order of 2 0 0 0 Ma Re n kT U a , one has to put expression (4.9) for  ( ) From the conditions for no heat flux q at the surface of the sphere (at r a = ) and at infinite distance from the sphere the expression follows (4.10) for  ( ) As previously, Equation (2.13) must be satisfied for each ( ) θ product of the order of 4 0 0 0 Ma Re n kT U a 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 kT p n = (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) Sixteen equations (П.8, П.9, П.13) from [8]  Numerical integration of the nonlinear set of 20 algebraic equations was carried out at ˆ100 γ = , and ˆ0.5 σ = .Calculations have revealed a great many roots.However, at 10 Re 129.1

< ≤
, 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 Re grows but its shape remains unchanged (Figure 3(а)).At * 0 Re Re 129.1 = = , the system becomes unstable.At Re 1  , the Barnett corrections [7] are known to be commensurate with the Navier-Stokes terms.For this reason the calculations [8] were interrupted at Re 10 = .

First Unstable Flow Regime
As the flow around sphere becomes unstable, the problem becomes nonstationary.As in the stationary case, the pair distribution function ( ) , , , Here, ( ) ( ) is given by Equation (2.2), and x G is given by Equation (2.5) with time-dependent coefficients klmn c .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 ( )   Re is accompanied by the ( ) , solution stability loss.Starting with 0 t = , the , small axisymmetric deviations begin to increase exponentially at * 0

Re Re >
. The solid curve in Figure 4 is the time history of ( ) ( )   Suppose that at time t t * = , 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 t t * = .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 t t * = .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 t t * = , 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 +v ij p and +Gv i q 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 20 S , we obtain a closed nonstationary set of twenty equations for coefficients ˆi , denoted hereafter by 20 S + .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 5 19 It turned out that the ( ) ( ) ( ) , solution to the 20 S + 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 ( ) ( ) A large spontaneous fluctuation causes the transfer of the system from the ( ) ( ) , solution is unstable.It follows that small axisymmetric deviations , begin to grow starting with time * 2 t t = .This pro- cess is repeated periodically.So, we obtained the solution ( ) ( ) 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 t , denote their number by K which can be infinitely large t are calculated within the classic mechanics.Fluctuation 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 ( ) Re < Re within which the equations for statistical coefficients have stable solution ( ) . Within this range, the overwhelming majority of dynamic coefficients t passing in the immediate vicinity of ( ) will remain in the vicinity of ( ) , describes most of the dynamic trajectories The situation in the unstable range ( ) Re Re > is radically different.Consider two ensembles of systems.The first α -ensemble at 0 t = incorporates systems with coefficients , here, ( ) ( ) , here, ( ) ( ) ˆD i j C t , 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 v ij p and Gv i q dissipation moments (2.7) are also true for the ensemble as a whole.Deviations , are nothing but fluctuations of stationary coefficients ( ) Ĉ , is preset at 0 t = , for example, ( ) ( ) 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 After the attainment of the first critical value * 0 Re , the 0 Sol 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 δ must be counterbalanced by the contribution of time and space derivatives of spontaneous fluctuations The i f 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 and l a   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.

Interpretation of System Stability Loss in Terms of Pair Entropy
Let us mentally circumscribe a sphere of radius А a > around the sphere of radius a .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 А radius of the larger sphere to infinity.In pursuance of the А a  condition, the number of particles clinging to the surface of the sphere of radius А at any instant is negligibly small compared to the total num- ber of particles in the system.The А a  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 А .In pursuance of the А a  , 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 Here, superscript (0) corresponds to the 0-solution to the multimoment hydrodynamics equations.Integrating the resulting equations with respect to x over the volume of the system V yields ( ) S t also convey the meaning of volume occupied by the system in the Г-space.
In terms of Equation (5.6), the x 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 ( 2Ma and Re ).The Re -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 2 Ma         upon integration with respect to x over volume V .That is why, in deriving (6.6), the integration limit for in- definitely increasing terms is changed by putting 1 2, π 0, 2π 0 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 ) compensates the entropy outflow through the surface confining the system 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 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 l a   and 0 Re a U τ  , 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 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 x x spatial variation of the pair entropy [19] has revealed that the   where the solution breaks, and the entropy of the system ( ) ( )

