Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hit-conf.imec.msu.ru/books/NeZaTeGiUs_2010.pdf Äàòà èçìåíåíèÿ: Fri Apr 4 12:06:29 2014 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:14:08 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 87 jet

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« » 01 - 07 2010 . . . . «» ­ : . 2010. - 196.

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, . .. (15 2009) .. (25 2008). , .. .. . - , . . . ..-.. ..

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. . 10. .., .., .., .., . . 11. .., .. . 12. .., .., .., .. . 13. .., .. . 14. .. . 15. .., .., .. . 16. .. . 17. .. - . 18. .., .. ­ . 19. .. .

6


20. .. . 21. .., .. . 22. .. . 23. .., .., .., .. . 24. .., .. - . 25. .., .. , . 26. .. . 27. .. - , . 28. .. . 29. .. . 30. .., .., .., .., .. .

7


31. .., .. , . 32. .., .. . 33. .. . 34. .. . 35. .. , . 36. .. . 37. .. . . 38. .. . 39. .., .., .. . 40. .. : . 41. .., .., .. .

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42. .., .. : . . 43. .., .. . 44. .. . 45. .., .., .., .. . 46. .. , . 47. . . . 1. .., .. . 2. .., .. . 3. .., .. 3He. 4. .. .

9


5. .. , . 6. .., .., .. . 7. .., .., .. . 8. .., .., .. . 9. .., .. . 10. .., .., .. - . 11. .., .. . 12. .., .. . 13. .. . 14. .., .., .., .. , . 15. .. .

10


16. .., .., .. . 17. .., .. . 18. .. . 19. .., .., .., .. . 20. .., .. . 21. .. . 22. .. . 23. .., .. . 24. .., .., .. . 25. .., .., .. .

11


26. .. . 27. .. . 28. .. - . 29. .., .. . 30. .., .. . 31. .., .. , . 32. .., .. . 33. .., .., .. . 34. .. . 35. .. .

12


36. .. . 37. .., .. - . 38. .., .. .

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.. , - . (). , , . -. . , . . . . , () ( ). , . . . , , , . , , . , . . . , . , . , . ,

14


. . () . . , . . , , . . . HYDRODYNAMICAL MODEL OF VACUUM Anatoly Abrashkin Institute of Applied Physics RAS, Nizhny Novgorod The model of a vacuum as a quantum-hydrodynamical continuum is proposed. According to the hypothesis a vacuum consists of virtual particles with Plank's scales named as etherons. Their length, lifetime and mass are expressed in terms of the velocity of light, the gravitational and Plank's co nstants. Particles form the Boze-condensate. Due to the Madelung transformation the SchrÆdinger equation for an e nsemble of non-interacting etherons is reduced to the system of hydrod ynamic equations for an ideal compressible fluid. The density of the fluid is propo rtional to the squared modulus of the wave function. The velocity is propo rtional to the gradient of the phase of the wave function. The dependence of the quantum mechanical pressure versus the fluid density is nonlocal. The equations of dynamics of a vacuum are invariant to the Galilei transfo rmations. From the hydrodynamic point of view they define potential flows only. Orig inal solutions for two -dimensional flows with the quantified circulation and the spherical stationary structures are constructed. These structures are the exa mples of the simplest elementary particles. It is shown that the quantum longitudinal waves of density propagate in a homogeneous vacuum as the gravity waves. According to the general relativity theory the gravity waves are periodical oscillations of metrics. Conventio nally it is assumed that they are similar to the electromagnetic waves, i.e. are transverse, non-dispersive and propagate with the velocity of light. We reject the idea of an absolutely empty space and assume that the gravitational waves represent the density oscillations of vacuum induced by the quantum pressure.

