Мокеjеv А. А., *Мокеjеv Аn. А icon

Мокеjеv А. А., *Мокеjеv Аn. А

НазваниеМокеjеv А. А., *Мокеjеv Аn. А
Дата конвертации10.08.2012
Размер96.98 Kb.



Мокеjеv А.А., *Мокеjеv Аn.А.

Private laboratory, (Vitebsk, Byelorussia, *Moscow, Russia)


The decrease of a tension of shift with speeds of shift which larger than velocity which is appropriated to energy of connection of fluctuation microcrystals (negative viscousity), leads to excitation of wave auto-oscillations of local velocity - to turbulent pulsations.

Their averaging by distribution of quasy-partcles - pulsons gives the square dependence of turbulent viscosity from average velocity, from distance up to a surface of a streamlined solid body, and its linear dependence on velocity of shift deformation.

Attenuation of pulsation waves in area, where the gradient of velocity is small expressed by a multiplier similar to Fermy distribution, in which the role of a Fermy surface plays a surface of a boundary layer. The turbulent viscosity is included, when the gradient of velocity will exceed critical meaning appropriate to critical local Reinolds number .

Decrease of structural viscosity with growth of a gradient of velocity ( to a wall of the channel) and increase of turbulent viscosity with growth of a distance from a wall of the channel leads to occurrence of the profile of average velocity, which is appropriated to experience.


The majority of researches assumes, that turbulence of an incompressible liquid can be described by the Navije-Stoks equation [1-5] with constant viscosity. It, however, does not leads to the occurrence of turbulent pulsations [6].

They arise only, when is supposed [7,8] negative value of index of attenuation in the wave decision, that is negative value of viscosity in perturbation equation [8], [9].

In the other approach [10,11,12] the pulsations of speed are entered as imposing of sound waves from primary perturbations of areas of superfluous pressure at a wall of the channel arising on the its unhomogeneity [2, 9-12]. However, all similar theories do not give practical results and do not allow to calculate a profile of velocity and a resistance to the turbulent current. There are separately physical "scripts" of occurrence of turbulence [7] and half-empirical models for technical accounts [1-4]. The opportunity of pulsations is considered as an opportunity of excitation of auto fluctuations of current, when the energy transmitted from the basic current to the perturbations will exceed some limit with large speeds [8]. "... The quantitative theory of turbulence does not exist yet " [13].

The present work is devoted to construction of the equations of a movement of a liquid which based [14] on exponent dependence of a shift tension from a difference of squares of velocity of shift and velocity, appropriate to energy of connection of fluctuations microcrystals of a liquid [15, 16], with a falling site, which accords to negative effective viscosity; is devoted to the finding of the auto-oscillations decisions of the Navije- Stocks equation, which represents a pulsations of velocity; is devoted to the definition of expression for the turbulent viscosity by means of the averaging of the motion equation, and is devoted to the account of a profile of the average velocity of turbulent current by a numerical method in the slot-hole channel.


The liquid with temperature which is close to the solidify temperature , consists of set of fluctuations microcrystals , which are separated by gaseous layers (loss of the distant order) [6, 8]. The dispersion interaction of microcrystals is described as an electrical current of polarization with density [18]


which in a supervision point with coordinates X, Y, Z at a moment of supervision t, is the imposing of the responses from all elements of environment in all previous moments of time t ' in all removed points r '. The environment creates a weak molecular electrical field with intensity Е (r-r ', t-t '), which is not destroying the environment and which depends on delay time t-t ' and distance r- r ', This is expressed through reason function f [17].

Therefore reason function f is represented in first terms of decomposition in a line under the small relations of distance to length of a wave of a molecular field, under the small relations of time of delay and speed of relaxation, which accords with plasma frequency, and under the small relations of intensity of a molecular field to intensity of forces of connection of system. It results to decomposition of density of a current on density of a current of polarization jp and density of a current of internal screning jc.