Re Re
< , 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, Re Re > , 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 ( ) ( ) 0 1 S t Boltzmann entropy behaves exactly as the ( ) ( ) 0 p S t pair entropy does, when the system passes through * 0 Re .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], entropy fluctuations with direct the system along extremely unlikely, i.e., impracticable 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 α-state with entropy ( ) p S α while entropy production in it exceeds entropy outflow through the surface confining the system for ( ) ( ) 0 p S t α δ ≤ 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 α-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 ( ) ( ) p S t α 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 . That is why, generally, the stability principle is formulated in terms of excess of the entropy production The α-state with entropy ( ) ( ) α 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 ( ) ( ) 0 p δS t α ≤ 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 α-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].

Interpretation of System Evolution in Terms of Pair Entropy
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 0 t t = , fluctuations find such a direction that is characterized by lower value of the second derivative of entropy: That is, at the time 0 0 t t = > , 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 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 λ be the set of parameters characterizing the jth stable stationary solution ( ) ( ) ( ) =  , to the multimoment hydrodynamic equations, and let an increase in λ be accompanied by the de- parture of the system state from the state of statistical equilibrium.In the simplest case, the entropy ( ) ( ) j p S λ is calculated in the entire space from one jth solution.Suppose that, at some value of the parameter 0 λ λ = , the pair entropy ( ) ( ) j p S λ , calculated from the solution ( ) ( ) λ is calculated based on the solution to the clas- sic hydrodynamics equations, which are valid in the 0 λ → 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 0 λ λ = 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 0 λ λ = , the priority lies with the kth approximate stable solution to the multi- moment hydrodynamics equations for which the derivative of the pair entropy with respect to λ 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 ( ) p S λ entropy, as is done in (7.2).Criteria (7.2), (7.3) are suitable for interpreting open systems with time-dependent boundary conditions.

Vortex Shedding Regimes
According to the 0 Sol solution, the recirculating zone is formed in the wake behind a sphere.After the attain- ment of * 0 Re Re = , 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 0 Sol solution does not de- scribe 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 , Θ R -variables.Distribution (8.1) defines a vortex ring at 0 r distance from the sphere center, Figure 3(b).The vortex ring cen- ter is located at the Z ′ axis, which forms the δ -angle with the Z -axis, Figure 7.The inflowing gas velocity 0 U is aligned with the positive direction of the Z 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 1 19 ˆ, , C C  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 r and trigonometric functions of the polar angle θ .To bring distribution (8.1) to this form, we change in Equation (8.1) from the , Θ R variables to the r', ' θ variables, expand these expressions in a Taylor series in powers of 0 0 , r r r r ′ ′ > , and retain in this series only the terms independent of and linear in 0 r r' : Then, we change from the spherical coordinates , , r θ ϕ ′ ′ ′ to the spherical coordinates , , r θ ϕ .The variables , θ ϕ and , θ ϕ ′ ′ are, respectively, the polar and azimuthal angles of the vector r in the Cartesian reference frame with the Z axis defining the direction of the flow incoming on the sphere and in the Cartesian reference frame with the Z ′ axis forming an angle δ with the Z axis, Figure 7.The transformations performed give rise to azimuthal angle ϕ dependent terms in the distributions of the particle-density flux  nU : es the distance from the vortex ring to the sphere surface, whereas the coefficient 22 Ĉ characterizes the devia- tion of the vortex ring center from the Z axis.Trajectory invariants for expansion (4.2), which lead to the dis- tribution (8.3), are not written in the present Section.C  should be supplemented with distribution (8.3).The presence of ϕ -dependent terms in Equation (8.3) for the distribution of the particle-density flux  nU leads to the appearance of these terms in the hydrodynamics equations.Taking into account the dependence on ϕ is beyond the accuracy limits of the approximation under con- sideration.
Expand expression (8.3) for the distribution of the particle-density flux  nU in a Fourier series in the azimuthal angle ϕ and substitute the zeroth-order Fourier expansion term in the multimoment hydrodynamics Equa- tions (2.9)- (2.11).Eighteen nonlinear algebraic equations are presented by relationships (A8)-(A10) in [8].Righthand sides of Equations (A8)-(A10) from [8] contain twenty coefficients ˆi C , 1, , 20 i =  . In accordance with the algorithm presented in [8], we supplement these equations by terms containing the coefficients 21 Ĉ and 22 Ĉ .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 ˆi C , 1, , 22 i =  . It turned out that, in the investigated range of Re , of great many so- lutions to the system S22, only two solutions correspond to such an entropy value that allows these solutions to compete with the solution 0 Sol .We denote these solutions as 1 Sol and 2 Sol .In the rearrangement of the distribution (8.1), the terms nonlinear in 0 r r′ were omitted.Calculations showed that the omitted terms have no significant influence on the solutions 1 Sol and 2 Sol .Substitute the full Fourier expansion of expression (8.3) for the distribution of the particle-density flux  nU into the multimoment hydrodynamics Equations (2.9)-(2.11)and integrate the resulting nonlinear system of differential equations over ϕ .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 1 Sol and 2 Sol .The subscript in brackets corresponds to the limits of spatial integration.The 0 Sol solu- tion describes periodic pulsations of the recirculating zone in the wake behind the sphere.The movement of the representative point over the curve (Figure 8 = , the entropy decreases permanently.This behavior of the entropy corresponds to the movement away of the system state that lost its stability from the state of statistical equilibrium.By the time * t t = , the degree of excitation of the recirculating zone reaches a maximum, which corresponds to the minimum value of the

