Shear Induced Phenomena тема автореферата и диссертации по астрономии, 01.03.04 ВАК РФ

Georgian Academy of Sciences Abastumani Astrophysical Observatory

George Khujadze Shear Induced Phenomena 01.03.01 - Plssisa Aitrophyjicj

Abstract cf Candidate Disjcrtsticu bi Physical and Mathematical Scienics

Tbilisi 1998

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The thesis was completed in: Georgian Academy of Sciences

Abastumani Astrophysics! Observatory

Superviser s

Expert:

Opponents:

Dr. J. Lominadze

Dr. A. Pataraia

Dr. A. Tugushi

Candidate of Pbys.Math. Sciences N. Shatashvili

Leading Organization: Institute of Physics, Georgian Acaderay of Sciences

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The date of Maiaism of the Tfeesas: 1998

Abasiisrasni Astrophysical Observatory Tbilisi, Kazbega Str. 2*

The thesis is accessible: in the library of Abastnimaisj Astrophysics!

Observatory

Secretary /\ Candidate of lPhys.Matfa. Sciences Q. Chargeishvili

General Description: Sheer flows are of a wide occurrcnce m astrophysics, in terrestrial environment and in laboratory and experimental studies. The existence of shear provokes a wide variety of high energetic processes. That is why considerable attention has been devoted to the stability problem of shear flows. How does the presence of the shear affect the stability of the flows? The prominence of this problem is enlarged also by its obvious relationship with the more global dilemma oi* the onset and maintenance of turbulence in shear flows.

The problem has a long history. The traditional and generally acknowledged paradigm is the eigenvalue analysis (also known as the normal modes approach). Usually, it proceeds in two stages: (i) linearization about the mean (laminar) solution and afterwards (ii) looking for unstable eigenmodes of the linearized problem [1]. The necessary condition for the flow to behave unstably (in the sense of the standard linear theory) is the existence of exponentially growing eigenmodes. For some processes (e.g., thermally driven instabilities in Rayleigh-Benard convection flow, or centrifugally driven instabilities in rotating Couettc flow) this approach is well matched with laboratory studies. At the same time, for other lands of hydrodynamic processes, especially those driv en predominantly by shear forces, the predictions of the normal modes approach fail to match most experiments [1]. For plane Couette flow, for instance, instabilities are observed to "switch on" for Reynolds numbers as low as R « 350, while the common eigenvalue analyses predicts stability for all R's. Traditionally, this anomaly was recognized as a failure of linearization and was attributed to nonlinear effects. Recently, it was argued, however, that the failure of ' the normal dmodes analyses should be attributed to step (ii) [1-3].

Standard linear algebra says that, even in the case when all eigenmodes of a linear system are stable, inputs to the system may be amplified by arbitrarily large factors, if the eigenfunctions are not mutually orthogonal [4]. The operator that arises in Poiseuille or Couette flow is not normal (i.e., its eigenfunctions are not mutually orthogonal) [5]. Small perturbations to these flows may be amplified by factors of thousands, even when all eigenmodes are exponentially stable [1-3]. Hence, in this context, the usage of the full spectral (Fourier or Laplace) expansion for this category of shear flows may be quite misleading: the study of a single normal mode evolution docs not give adequate description of the system behaviour. These difficulties provoke a change in the paradigm. The so called "nonmodal approach" to the study of waves and instabilities in shear flows (stemming out of the 1887 paper by Lord Kelvin [6]) has recently become well-established and has been extensively used [2,3,5,7-9]. In the Kelvin formalism one considers the temporal evolution of spatial Fourier harmonics ("Kelvin modes" [7]) of perturbations without spectral expansion in time. The wave number of the spatial Fourier harmonic becomes variable in time: there exists a "drift" of the spatial Fourier harmonic in the phase space of wave numbers (k-space)[7,10.11 ]. The nonmodal approach greatly simplifies the mathematical description and helps to grasp phenomena that were overlooked in the framework of the modal approach. By this method some unexpected results on time evolution of both vortex [2,3,7,9,10] and sound-type perturbations [12] in shear flows were already obtained. This approach was successfully appb'ed to the study of magnetobydrodynamic waves [11]. It helped to formulate a new conjecture of transition to turbulence [ 10] and lead to the discovery of the new linear mechanism of the mutual transformation of waves in shear Sows.