The current of polarization jp is formed by the polarizing charges with density


Where - potential of an electrical field, q - charge of particles making a microcrystal, - factor of its screening, kB - Bolzman constant , Т - absolute temperature, - dielectric permeability of vacuum.

Density of a current of internal screening is equal to product of average velocity v of the charged particles in the volume of correlation of their movements and the density of internal screening charges [10]


The Gauss theorem with these density of a charge gives the equation for potential. Through the decision of it the expression of force of interaction of microcrystals adjoining through a gaseous layer by thickness , is expressed as a Maxwell forces of a tension, under action of an electrical field with intensity Еа, which arises as an adhesion force of a liquid to a surface of a streamline solid body, through distance between surfaces of microcrystals .


With dielectric permeability of a gaseous layer , with concentration of the charged particles N, with their charge q, with distance between surfaces of microcrystals , with which this force is maximal, is equal

The energy of this interaction U (Ea) as Maxwell forces of a tension under action of an electrical field with intensity Еа, which arising as force which keeps a liquid close to a surface of a streamlined solid body with effective dielectric susceptibility , is equal to energy of an electrical field in a layer between microcrystals by the area So, [17-19].


In a layer of a liquid by thickness h, the walls of the channel is connected through microcrystals circuits, the lengthenings Y = h/r of which are sum of distances between microcrystals . The lengthening, with which is started the break of circuits, that is the sum of thicknesses of gaseous layers between microcrystals D = (h/r) d is the sum of the equilibrium distances between microcrystals d.

The force Fy1 of interaction of walls of the channel through one circuit of microcrystals is expressed through the module of elasticity of a circuit


Of the number Ns of the circuits connecting a walls of the channel by the area S, Ns = S/So a share of circuits, broken off by displacement, Wp and a share of circuits, which are not broken off by displacement, Wn = 1- Wn determine force Fy of interaction of walls of the channel through the not broken off circuits of microcrystals


For shift deformation of layers X of a liquid across intensity of a field Еа, across circuits of particles,the lengthening of a circuit consisting from h/r of microcrystals is equal Y = X2 /h, critical lengthening of break of a circuit Yo = d h/r = Dx2 /h,

Force of a tension of one circuit [10,11]


The force of interaction of walls through Ns circuits [10,11] that is component along shift, determines weak shift elasticity of a liquid disappearing with X > D = 10 A, is found out on experience [20].


With displacement X > D the circuits of microcrystals are broken off and the current , relative movement of microcrystals with speed Vr, shift of layers of a liquid by thickness r, causes the collisions of microcrystals . With collision between them there is a circuit of molecules, which connect the microcrystals, with the size of a microcrystal r, with number of particles Nc close to number of particles in a microcrystal, which is broken off with relative displacement of microcrystals Xo = r.

Transmitted through it a tangent momentum during its existence Xo /Vr, is equal


where М - mass of a microcrystal, Da - relative displacement of microcrystals, when their connection is broken off, I - integral in last expression . Kf (Ea) = So K (Ea) d. During the collision the momentum is transferred.


The frequency of collisions of one microcrystal is equal to number of microcrystals in volume of the cylinder with effective section and with the length Vr, and which is directed along speed of their relative movement Vr and which is parallel to speed of current.

where - concentration of microcrystals.

With each collision through the area of longitudinal section of this cylinder normally to plane of current the momentum is transferred during the time dt

The number of such microcrystals on the area of longitudinal section of the cylinder Soo = 1 m2 which contain a velocity vector, is equal to number them in volume of a layer by thickness which is equal to the size of the microcrystal r, .