Selecting the Direction of Instability Development
At the lettime * t t = , the 0 Sol solution to the multimoment hydrodynamics equations breaks.The movement of the representative point over the curve (Figure 8) corresponds to the return of the most excited recirculating zone to its original position, i.e., the position corresponding to the time 0 t = .Since the time * t t = , the movement of the representative point is described by the reverse equations of multimoment hydrodynamics [6].The reverse equations are solved with regressive timing along the time axis, from the past to the future.This timing order is represented on the axis beneath the abscissa in Figure 3 . Relationships (5.4) between the coefficients ( ) ( ) and ( ) ( )  Re , the periphery of the recirculating zone begins to be periodically shed from its core and moves downstream.The shed vortex structure has the shape of ring.Separate vortex rings depart from a sphere downstream and move along the spiral path ( ) 0 , exp t W x [11].The attainment of *** 0 Re is accompanied by the change in the regime of vortex shedding from sphere.The frequency, which characterizes the vortex shedding, monotonically increases as Re grows starting with *** 0 Re Re = .The increase in the frequency causes the complete disappearance of intervals between vortex rings.Vortex rings penetrate into each other and form the ( ) 0 , exp t Q x continuous spiral sheet in the wake behind a sphere [11].According to the measurement data reported in [10], * 0 Re 130 = .In [11], the x continuous spiral vortex sheet was observed in [11] over the whole range of Reynolds numbers studied, up to Re = 30,000.
The simplest solution to the multimoment hydrodynamic equations gives the distribution (3.6-3.7).This distribution coincides with the Stokes solution to the classic hydrodynamics equations in the Re 1  limit.The Re , the entropy outflow through the surface confining the system begins to exceed the entropy production in the system.Such interpretation corresponds to the principle of retention and loss of the open system stability (6.11).The 0 Sol unstable axisymmetric solution reproduces satisfactorily the ( ) After the attainment of the second critical Reynolds number value ** 0 Re Re = , the 0 Sol solution for the pe- riphery of the recirculating zone and in the far wake is replaced by the 2 Sol solution to the S22 set, which de- scribes a vortex ring moving downstream.The cause for the replacement is that the combination of the solutions 0 Sol and 2 Sol becomes more preferable in comparison with the 0 Sol solution.The combination of the solu- tions 0 Sol and 2 Sol provides a sharper drop in the entropy in the course of evolution than the 0 Sol solution does (Figure 10).The criterion (7.2) dictates the choice of the direction of development that is given by combination of the 0 Sol and 2 Sol solutions.The positions of detached vortices at the Z axis at Re Re = , the 2 Sol solution at the peri- phery of the recirculating zone and in the far wake behind the sphere is replaced by the 1 Sol solution to the S22 set, which also describes a vortex ring moving downstream.The cause for the replacement is that the combination of the solutions 0 Sol and 1 Sol provides a sharper decrease in the entropy in the course of evolution than the combination of the solutions 0 Sol and 2 Sol does (Figure 10).Criterion (7.2) dictates the choice of the di- rection of development that is given by combination of the x .The observed process of formation and separation of the vortex structure from the recirculating zone in the wake behind the sphere is shown in photographs 40 and 41 in [21].A vortex structure arising near the surface of the sphere moves inside the core zone of the recirculating zone to its periphery, increasing in size and acquiring a definite shape.At the periphery of the recirculating zone, the vortex structure is separated from this zone and moves downstream.
In the calculated flow pattern, a formed vortex structure appears in the recirculating zone instantly at the time 1 t t = .This simplification of the observed complex process of formation and separation of the vortex structure is motivated by a lack of computational resources for simulating the process of vortex shedding.