15


As a result, our equation differs from the traditional wave equation. It is of the fourth order and describes the longitudinal waves. The dispersion equ ation for the waves contains the spatial dispersion. The value of their group velocity can't be higher than the velocity of light because the maximal wave number is limited by the inverse Plank's length. The state of the homogeneous and isotropic vacuum turbulence is stu died. The etherons play the role of pulsations. The spatial scale of average tu rbulent fields is much greater than the Plank's length and temporal scale of the fields considerably exceeds the Plank's time. At these scales the density is constant and the pressure is equal to zero. It is shown that the transverse ela stic waves can propagate in such a medium. They are identical to the electr omagnetic waves. The magnetic and electric fields relate to small perturbations of the average vorticity and to the divergence of Reynold's tensor respectiv ely. The velocity of the wave is equal to the velocity of light. Thus, the photon is interpreted as a quasi-particle of a turbulent vacuum. Cosmological applications of the model are discussed as well. It is a ssumed that the dark matter is a real continuum of etherons in the laminar state. The turbulent continuum radiates photons. But due to the turbulent di ffusion the wavelength of the photon acquires the red shift. .., .. , ­ ( , - , , - ) [1]. 1931 . . : . , 1946 . , , , .

16


, . ­ . - -. , [2]. .

. 1.

1. .. . ­ ., , 1969.-183. 2. Tarunin E. L., Alikina O. N. Calculation of heat transfer in Ranque ­ Hilcsh's vortex tube. International journal for n umerical methods in fluids, 2005, 48, P. 107 ­ 113.
TURBULENT REGIME OF THE HYDRODYNAMICS AND HEAT TRANSFER IN THE RANQUE - HILSCH VORTEX TUBE Antiipina N.A., Tarunin E.L. TThe Perm state university (Perm, Russia) Due to features the vortex effect finds practical application in many various areas of technique and manufacture (vortical refrigerating chambers, thermostats and vac uum pumps, driers, separators, elements of instrumentations) [1]. The effect of temper ature division of gases was open by French scientist Ranque in 1931. According to his experiences a twirled stream of a compressed gas was separated in a vortex tube on two: cold and hot. After experiments of German scientist Hilsch (1946) which have confirmed experiences Ranque, the vortex effect beca me object of researches of scientists of different countries.

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Difficulties of studying the processes occurring in vortex devices consist of h ydrodynamics and heat exchange in them is described as a complex system of nonlinear equations in partial derivatives. The current in real vortex tubes is three-dimensional and turbulent, pressure difference is great and effects of compressibility are essential. For this reason at numerical researches full Navier ­ Stokes' and energy equations were used. One example of current received as a result of calculations at the assumption axial symmetric of a vortex tube, is shown in figure 2. The main feature of the flow is the vortex zone of the recurrent flow, the existence of which was doubted in many papers [2]. Numerical experiences were fulfilled with different values of the parameters of the problem.

Fig. 1. One example of axial symmetric current

References
1. Merkulov .P. Vortex effect and application in a technics of it. ­ Moscow., Machine-building, 1969.-183p. 2. Tarunin E. L., Alikina O. N. Calculation of heat transfer in Ranque ­ Hilcsh's vortex tube. International journal for numerical methods in fluids, 2005, (48), p. 107 ­ 113.

.. . .. . [1] 2010 . ,

18


. , . , . . ParJava (4D) (), . , , . , . 1. .., .., .., .. . . . .. . -: , « ». 2010. 362 . ABOUT MESO-SCALE TURBULENCE IN THE OCEAN-ATMOSPHERE DYNAMICS S..Arsen'yev O.Y. Schmidt's Institute of Physics of the Earth. Moscow This talk presents the book [1] which is publishing in 2010 year [1]. Developed in [1] the theory of meso -scale turbulence is based on the spatial averaging method that leads to asymmetry of the Reynolds's tensor of stresses and necessity to use the balance equations for angular moments of the ave raged flow and meso-scale turbulent eddies. The system of equations is closed by the law of conservation for moment of inertia which is proportional to square of scale of turbulent eddies.

19


When applying the meso-scale theory to physics of atmosphere, the problems of turbulent wakes behind a body are considered including effects of suspensions and intermittency of flows. The geophysical applications include the problems of formation of ocean currents on equator and global oceanic circulation. The problem of the magnetic dynamo of the Earth is also touched upon. Two chapters are devoted to turbulent currents and heat exchange in pipes and channels where application of the meso-scale theory has allowed to completely describe the classical turbulent velocity structures as well as to find out some new effects. Implementation of the Par Java parallel program environment for cluster systems has allowed to receive four -dimensional (4D) numerical solutions of formation and evolution of tornado, which illustrate the kinematics of processes and the adequacy of calculation to natural observations. The book is offered to specialists in geophysics, hydrodynamics, meteorology, and computer implementations, as well as to students and postgraduates. LITERATURE 1. Arsen'yev S.A., Babkin V.A., Gubar' A.Y., Nikolaevskiy V.N. The theory of meso-scale turbulence. The eddies of atmosphere a nd ocean. Ed. by academician G. S. Golitzin. Moscow- Izhevsk: Institute of the computer calculations, Scientific Research Center "Regular and chaotic dynamics". 2010 year. 316 pp. , .. . .. . ­ ( 2 30 ) , . . , ( ) , :