It has been shown that there can be substantial transient growth in the energy of small perturbations to plane Poiseuille and Couette flows if the Reynolds' number is below the critical value predicted by linear «ability analysis. This growth, which may be as large as 0( 1000), occurs in the absence of nonlinear effects, and can be explined by the non-normality of the governing linear operator - that is, the non-orthogonality of the associated eigenfunctions. It has been recognized in recent years that the linear stability operators for plane shear flows can support solutions that exhibit large tranzient growth in energy, although the eigensohuions of the operators are dumped. This transient growth mechanism is attracted the attention of scientists a the field of the hydrodynamical stability [1-3,5].

t3

Theoretical science up to the end of the nineteenth century can be viewed as the study of solutions of differential equations and the modelling of natural phenomena by deterministic solutions of the« differential equations. It was at that time commonly thoght that if all initial data could only be selected, ono would be able to predict the future with certainly. We now know this is not so, in at least two way*. Firstly, the advent of quantum mechanics within a quarter of a century gave rise to a new physics, and hence a new theoretical basis for all science, which had as an essential basis a purely statistical element. Secondly, more recently, the concept of chaos has arisen, in which even quite simple differential equation systems have the rather alarming property of giving rise to-essentially unpredictable- behaviour. To be sure, one can predict the future of such a system givne its initial conditions, but any error in the initial conditions is so rapidly magnified that 110 practical predictability is left. In fact, the existence of chaos is really not surprising, since it agrees with more of our everyday experience than does pure predictability — but it is surprising perhaps that it has taken so long for the point to be made [13].

Fluids consist a large number of identical or similar particles moving in an extremely complicated way, far beyond any possibility of calculations. Yet, on macroscopic length and time scales (the so-called hydrodynamic regime), their dynamical behaviour is well described in terms of a few hydrodynamic variables which obey a set of relatively simple phenomenological laws leading to the hydrodynamic equations. These are deterministic ones, in the sense that they determine the future values of the hydrodynamic variables entirely from their initial ones. The approximate character of the hydrodynamic equations shows up in the fact that the hydrodynamic variables fluctuate about their deteiministic values. These fluctuaiions appear even on hydrodynamic scales as collective phenomena which involve the coherent motion of a large number of particles. Of course, the fluctuations cannot be computed exactly, that would be require solving the microscopic equations. However, in the hydrodynamic regime they can be decribed as a stochastic Markov process which obeys again simple hydrodynamic laws.

To describe hydrodynamic fluctuations Landau and Lifshitz have proposed adding a purely random stress tensor and a purely random heat flux vector to the Navier-Stokes equations. Nonequilibrium steady state (near full equlibrium) arise in open systems that are in contact with several reservoirs not being in equlibrium among themselves. This couses the presence of permanent hydrodynamic forces in the system, which prevent it from relaxing to equilibrium. It will be assumed here that the fluctuations are small so that they can be neglected beyond the linear order. Large fluctuations became important only when the steady state is close to a critical point, afler which the flow transition to turbulence happaned.

To avoid the difficulties arising from the slow change of the hydrodynamic fluctuations, the proper «arting point for a generalization of the theory is the Landau-Lifshitz theory In fluctuating hydrodynamics the stochastic properties of the hydrodynamic variables are implied by those of the rapidly changing random currents. To establish these in the nonequilibrium case one assumes that the fluctuation-dissipation theorems remain valid. This assumption is again based on the Markov property: Synce the correlation length of the random currents is of microscopic order, they do not "feel" the applied gradients and behave locally as in equilibrium.