The tension of resistance to current is equal


The substitution in this expressions the expression for from (9) gives


The relative velocity Vr of microcrystals with shift current is equal Vr = r dV/dz = V/h, so

is expressed through effective viscosity

. (13)

The probability of "collision" of microcrystals Wa is probability that, by N molecules is formed a microcrystal near to other in the next layer. For it the potential energy U of interaction of these molecules should exceed energy Er of their relative movement together with the next layers of current. Representation of free energy through entropy S (0), through number of microconditions which are carrying out macrocondition with interactions energy, appropriate to infinite distance of molecules, that is energy of connection of a microcrystal, the replacement the of free energy through speed of shift G appropriate to energy of connection of a microcrystal, and the replacement of the thermal energy Uav through the velocity Q appropriate to it,

, ,


If viscosity of a gaseous layer , the substitution of this expression at the (12) gives

structural viscosity,


and tension of resistance to current


With g << Н the tension increase linearly with growth of speed of shift, and with g> > H it decrease to a tension in a gaseous layer.

Dissipative characteristic of a flow has a falling site, appropriate to negative friction.


The law of a movement of a liquid (profile of velocity) is determined by its equation of a movement [7]. For current with the axis symmetry [1]


Owing to existence of a falling site of the rheological characteristic appropriate to negative

viscosity, the energy of current of a liquid can be transferred to auto fluctuations, which cause deviations V1 of speed from its average meaning V and create of perturbati

ons of pressure gradient relatively to the average ,

which are determined by the self-coordination decision of the equations of a movement


Then movement equation breaks up to two. The equation of the first approximation for V


The perturbations equation


In this approximation the perturbations equation in the narrow symmetrical channel of average radius r1 has almost constant factor and is the equation with constant factors. Designations


lead the equation of perturbations to a kind


With average speeds of deformation in the field of a falling site of the rheological law and negative differential viscosity

critical Reinolds number Rek corresponds to the kinetic energy of a relative movement of micro-crystals, which is greater them the energy of their connection, with which force of inertia surpass the dissipative force of "collisions" of microcrystals, the effective viscosity is negative, реологический the law has a falling site

The equation of a movement (18) admits division of variable

It is equivalent to the two ordinary equations

The second of them has the decision

Together with the decision of the first equation in which - the root of characteristic equation

with gives the decision as an "standing" wave across current which increases , and which represents the multi-streams current


while will not leave for a limit of a falling site of the rheological law and the effective viscosity = / will not become positive and the stationary amplitude of pulsations of speed will be established. The existence of spatially oscillatying movements represented by the decisions of the perturbations equation allows to admit and existence of the wave decisions [8], being valid, if is satisfied the dispersion equation

This decision represents a not fading wave extending with speed C an across flow. Imposing of these waves


represents casual pulsations of speed about the first approximation which is the locally average approximation, that can be submitted a as quasy-termal movement of the quasy-particals - pulsons, which form the gaseous system. Average length of the free run of the pulson coincides with length of a Prandtl way of mixing [1-3].

With small and large speeds of deformation in the fields of current about a surface of a streamline body and near to a surface of a boundary layer the factors of the perturbations equation are positive and it has the decision as the relaxation law , which represents a subsiding standing wave [8].


With averages Reinolds numbers Re> Rek the movement of a liquid is the imposing of the first ordered approximation V and a casual turbulent pulsations V1, which is determined by the distribution of probabilities in locally equilibrium and is proportional to concentration of pulsons n (r),


so the average meaning of local pulsation's velocity is equal to zero

<V1> = 0 (23)

Averaging term by term equations (16) first approach(approximation) and the equations (17) возмущений раy in zero composed, containing odd degrees of indignation V1, according to Prandtl rules of averaging [1,2]. Then within fourth order of a little in the right part of equation (16) remains the components, for which with the account of non-elasticity is turned out expression

In it in last angular brackets the gradient of energy of pulsations in stationary current is equal to zero, just as a gradient of a flow of a momentum along current.

The flows of a radial momentum along radius to walls of the channel, are counterbalanced by reactions of walls. The stayed flow of a tangent pulse along radius to walls of the channel creates additional, турбулентное a pressure of resistance to current.