u ( / H ) sec h2 ( / ) , .
P p0 g z g

(1 ) (2 )


gH

sec h 2 ( / ) ,

20


= H (1-n) (4 AL/3A)1/2 ­ (1),(2), AL A ­ , . (1),(2): P ­ , p0 ­ , H ­ , g ­ , - , z ­ , = x + Vt ­ , n ­ , V = (gH)1/2 ­ . . , , . , ( ). , . ON THE THEORY OF SQUALL STORMS AND TORNADOES Arsen'yev S.. O.Y. Schmidt's Institute of Physics of the Earth. Moscow By definition, the squall storm is brief (from 2 to 30 minutes) increase of wind velocity down to hurricane values without rotatio n, which is observed in the zone of severe thunderstorms or on propagation of rapid cyclones carrying cloudiness and rains. In present work the soliton theory of squall storms is constructed. It is shown that the squall storm is dissipative (turbulent) aut osoliton of the wind velocity and pressure in lower troposphere. This auto soliton is exited on the under cloudy temperature inversion by hurricane winds in the middle and upper troposphere

u ( / H ) sec h2 ( / ) , .
P p0 g z g

(1 ) (2)


gH

sec h 2 ( / ) ,

where = H (1-n) (4 AL/3A)1/2 is width of solitons (1),(2), AL and A are coefficients of horizontal and vertical viscosity, correspondingly . In formulas (1),(2) also: P is air pressure nearby Earth's surface, p0 is air pressure on in-

21


version, H is height of inversion, g is acceleration of gravity, is air density, z is vertical coordinate, = x + Vt is progressing coordinate, n is relative roughness on the Earth's surface, V = (gH)1/2 is velocity of soliton motion. The theory corresponds to observations of squall storms and it allows calcula ting of them. If the progressing storm soliton hits into weakly rotating tornado gene thunderstorm supercell, then it stimulates additional collapse of air pressure and increases wind rotation, generating tornado. In present work we discovered analytical formulas, which describe azimuth component of wind v elocity in tornado (depending on radius). Theoretical calculations are compared with observations the wind velocity inside tornado, which was obtained with the help of mobile Doppler's weather radar in USA. ( , ) .. ­ - () () (), -- - . , () - . , , : , , -- , () . : - , , -- , . , - -

22


-- . . , , - . , 60% -- , 5 15 . .. . . ( 80 ) () -- , , , . ( , , ) -- , , ( , ). -- - -- . -- (, ) , , , , . ( ) , , 70­75 . . , Alberto 30% . -- 2 10 . -- .

23


, -- , -- ( ) . -- , , , , .. . , , , , , . , , . , , , . , . , , , , , . «» .. (1), .. (2), .. (2), . . (2) (1) , (2) " . . .. ", , , , / , . -

24


-- , , . , , . ( ) ( ) . -- , . -- , , , , . . , . ( GLOBAL -Field, http://www.iki.rssi.ru). , . , "" . , , , , , , - , . , . , , , , -- , . 07-02000294.

25


.. , , , , , . . , - - , . , . , , . - , . . : , 10 . , , . Solfatara (Campi Flegrei). , 9 . , . ,

26


, 4 . . . , . , , , . , , , . , . , ­ . , , . ( 09-01-92434) « » ( 1959.2009.1). HIGH-TEMPERATURE WATER-CARBON DIOXIDE FLOWS IN POROUS MEDIA Afanasyev A. Institute of mechanics MSU, Moscow Coupled water and carbon dioxide flows in porous media can take place both in natural flows like those forced by volcanic eruptions and in man -made processes like underground carbon dioxide storage or geothermal energy r ecovery. The pressure and temperature in these flows can consider ably exceed their values in water critical point. Today there are no adequate mathematical models that can in aggregate describe both water and water -carbon dioxide mixture properties in sub- and supercritical regions and the dynamics of their flows in such conditions. Thereby the influence of critical p oint on water flows in porous media is not well understood. In the paper cubic equation of state is used to describe water -carbon dioxide mixture in wide range of conditions including critical conditions for