The first to obtain the correct stochastic evolution equations and fluctuation-dissipation theorems in arbitrary nonequlibrium states was Keizer in 1978. Keizer gets nonlinear hydrodynamic equations for the average values, and 8 nonstationary, Gaussian, Markov process for the fluctuations end up obeying linear equations. Having established the evolution equations for the correlation functions, it is still difficult to obtain explicit results. In the first place one has to solve the nonlinear hydrodynamic equations to obtaia the steady state, which is only possible for special geometries. Examples include a fluid between a pair of parallel planes that may have different temperatures or/and perform a ample shearing motion, or a fluid between two concentric cylinders rotating with different velocities. Moreover, in solving tlie equations for the correlation functions in a given steady state, complications arise because the spatial inhomogeneouty of the system have to be taken into account. A simple fluid in uniform shear is characterized by a velocity field with conaaat gradient orthogonal to the flow. The magnitude, or shear rate, provides a single control

V,

aramcter to measure the departure of the fluid from its equilibrium state. For planar geometry, the ressure is spatially constant and the temperature is either temporally, so the macroscopic state of the fluid an be very far from equilibrium, for large shear route, but still structurally quite simple. In recent years lere have been several attempts to calculate the transport and fluctuation properties for shear flow both leoretically (using modal approach) and by novel methods of nonequlibrium computer simulations.

The purpose of this work is to outline a new linear mechanism of mutual transformations of waves I smooth shear flows: The linear evolution of acoustic waves in a fluid flow with uniform mean density nd uniform shear of velocity is investigated; The process of the mean flow energy extraction by the three-imensiooal acoustic vv'Sves, stimulated by the non-normal character of the linear dynamics in the shear low, is analysed; The fluctuation background in the two and three dimensional Couette flow is studied; lie Brownian motion of macroscopic particles in such flows is considered; The astrophysical shear flows re investigated (relativistic jets from compact objetcs and accretion disks onto Kerr black boles).

Mail description of the work: In the first subsection of the work the methods of shear flow lvestigations is described. In the second subsection the transient growth of the perturbations is studied; In se third subsection of the first section the mutual transformation of waves in smsoth shear flows is utlined. The mechanism is closely related to non-normality of linear dynamics of waves in shear flows and i best interpreted in the framework of a nonmodal approach - by tracing of evolution of spatial Fourier armonics (SFlfs) of waves in time. The core of the phenomenon may be specified as follows: Wave ■equency is a certain function (specified by its' dispersion) of the wave numbers. Mean flow velocity shear ssuhs (even in linear approximation) the variation of a wave numbers) of each SFH in time, due to the fleet of the shearing background on the wave crest. With this variation it follows that tlie frequency of iFH varies in time - it "slides along" the dispersion curve of the wave mode. As a result, for definite urometers of t[ie system, if the dispersion curves of the wave modes have pieces passing nearby one aother, wave frequencies will be closely related in a limited time interval. It is this fact which causes the j wtual transformation of the waves in case of a slow variation of waves frequencies. Waves linear • ■ansformation phenomenon should be realized in a wide variety of terrestrial and astrophysical shear flows nd grants significant diversity to processes taking place in these flows. Let us concider 2-D flow with onstant shear of velocity U0=(Ay, 0). Where A is the shear parameter. Disturbances cannot have a form fa simple plane wave due to effect of the shearing background on wave crests in such a llows (see Refs. 3]). As a result wave number of SFH depends on time: If at initial moment of time SFH with wave-umbers k„ and kv(0):

y(0) = v(k„ M0),0)exp(i k, x + ik, ((%), ; generated (where v is a general vector composed from disturbed quantities), then the phase evolution i defined by equations:

v|/(t) - y<k„. k,(t), t)exp(i k, x + i k, (t)y), kv(t) = k,(0) - k„At. .«., the wave-number of the SFH varies in time along the flow shear — in linear approximation there is a drift" of the SFH in the k -space (in wave-number space). The physics of this phenomenon can be easily nderstood by mechanical analogy of the oscillating system. Let us consider two weakly ocrupled endulums with slowly (adiabatjcally) varied lengthes in time. Equations describing the oscillations of such nes are:

5,JX,+cjJ,(t)X,=x(t)X!, d,2 Xi + t02J(t)X2 = x(t)Xi,

,-here x(t) is the interpendulum coupling coefficient. If the frequencies of these pendulums differ strongly om one another, then despite of the coupling, there is virtually no exchange of energy between them. fTective energy exchange starts when the frequencies of the oscillators close to one another. Necessary onditions for the efficiency of the energy exchange are:

I existence of the so-called "degeneration region", where | e>iJ - raj' | | x(') I. Z slow passage through the degeneration region:

55

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If only the first pendulum oscillates initially, then because of the change in its length,, the frequencies Oi(t), and o>2(t), may come close to one another and the conditions 1 and 2 will be satisfied during a certain limited interval of time. A large (and possible the main) part of the oscillational energy of tbe first pendulum is transferred to the second pendulum. As a result the second pendulum begins oscillations. In the process, tbe first pendulum may stop completely. This scenario may straightforwardly be applied in the analyses of linear coupling and transformation of waves due to the similar nature of equations, describing the linear evolution of wave SFH In smooth shear flows.

The linear evolution of acoustic waves in a fluid flow with uniform mean density and uniform shear of velocity is investigated in the second section. The three dimensional Couette flow is considered: U = (Ay,0,0); One can obtain the following set of ordinary differential equations: D'" = v4+(Хт)\ч+ууг v((1> = - Rv, -D v,(" = -P(t)D v,(1) = -yD

where D = ip'/p,,; v = u/c,; R = A(c.k,) ; р(т) = кЛ. - Rt ■ 0„ - Rt ; у - I.,' k,. t«c,k,t; It is quite important to note that between perturbed quantities such algebraic relation holds:

(1V) v, - P(tX v, + у v.) = RD+const; where const is some constant of integration. Using this expression and defining an auxiliary function: £(т) ev,(i|+] v,(t) (one can obtain the exprssions for other perturbed ones using this variable md its Erst derivative) one can get the second order equation:

4'2'+o2(t)4 = -(5(t) const; where со:(т) - 1 + рг(т) * f - (kx2+к,г(т>+k,2yk,\ kv(t) = k, - k,Rr:

As it is seen from these equation the SFH wave-number vector component aloug the Y axis varies in time. Thus, in the linear approximation, there is a "drift" of SFH in к - фасе. The variation of wave cumbers of SFH is well-acknowledged in papers cultivating the nonmodal approach. Tbe spatial characteristics of the SFH defme[k4, к»(т), к,] its frequency and, as we shall see below, the intensity of energy exchange between the SFH and the background Dow. The crucial stage of the solving process should be the solution of the equation for i,. Its general solution is the sum of the special solution of this inhomogencous equation and the general solution of the corresponding homogeneous equation (with const=0). h may be said that this equation describes two different types of perturbationi. 1) aperiodic vortex perturbations—which correspond to the case (const * 0) and 2) sound-type perturbations—which correspond to the case (const = 0); This classification is strongly justified for flows with R ''' I ; When Rsl sound-type perturbations acquire vortical features, wliile vortex perturbations gam potential qualities, so that the classification becomes ill-defined. In this section the detailed study of the sound-type perturbations is dedicated. When const = 0 the second order equation becomes a homogeneous one that is well-known in mathematical physics. It describes the dynamics of a linear oscillator with a variable frequency to(t). The equation is solved approximately when to(t) depends on * adiabaticaDy. Mathematically this condition is written as

R|PO)1«V<t)|;

then

i(t) « (C/oirM'^exp (i[q>.(t) + <(.„]); where <р^(т) is the phase of and q>„ is the initial one. In the zero shear case (R - 0) all physical quantities oscillate uith identical phases, while for R*0 a phase difference appears.

Note that left hand side of the algebraic relation is equal to [kx(kxv)],/k,2. Thus, it reveals an important quality of sound in shear flows, there are no purely potential perturbations in shear flows (acoustic waves acquire a vortex behaviour). This character is weekly pronounced for small R. Generally,

to

vorticity of the perturbation depends on the angle between the vectors k and v. When R << 1 (const = 0), right hand side of algebraic relation is small, and v and k are almost parallel to each other (vorticity of the SFH is negligible). When RSI, the direction of the velocity field of the given SFH varies considerably from point to point, tlie angles between v and k change and are nonzero in the wide region of space, i.e., velocity field of the SFH gains a vortical character. -

The expression for the perturbation energy in the adiabatic case has the following form: E(x)»[C/(lV):Ht).