Moreover, only one of these components coupled with the 20 С coefficient dominates the particle density flux (4.8), reproducing recirculating zone in the near wake behind the sphere.The z F drag coefficient of the sphere also depends only on one component coupled with 7 С : Further, the dominant component in the distribution (4.8) that is coupled with the 20 С coefficient is used to reproduce a single downstream-moving vortex structure unrelated to the sphere, expression (8.1).Thus, a detailed description of the observed pattern of nucleation and separation of the vortex structure can be achieved by increasing the number of trajectory invariants involved in calculating the particle-density flux distribution.This increase is necessary both in the formulation of the set S20 describing the pulsations of the recirculating zone in the near wake and in the formulation of the set S22 describing the motion of single vortex structure outside the recirculating zone.Moreover, constructing the set of equations suitable for describing the far wake only (system S22) is barely an approximation of the calculation procedure.A further improvement necessitates combining sets S20 and S22.The solutions to the combined set must describe the movement of the single vortex structure not only in the far wake, but also in the recirculating zone.It is these solutions to the combined set of equations that must compete with the solutions to the set S20. Forming the set of equations S20 and S22 was conducted so that these sets would be able to reproduce the time behavior of the recirculating zone and the evolution of the vortex structure separated from it in the wake of the sphere.The terms of expansion (4.2) proportional to 19 c , 20 c , and 21 c enabled us to describe the wake behind a sphere.By virtue of the proportional to 19 c , 20 c , and 21 c terms, however, yet another physically improbable vortex configurations arise in front of the sphere at its surface at a sufficiently high Re [8].The appearance of physically improbable vortex configurations in the distribution of particle density flux (4.8) generally does not allow of using this distribution outside the wake.To obtain the distributions of the hydrodynamic values outside the wake, it is necessary to formulate a different set of the multimoment hydrodynamics equations.The boundary separating solutions to different sets should be sought based on the principle (7.3).In other words, the boundary separating the solutions should be positioned in space so as to ensure the most rapid decrease of the entropy.
In the Section 6, in calculating the pair entropy, the 0 Sol solution to the set S20 was used both in the wake behind the sphere and outside the wake.At Re close to * 0 Re , physically improbable vortex configurations in front of the sphere are very weak.That is why, these configurations distort the calculation very little at * 0 Re Re ∼ .Another result of going with the 0 Sol solution trace beyond the wake produced divergent terms in the course of spatial integration over the volume V.The mergence of divergent terms forced us to restrict the in-tegration limits in the Section 6 to a spherical concentric layer directly adjacent to the surface of the sphere.Namely, in deriving Equation (6.6), the integration limit for indefinitely increasing terms is changed by putting 1 2, π 0, 2π 0 r θ ϕ ≤ ≤ ≤ ≤ ≤ ≤ .As revealed by calculations, the appropriate numerical coefficients in Equation (6.6) change under changes in the integration limit, however, characteristic features of variation of ( ) The approximate calculation of the pair entropy in Section 9 was performed within hemispherical concentric layers 0 H , 1 H , and 2 H (Figure 9).This approximation was used because the dominant contribution to the deviation of the entropy from its equilibrium value comes from processes in the near wake behind the sphere.These are the processes of excitation of the recirculating zone and of separation of the vortex structure that appeared in this zone.The outer boundary 2 r of the layer 0 H is specified by the position of the separating vor- tex structure at the time of its detachment, 3 t t = , from the recirculating zone.Estimates show that the approxi- mation of the boundaries of the near wake behind the sphere by the boundaries of the hemispherical layer 0 H does not lead to a qualitative distortion of the results.