27


mixture. The equation generalizes well known Peng-Robinson equation and can be used to describe properties not only of hydrocarbons but also of ca rbon-dioxide and water. The real mixture properties measurements are used to determine the equation coefficients. Comparison between experimental mea surements and data calculated via the equation of state shows a good agre ement between the data. For example the error in water density calculation is less than 10 percent. There are worked out effective and fast algorithms for phase equilibrium calculation via pressure, enthalpy and mixture composition ­ those thermodynamic variables that are most suitable for near critical flows calculation. The mixture properties simulations are used in research of it flows in p orous media with application to flows in Solfatara volcano (Campi Flegrei). The flows in porous media that take place in the system are forced by ma gmatic chamber located at depth of 9 km. Magma degassing make hot supe rcritical plume of water-carbon dioxide mixture ascend to shallow layers where the fluid from the chamber mixes with cold meteoric water. In the paper the dynamic of high-temperature water plume is studied that is developed because of high-temperature water source located at depth of 4 km. The source sim ulates the mixture flux from the magmatic chamber. The problem is studied in one-dimensional and three-dimensional cases. There is discovered that in one dimensional case there exists a single temporal value when water critical co nditions are reached. At this time intense phase transfer process s tarts and twophase flow region rapidly develops and expands both in upper and lower d irections. In three-dimensional case the water critical temperature and pressure exist permanently after the moment they are reached in the flow. There is di scovered that there exist not a single spatial point but a whole line where the critical conditions are reached. There is shown that as the system tends to steady state there is vapor single-phase flow exactly over the source. The two phase flow zones develop only in peripheral regions where the hot plume interacts with cold meteoric water. The work is supported by Russian foundation for basic research (N 09 01-92434) and grant for leading scientific schools (1959.2009.1).

28


.. , , 10 . , . - . .. . . .. , . .. , - . , . , , , . ( ), , , ( ), .. . . , «» , -

29


(- ). , , , (Voyager 1 2, Hubble Space Telescope, Ulysses, IBEX .). Voyager 1 2 2004 2007 .., , . ( ), . ( , , .) . .., .. . .., , , . . 1.1-1.5 ( 1600 /). ~30 , ~0.5 . [1]. ( ). 200 , ; ~0,4 . . -

30


E/N ( ­ , N ­ ) . , : , , [1, 2]. . - ( . 1 ). , - . , ~6 , . . 08 -08-90003-_.

31


. 1 - : ) , ) .

1. .., .., .. . // , 2008. . 34. . 15. . 75-80. 2. .., .., .. .// , 2009. . 427. 1. . 32-34.

32


. .., .., .. . .. , , , (Pr=0.05) (. [1]). , «Instabilities and Bifurcations in Fluid Dynamics 2009». Pr=9.2 , . R=1, H=0.92 Rx=0.5 [2]. ­ T. Rez. : 1 ( ) 3 ( ). , , ( ) . , . ­ , . , , , , . (Gr = 1.9105T, Mn = 586T, T 0.15K 1.0K , Re x 0 1500) . . [2], . 33


, , [2] ( ) . [2] T=1.0 K, Rez = 0 "false transient" . , [2] . , ­ [3], .

, . (. [4]). 2496.2008.8 ( 09-08-00230).

34


. 1. V. Polezhaev, O. Bessonov, N. Nikitin, S. Nikitin. Three -dimensional stability and direct simulation analysis of the thermal convection in low Prandtl melt of Czochralski model. The Twelfth International Conference in Crystal Growth, Jerusalem, Israel, Abstracts, 1998, 178. 2. N. Crnogorac, H. Wilke, K.A. Cliffe, A.Yu. Gelfgat, E. Kit. Numer ical modelling of instability and supercritical oscillatory states in Czochralski model system of oxide melts. Cryst. Res. Technol., 2008, 43(6), 606 -615. 3. O. Bessonov, V. Brailovskaya, L. Feoktistova, V. Zilberberg. Numer ical simulation of 2D and 3D convectio n in water-soluble crystal growth processes. International Conference "Advanced Problems in Thermal Conve ction", Perm, Russia, Proceedings. Perm, 2004, 325-330. 4. .. , .. , .. . . , 2003, .2, 4 , .63-105. ..1, ..1, ..1, ..2, .3 1 , 2 , , 3 , , () ­ 10 . . -, , . -, , , «» . . , , ). , -