The substitution of these expressions in the average equation of the first approximation lead this equation into a kind


Then the pressure of turbulent resistance to current is equal


and is proportional to concentration of pulsons, which is assembled by pulsons concentrations, which are irradiated by all layers of a flow from a surface of a streamline body. The power of radiation (the speed of increase of pulsons number) in each layer, the pulsons concentration is more, than more layers between the given layer and the streamline body, and than more the distance r up to body.

Absorption of pulsons in the field of current with small gradients of speed

V ' < V'k, with small local Reinolds numbers Re (r) < Rek near to a surface of a boundary layer on distance r = , which is equal to thickness of a layer, is expressed by a multiplier in distribution of pulsons concentration, similar to Fermy distribution, in which the role of a Fermy surface plays a surface of a boundary layer, and which is approximated by the generalized Heavisid function Ф (r- )

For current in the channel between two walls of width a << b, much smaller " depths "b", crack) the resistance to current is assembled by resistances which created by each wall, so

The turbulent viscosity [1,2] is included by relaxation function Ф, when the local Reinolds number will exceed critical value or the local speed of shift will exceed appropriate critical value, the falling site of the rheological characteristic, Ф(V'k-V ')=Ф(H-V ') = Ф(Rek (r) -Re (r) is reached. In the flat slot-hole channel of width a> > b, much more depth, with an axis OZ of the rectangular coordinates which is directed along current, and with an axis OX which is perpendicular to current

the turbulent viscosity is equal


In the averaging equation of turbulent current the viscosity is the sum of structure viscosity (13), which decreases with growth of a gradient of speed with approach to walls of the channel and creates a profile of velocity which is appropriate to the viscous stream in to fluid environment, and turbulent viscosity , which growes closer to middle of the channel


This equation is reduced to a kind, convenient for the numerical decision


With boundary conditions of a adhesion [1]

and with the initial conditions .

The equation of a movement reduces to implicit scheme of differences[11] with a step of integration on width of the channel h and of time

which is


a thridiagonal system of the algebraic equations


with variable coefficients.

If the values are calculated by substitution in them of meanings of speed Vi from the profile, which previous in the time , this system becomes linear with constant factors and is solved by a method of driving [10,11] with the help of the COMPUTER in algorithmic language of system MathCad 2000 [13]. The program of the decision is similar it which is given in [12]. The decision is represented by a sequence of the profiles diagrams through equal time intervals almost up to an establishment of stationary current

The details of calculations are given on Internet-site www.avtoferelrheo.narod.ru

Рис.4. A sequence of an establishment of structures of speed of average turbulent current.


1. The rheological law with a falling site with speeds of shift deformation, which are larger than the speed, appropriate to energy of connection of the fluctuation microcrystals, leads to the nonlinear equation of a movement with negative viscosity on this site, which breaks up on the averaging equation of first approximation for averaging current and perturbations equation

2. The decision of the perturbation equation by a linearisation method with small and large speeds fades, so the decision of the equation of the first approximation appears steady.

3. With average speeds of deformation, when the falling site of the rheological law begins, the decision of the perturbations equation is the nonhogerrent imposing of monochromatic waves of every possible frequencies extending across current and represents casual turbulent pulsations of speed.

  1. The averaging equation of a movement gets composed, appropriate structural and turbulent viscosities.

5. The structural viscosity decreases as exponent with large speeds of deformation closer to walls of the channel with growth of speed.

6. The turbulent viscosity proportional to speed of deformation of shift and a square of speed, is included when the local speed of shift achieves the critical meaning which is appropriate to local critical Reinolds number and the beginning of a falling site of the rheological law.

7. It is grows proportionally to square of distance from a wall of the channel, and from a surface of a boundary layer (on middle of the channel) and from a surface of a streamline body in the laminar sub-layer is switched off by a relaxation multiplier similar to distribution of Fermy.

8. The numerical decision of the equation of a movement for averaging current leads to

the profile of speed of the turbulent current which is filled, and satisfactorily conterminous with received on experience.


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