Discussion
Let us substitute the inequality (6.11a) into Equation (6.1) that is written in ( )  Behind the modern concepts of open system stability is the Glansdorff-Prigogine theory [12].This theory, in turn, is formulated in terms of nonequilibrium thermodynamics and, therefore, allows for the possibility of constructing the universal function ( ) , Z t x with non-positive fluctuation at any fluctuations of hydrodynamic val- ues The fluctuation of ( ) , Z t x was used as Lyapunov function.The set of stability conditions derived in [12] in terms of this fluctuation involves inequality (1.11b) and ( ) ( ) Based on inequalities (11.1b) and (11.2),P. Glansdorff and I. Prigogine formulated the condition for stability of open system state in terms of entropy production and entropy outflow [12].They argue that the main quality controlling the stability and evolution of the system is excess of entropy production . The Glansdorff-Prigogine stability condition states that for the system state to be stable, the entropy excess must be generated in the system rather than being absorbed In accordance with Glansdorff-Prigogine theory [12], for the α -state to be stable, the effect of excess of en- tropy outflow on the system must be equivalent to the effect of excess of entropy production.In other words, for the state to be stable, entropy excess must flow into the system through the confining surface, thus augmenting the entropy excess generated in the system  Since the time of Boltzmann, the responsibility for directing the evolution of the system rests with the initial conditions realized in the system, namely the set of initial values of the coordinates and velocities of particles [22] [23].For a given mutual arrangement of the particles, the system evolves in the direction that we see everywhere and every second.However, there are such arrangements of particles that direct the system in an extremely unlikely, rarely realized direction.Even weak disturbance l δ of the configuration of particles ( ) can change the direction of evolution ( d is characteristic size of particles) [24].
The local pair entropy corresponding to the direct equations for pair distribution functions (1.12) from [6] and the multimoment hydrodynamics equations they yield can only be produced in the system due to binary collisions at any space point x and at any instant t , ( ) , 0 . Thus, at any instant binary collisions merely rise the pair entropy of the system, ( ) 0 IN p S t ∆ ≥ .Such behavior of the entropy is in full accordance with the second law of thermodynamics [25].The solutions to the direct multimoment hydrodynamics equations describe the direction of evolution of the system that is everywhere and every second is found in nature.
The local pair entropy corresponding to the reverse equations for the pair distribution functions (1.15) from [6] and the reverse multimoment hydrodynamics equations they yield can only be absorbed in the system due to binary collisions at any space point x and at any instant t + , ( ) , 0 . Thus, at any instant binary collisions absorb the pair entropy of the system, ( ) 0 The solutions to the reverse multimoment hydrodynamics equations describe the evolution of the system in the opposite direction, which, as is commonly believed, is extremely rare in nature.
The direct multimoment hydrodynamics equations are valid for the progressive direction of timing on the time axis pointing from the past to the future.The reverse multimoment hydrodynamics equations are valid for multimoment regressive direction of timing on the same time axis [6].Processes occurring in nature are objective events, while the choice of the direction of timing on the time axis is subjective process.Time is counted by an observer, while processes occurring in nature absolutely insensitive to the direction in which the observer counts the time.Let two observers, agreeing upon the origin, began to observe some phenomenon.Let the first observer counts time in the regressive direction at the time axis, directed from the past to the future and let the first observer managed to describe the phenomenon he observed using the reverse equations.Let now the second observer counts time in the progressive direction at the same time axis.Then, the second observer can claim that the reverse equations described phenomena for progressive direction of timing.
This means the following.On finding the solution to the inverse equations, the first observer obtained the distribution of all hydrodynamic variables and their spatial and temporal derivatives.The first observer found agreement between the calculated and measured values.The second observer used the solution to the reverse equations to compare with his observations and, on establishing the appropriate relationship between the two time scales, found that the reverse distributions agree with the observed values in their time scale.Changing the sign of the time derivatives, the second observer also obtained an agreement with his observations.However, the second observer failed to derive direct equations that would satisfy altered distribution.
This example returns us to the Boltzmann time, when there were heated debates about the correctness of the Boltzmann equation and the H-theorem.Boltzmann opponents, E. Zermelo and J. Loschmidt, gave examples of processes that are not described by the Boltzmann equation [23] [26].Basically, L. Boltzmann acknowledged the objections of opponents, agreeing with the existence of processes that can not be governed by his theory.However, L. Boltzmann argued that such processes was extremely unlikely, i.e., impracticable [22].Probably, at a weak deviation of the system state from the state of thermodynamic equilibrium, as predicted L. Boltzmann, conditions guided a system in the unlikely direction are extremely rare.However, as the degree of nonequilibrium increases the probability of occurrence of such conditions may grow.The penetration into the instability field confirmed this assumption.It turned out that the motion of unstable system along the unlikely direction becomes an event that repeats periodically.