35


, ­ . , . -, - , , , , , . , , . , -. , . , , ; . 07-08-96039. CONVECTIVE FLOWS IN MAGNETIC NANO-SUSPENSIONS A.A. Bozhko.1, A.F. Glukhov.1, G.F. Putin.1, S.A. Suslov.2, T. Tanjala3 1 Perm State University, Perm, Russia 2 Swinburne University of Technology, Melbourne, Australia 3 Lapeenranta University of Technology, Lapeenranta, Finland Magnetic nano-suspensions (MNS) belong to one of the types of nano fluids ­ colloid suspensions of single-domain particles with the average size of 10 nm. Two types of body forces arise in a non -uniformly magnetized nonconducting fluid placed in a magnetic field. First are the driving magnetic forces that can induce convective motion under certain conditions. The second type are the "resistive" forces which arise because of the distortion of a ma gnetic field due to the fluid motion. The magnetization non-uniformity in MNS can be caused by the temperature variation in the medium and by the non uniformity of the volumetric distribution of magnetic particles. Therefore

36


there exist two main mechanisms of magnetic convection: thermal and co ncentration-driven (magnetophoresis, thermophoresis). In addition, one has to take into account gravitational mechanisms of convection that also have two origins: thermal expansion and variation of concentration of a solid phase due to both the thermal diffusion and gravitational sedimentation of particles. It has been shown that convection in MNS is of an oscillatory typ e. Irregular oscillations were observed in one -, two- and tree-dimensional flows arising in a convection loop, spherical cavity and horizontal, inclined and vertical layers, respectively. A spontaneous excitation and decay of convection have been observed and regimes of chaotic localized states and of standing and propagating waves have been detected. The influence of longitudinal and transverse magnetic fields on the mechanical equilibrium and flows of fluid and the heat and mass transfer has been investigated. The interaction of thermo-magnetic and thermo-gravitational flows has been considered in order to determine the evolution of convective patterns. Understanding of the nature of magneto -convection is required for the use of MNS in such applications as energy conversion devices, various sensors, alternative cooling in micro-electronics and micro-gravitation and in crystal growth control. The work was supported by the grant 07 -08-96039 from the Russian Fund for Basic Research. 3 .., .. , , , 3He [1]. 1) 3He . 2) . 3) . 4) . , 3He , . , , 3He -

37


( , , K , .) , ( v / t ) , v , K E [2]. , rotE , E, . 1. Salomaa M.M. & Volovik G.E. Quantized vortices in superfluid 3He. Reviews of Modern Physics 59 (1987) p.533 2. Boldyreva L.B. & Sotina NB. Superfliud Vacuum with Intrinsic D egrees of Freedom. Physics Essays 1992; 5: 510. 3. .. . . 2002 . .. . .. , ( ) . . , , . , . , , [1-3]: dv dv 1 div (1 C ) v C v s 0, 1 C C s s p j ij e i dt dt Re





38


dsvs 9C ( v v s )1 dt 2 s , , ­ , d s / dt . : U ­ , L ­ , 0, 0 ­ . : Re UL 0 / 0 - , , 60 L /(mU ) - C div C v s 0, t
0 s / 0 - .

0 ( [3]). : 1, 2 ­ , ­ , y0 ­ . . [4]. [5] . , , ( > 0.23). , [5], C2<<1, C2 ~ 1 y0 = 0. [5]. , y0 > 0 . max i(k) Re ~ 10. , .