It is seen from this expression that when k/kx > 0 the SFH give energy to the mean flow for times (Vct<t..a (50 /R. The SFH energy decreases and reaches minimum at t*r. , Afterwards, when the SFH "emerge" into the area of k - space where k,(r)A, < 0 (the "growth area" for the acoustic waves, (T.<T<c°), the energy of SFH begir,s to increase. Evidently, if the SFH is situated in the "growth area" from the beginning ( p0 <0) then its energy increases during the whole process of evolution. The energy exchange process between SFH of acoustic wave and mean flow is symmetrical about the point t=t. .

In the second chapter of the second section the acoustic waves are studied when R =1. In this case the adiabatic character of the frequency variation still holds if y » 1 a nil/or [Mr) » I When y » 1 adiabatic condition is satisfied and the adiabatic behaviour takes place during the whole process of the evolution. As regards to small values of Y (for simplicity, let ustake that y =0) for the SFH with [in » I' the process evolves in the following way: At the initial stage of the evolution it has an adiabatic character. Afterwards, during the certain interval of time the adiabatic condition is violated and the frequency of the acoustic wave does not vary adiabatically. Later on, when (5(t) attains large enough values the evolution of the SFH becomes again adiabatic. The presence of the nonadiabatic stage in the process of the evolution essentially influences the temporal variability of the SFH energy. Finally, we would like to itemize the main issues of the present study:

The dynamics of acoustic waves in shear flows is determined by the non-orthogonal character of i the linear problem and is described in the masl convenient way using the "language" of notiniodal' approach;

Acoustic waves can extract shear energy as intensively as vortex perturbations do. However, in the latter case the energy exchange is effective only for those SFH that satisfy the condition \ky(r)!kx | < 1, for the larger values the energy exchange efficiency heavily decreases, while in the case of acoustic waves, the intensity of energy exchange with the mean flow is substantial even for \ky(r)/k%\» I.

For RrO displacements ofphases of different physical variables appear.

In the flows with weak shear (R << 1) the evolution of the SFH of acoustic waves is highly adiabatic, while in the flows with moderate values of the parameter IRslJ, the adiabatic character of the SFH evolution is preserved only for y » 1 and/or /3(r) » I

At the nonadiabatic stage of evolution the energy exchange process (.y S 1 .»K Pit) < \ R S 1) has rather complicated character and strongly depends on the phase of the SFH at the moment of "entering" of the nonadiabatic region.

In the third section of the work the fluctuation background in shear flows is studied. -In the iatroduction the history of the problem is outlined. In the first chapter of the third section two dimensional fluctuation background is investigated in the Couette flow. To describe hydrodynamic fluctuations Landau and LiAI'.itz have proposed adding a purely random stress tensor and a purely random heat flux vector to the Navier-S[okes equations

f= dSyldx,, ij = x,y.

where S,j is the spontaneous strain tensor; Statistical properties of which are modeled by the following correlation function in accordance with the Fluctuation-Dissipation theory:

<S,j(t,r)Su(t',r')> = 2Tp0v[5iS), + 5U<V 2/3 5^5u]S(r-r')5{t-t");

After some mathematics one can obtain the following dynamical equation for perturbation eacrgy denscy (in the stationary case):

Akx<?et«k, + (2Akxkv/(k,: + kv:)lev-2v( k,! + k.!)ei + 2v< + k/)T=0

The terms on the left-hand side of this equation correspond to the processes responsible for formation of the fluctuation background. The first one describes "linear drift" of SFH ia the k -space; The second term represents the energy exchange process between mean flow and SFH. Third and fourta ones describe the dissipative and sto .hastic forces respectively. As it seen from this equation in the uniform flow (A=0). fluctuation background is formed by the las! two terms only, because the first two term are equal to zero. !n this (equilibrium) case, in stationary limit, dissipation and stochastic forces lead to ¡be thermal noise. Ci =T;