Conclusions
The experiment records two stable stationary medium states represented by the x velocity distribution.Each of these three stable flows begins to develop in its own direction qualitatively different from other flows when it loses stability.The development occurs through a sequence of regular nonstationary periodic states schematically shown in Figure 1 from [3].Each of these three experimentally observed directions inevitably reaches the periodic vortex Re is accompanied by the stability loss.The system loses its stability when entropy produced in the system can not compensate entropy outflow through the surface confining the system.Such interpretation follows directly from the principle of retention and loss of the open system stability formulated in Section 6.In accordance with solutions to the multimoment hydrodynamics equations, the system, when loses its stability, remains further unstable.One unstable flow is replaced by another unstable flow as Re grows.The replacement of one unstable flow regime by another unstable regime is governed the tendency of the system to discover the fastest path to depart from the state of statistical equilibrium.This striving follows directly from the evolution criterion formulated in Section 7.
Figure 8 demonstrates the behavior of pair entropy of the system losing its stability.According to the 0 Sol solution, pair entropy describes periodic pulsations of the recirculating zone in the wake behind a sphere.Starting with 0 t = , pair entropy begins to decay monotonously.The monotonous decay is replaced by surge.How- ever, the sharp decrease of entropy is not unlimited.At * t t = , the decrease of entropy finishes.Starting with * t t = , pair entropy begins to increase up to time * 2 t t = .At * 2 t t = , the system returns to its original position.Pair entropy begins to decay again starting with * 2 t t = .The process is repeated with the period * 2 t t = .A similar behavior demonstrates pair entropy, which is associated with periodic vortex shedding, Figure 9.
The sharp decrease of entropy is completed at * t t = .Starting with * t t = , pair entropy begins to increase up to time 3 t t = .At 3 t = , the system reaches the position that it occupied at 1 t t = .Pair entropy begins to decay again starting with 3 t t = .The process is repeated with the period T t t t = − − ∆ .The cut-off of the 0 Sol solution is the reason for the completion of the sharp decrease of pair entropy at time * t t = .It turned out that at * t t = the multimoment hydrodynamics equations [5] become unsuitable for modeling evolution of the system.However, in the neighborhood of the cut-off point there is a solution to the reverse multimoment hydrodynamics equations [6].The solution to the reverse equations changes the direction of evolution, striving the system to the state of statistical equilibrium.The pair entropy corresponding to the direct multimoment hydrodynamics equations can only be produced in the system due to binary collisions at each time point.Such behavior of the entropy is in full accordance with the second law of thermodynamics.The pair entropy corresponding to the reverse hydrodynamics equations can only be absorbed in the system due to binary collisions at each time point.The solutions to the reverse multimoment hydrodynamics equations describe the evolution of the system in the opposite direction, which, as is commonly believed, is extremely rare in nature.
At his time, Boltzmann, defending his point of view in disputes with opponents, suggested that the exclusive conditions that guide the system in a highly unlikely direction arise very rarely.Apparently, Boltzmann's assumption is correct for a weak deviation of the system from the state of statistical equilibrium.However, after crossing the border of the instability field, exclusive conditions arise with periodic regularity.This regularity manifests itself in each of three unstable modes that represent the flows x .The tendency of an unstable physical system to find the fastest path to recede from the state of statistical equilibrium does not lead the system to infinity.At time * t t = , the entropy stops decreasing.The multimoment hy- drodynamics equations are unable to provide solutions that would continue to divert the system from the state of statistical equilibrium.Time intervals during which the system moves away from the state of statistical equilibrium are periodically followed by intervals within which the system tends to equilibrium.It is this periodicity that permits to interpret vortex shedding, a graphic example of periodic unstable phenomena.