39


0(y). ­ ( -622.2009.1) ( 08 -0100195). 1. Marble, F.E. Dynamics of dusty gases // Ann. Rev. Fluid Mec h 1970, v.2, 397 446. 2. .. // . . . 2008. 6. . 40­53. 3. .. // , 2009, . 429, 4, . 477­480. 4. .. // . . 1961. . 16. . 3. . 171 174. 5. .., .., .. // . 1998. . 24. 5. . 76 ­ 80. STABILITY OF SUSPENSION COUTTE FLOW UNDER THE PRESENCE OF PARTICLE CONCENTRATION GRADIENTS Boronin S.A. Institute of Mechanics of Lomonosov Moscow State University, Moscow We consider the stability of the dispersed flow between two plates mo ving relative to each other (plane Couette flow) taking into account the finite volume fraction of particles. The suspension flow is described within the framework of a modified two -fluid model. It is assumed that the suspension effective viscosity as well as the interphase force depend on the particle vo lume fraction. The study is aimed at analyzing instability mechanism of part icle-laden flows which arise due to a stratification of the concentration of i nclusions. The system of governing equations for the suspension flow in the nondimensional form is as follows [1-3]:

40


div (1 C )v C v s 0,

1

C



dv dv 1 C s s p j ije dt dt Re



i



dsvs 9C ( v v s )1 dt 2 Here, parameters of particulate phase are denoted by index s; is the volume fraction of particles; individual derivative d s / dt is calculated along the particle trajectories. The scales of the flow are as follows: U is the half of the relative velocity of plates; L is the half of the height between the plates; 0, 0 are viscosity and density of the carrier phase. Non-dimensional parameters include: Re UL 0 / 0 is the Reynolds number, 60 L /(mU ) is C div C v s 0, t
0 the particle inertia parameter (inversed Stokes number) , s / 0 is the particle-to-fluid substance density ratio. Main-flow particle concentration profile 0 is specified analytically and corresponds to the particle accumulation in the vicinity of middle -plane of the flow (the same as in the study of a stratified suspension flow in the plane channel [3]). The particle concentration profile is specified by the following non-dimensional parameter: 1, 2 are particle volume fractions in the vicinity of plates and in the vicinity of the flow middle -plane respectively, is the particle concentration gradient, y0 is the width of the zone of uniform particle distribution. After the linearization, the problem of linear stability is reduced to the e igenvalue problem for the fourth-order ordinary differential equation in terms of the magnitude of the stream-function perturbation. The eigenvalues are calculated by means of the orthohonalization method [4]. In the study [5], the stability of plane Couette flow of the dusty gas with the non-uniform particle concentration profile was analyzed. It was shown that the dispersed flow is unstable in the range of Reynolds number values if the magnitude of the particle mass concentration exceeds some threshold va lue ( > 0.23). The problem formulation presented in [5], in the framework of the present study corresponds to the limiting case C2<<1, C2 ~ 1 and y0 = 0. The results of the calculations is in agreement with the study [5]. It is found that in the case of y0 > 0 the growing waves exist even at low Reynolds numbers. Maximal increment of the wave growth max i(k) is reached at Re ~ 10. Numerical calculations show that both particle-to-fluid density ratio and particle inertia parameter do not affect the stability of the suspension flow significantly. Major effect on the stability is produced by the variation of shape of the particle concentration profile 0(y).

41


The work is supported by the grant of the President of Russian Federation for the support of young Russian researchers with Ph.D. degree ( 622.2009.1) and RFBR grant ( 08-01-00195). REFERENCES 1. 2. 3. 4. 5. Marble, F.E. Dynamics of dusty gases // Ann. Rev. Fluid Mech 1970. V.2. P. 397 446. Boronin S.A. Investigation of the stability of a plane -channel suspension flow with account for finite particle volume fraction // Fluid Dynamics. 2008. V. 43. 6. pp. 873­884. Boronin S.A. Hydrodynamic stability of stratified suspension flow in a plane channel // Doklady Physics. 2009. V. 54. 12. P. 477-480. Godunov S.K. On the numerical solution of the boundary-value problem for the solution of the system of linear differential equations // Uspekhi Mat. Nauk [in Russian]. 1961. V. 16. Iss. 3. P. 171 174. Rudyak V.Ya., Isakov E.B., Bord E.G. Instability of the plane Couette flow of two-phase fluids // Letters to ZhTF [in Russian]. 1998. V. 24. 5. P. 76 ­ 80.

.., .. . .. , - . -326 . M = 6, Re1=11.9106 -1. . 500 7. . , . : ) , - 50 42


­ 100 ; ) 30 . . f1, f2 f3 = f1+f2. 0 ( ) 1 ( ). . . , . , . , [1]. ( 50 ) . . , ( ) . , [1]. . , . , . , (. 1) [ 2,3]. , .

43


. , , , , , .

. 1. () (d). f1 + f2 = 300


1. Bountin, D