0Re * is accompanied by the 0 SolRe * * * , the 2 Sol
solution stability loss.The 0 Sol solution, when loses its stability, reproduces periodic pul- sations of the periphery of the recirculating zone in the wake behind the sphere.The the recirculating zone periodically detached from the core and moves downstream in the form of a vortex ring.After the attainment of the third critical value 0 solution at the periphery of the recirculating zone and in the far wake is replaced by the 1 Sol so- lution.In accordance with the 1

,Figure 1 .Figure 1 .
Figure 1.The XYZ frame with its origin at the center of the sphere.The

Figure 2 .
Figure 2. Regions of integration with respect to G.
Each of three terms of series (3.2) can contribute to each principal hydrodynamic value component of expansions(4.

Figure 3 .
Figure 3. (а) Schematic representation of recirculating zone in near wake behind the sphere; (b) Schematic representation of vortex ring in far wake.

20 Ĉ
in[8]).It turned out that at * t t > , the20  S    set becomes unsuitable for modeling evolution of the physical system.Time history of the ( ) ( )0 t coefficient at * 0 t t< < 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 0 t = , and up to t t * = , the periphery of the recirculating zone in the wake behind a sphere moves translationally, receding from the sphere.Starting with t t * = the periphery of the recirculating zone moves back towards the sphere[10] [11].

Figure 4 .Figure 4
by a solid line corresponds to initial deviation The equations of the 20 S , 20 S + set were integrated with the accuracy ε ∆ of the order of of fluctuations of hydrodynamic values at thermodynamic equilibrium.The time history of the coefficients ( ) ( ) corresponds to axisymmetric pulsations of the periphery of recirculating zone in the wake behind a sphere.The half-period of pulsations * t was estimated at approximately one minute at ~1 a sm and 0

at time 0 tFigure 4 (
= .Time history of the coefficients ( ) ( ) solid and dashed curves).Initially close trajectories diverge.It immediately follows that each dynamic j -coefficient, , ,18, 20 i =  , of other coefficients are calculated from the 20 S set, in which Equation (5.2) is substituted by equation Time evolution of the so formulated fluctuation is approximated by solution of the 20 S , 20 S + set.Fluctuation of any hydrodynamic value at any instant t and any point x is calculated from Equations (4.4)-(4.10).The fluctuations are interre- lated in time and space.
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 τ and l  .Possible large-scale spontaneous fluctuations with 0 Re a U τ  and l a   are of the order of large-scale regular fluctuations.Realistic small scale spontaneous fluctuations with 0 Re a U τ  of hydrodynamic values (3.6), (3.7), (4.4)-(4.10)into (А.7),(А.10) and integrating the resulting expressions with respect to x over the vo- lume V yield

5 )
incorporates the total number of particles in the system N and the total thermal energy E .Owing to aforesaid condition ( ) А a  and the boundary conditions (2.16) adapted to an individual system, fluctuations of these quantities ( N and E ) can be excluded from consideration.Thus,

Re are plotted at Figure 5 .is illustrated in Figure 6 .<<
removal from the system exclusively through the surface of the solid sphere of radius a .By virtue of the А a  condition, the 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 0 t = , the system produces an entropy fluctuation As revealed by calculations, the system tends to fade out, whereas at From Figure6follows that ( ) ( ) generated by the system die down monotonically, and the entropy of the system ( ) ( ) 0 p S t tends to its stationary value ( ) produced by the system build up

Figure 6 .
Figure 6.Time history of the pair entropy derivative Along with the coefficient20 Ĉ , the transformed expression (8.3) for the distribution of the particle-density flux  nU contains two additional coefficients:

Figure 7 .
Figure7.The Z axis of the XYZ frame with its origin at the center of the sphere; , , r θ and ϕ are the spherical coordinates of some point x.

Figure 8 Figure 8 .
Figure8shows the time dependence of the

1
Sol stationary axisymmetric solution to the S20 set of the multimoment hydrodynamic equations satisfactorily reproduces the around a sphere.The attainment of the first critical Reynolds number loss.The cause for stability loss is as follows.After the passage of * 0

2
, are shown in the upper branch in Figure12( ) Sol solution, there are significant intervals between neighboring vortex rings.The combination of the solu- According to the Lyapunov theorem, the condition (11.1b) is compatible with the fundamental Lyapunov definition of stability.Inequality (11.1b) makes up a formal mathematical stability criterion.A formal mathematical criterion, however, gives no way of revealing the physical cause of stability or instability of an open system.

Figure 13 .
Figure 13.The excess entropy production component of the order of 6 2 0 0 0 Ma Re kn v τ

Figure 14 .:
Figure 14.Time history of the distributions, and a stable state of the central type with the ( ) Consider the flow regime at small Reynolds number Re 1  .At the hydrodynamic stage of description cha- racterized by small values of the Knudsen number Kn 1  , Ma Kn Re 1  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 2 l ≤ − .G ij S x in the flow regime in question.In expansions (3.1) of hydrodynamic values n , v p , and G ij S , it will suffice to retain only( When constructing the distributions of hydrodynamic values in Section 3, the series (2.5) was truncated to the terms that contribute linearly in cosθ and sinθ to these distributions.Going beyond the limits of the Re 1 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)thatareproportional to 1, cosθ , and 2 cos θ .The expansion (4.2) is used to calculate the first-and the third- order moments and gives rise to the r -components of Equations (4.8)-(4.10)proportional to cosθ and 3 cos θ .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 cosθ .We were compelled to make one exception associated with re- , in the distribution of particle-density flux (4.8).The terms of this order dominate nU at Re 1  .
[8] 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 20 S .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.
[18]ems as a whole.The second term of the left-hand side of Equation (6.1) is integrated over surface confining volume V (over the surface of the spheres of radii А and a ).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 In accordance with (6.1), evolution of the ( ) ( )0 p S tpair entropy is defined by two factors, by the ( ) ( ) pp S t , and deviates increasingly farther from its stationary value( ) * 0 )

1
S t terms.We obtain that the α - state of the open system remains stable if 1S t α δ serves as Lyapunov function, and then, the first of inequalities (11.1b) is a sufficient condition for the system stability in re to