It is obvious from the dynamical equation that the various efficiency of the al>ove-menticned processes is observed in different regions of the k -space during the formation of the fiucluation background. The last two terms in this equation are dominant for SFH with large wave numbers, whereas the first two terms are - for small ones. It is worthy to note that energy spatial spectra) density of incompressible fluctuations in laminar Couette shear flow is anisotropic and significant differs bom thermal noise; Particularly, its level by far exceeds thermal noise m the certain regions of vvav^-number space. Moreover, from the dynamics of SFH follows that the permanent transformation of the mean flow energy to background (incompressible, vortex) perturbations and at last — to the flow thermal energy is realized. It is obv ious, that a new, indirect way of the flow energy conversion to heat depends on rate of the mean flow energy extraction by the background perturbations and increases with increase of Reynolds number. Consequently, the efficiency of a new way of shear flow energy conversion to heat tncreists with increase of Reynolds aumher{!). The possible influence of fluctuation background on transition to turbulence in some shear flows is discussed at last. For example, it is well known, that Couette flow remains insensitive to infinitesimal perturbations for any Reynolds numbers, but its transition to turbulence occurs at finite ones. In according to concept developed in recent years, transition to turbulence of Couette flow occurs due to the "positive feedback" (regeneration of SFK that can extract mean flow energy) that in turn is caused by nonlinear effects. Consequently, existence of finite perturbations in these flows ia necessary for transition to turbulence It is obvious that such perturbations should be rised from the external force. Actually, level of fluctuation background in small wave-number region at high Reynold, numbers by far exceeds thermal noise. This circumstance, ia turn, causes tfci "switch" the nonlinear processes at certain Reynolds numbers and at "positive feedback" — flow transition to turbulence.

In the second chapter of third section the fluctuation background of incompressible three dimensional Couette flow is studied. In this case the it is not possible to give the dynamics! equation for the spectral density of energy. In the three dimensional case the maximum of this function leads in the region, where K/K, < 0, that means that the energy of perturbations continue to increase even after pass the point t. .

hi the third chapter of third section the new way of energy transformation bto beat is discribed The mean flow energy transformation into perturbation energy and at last its transfer into heat exists ia shear flows and:

E(r,t) = i K * (E(K,t) - 1) dK, where E - E/T, (T is the absolute temperature in the energetic dimensions). Afler numerical caiculatioa one can obtain the following result: £=1,5.

In the fourth chapter of third section Brownian motion in Couette flow is investigated. As it was shown in the previous section, the fluctuation background in shear flows differs from white noise, and has anisotropic character in wave number vector space (is has the maximum m the defined region of this space). It is shown that the fluctuation background exerts on Brownian particles is Couette flow. Because of shear character of flow Brownian particle has no more diffusion behaviour, that is shown in others papers, who study this problem using model approach, but as it was shown ia the previous sections this approach is do correct in such flows. Becouse of this fact it is expected to obtain many new results by using nonmodal

approach The equation which discribes the motion of Brownian panicle, so called Langevin equation, has tie fcrru

-rd'-V) 1- Mt) - 0.

wfcere 7 - ( 6*R/m)ii, r| is the fluid viscousity, U — Brownian particle velocity; V = V, + u — fluid veSocty. o — the fluctuation of fluid velocity, and V,«V,( Ay, 0,0) is the mean flow velocity. Mean square displacements of Brownian particle along axis X, Y, Z has the form (in the dimensionless variables): <XX> =■ 2r + 6Jt/?/„t! + (2/3 +• 2kR/„)i' + V2kR /,Tt4 , <YY> = 2T +■ 6itRf„t2, <ZZ>« + 67t/?/„tJ, <XY> - (1 +SnR/„)z' + 2kR A»T'; with R " (Re)iriR/Ui (R radii of Browtiian particle), <XX> = <x(t)x(t)>/DA, D= 7K«T/m is the intensity of Lmgtvia force /„./»,/„ are constants:

a) Lo= 1cm. Re » 300;

/„ - 2.23. /„ » -0.65. !„ = I.5S. /« =■ 1.86

b) Lo= Icm. Re = IOO;

1.76. ¡„ =-0.57. I„ = 1.53. /„ = ¡.68 fbr the compare with our results let us display here the general accepted results, from the paper 14: <XX>„ = 2T + 2/3 T\ <YY>, -<Z2>, - 2t, <XY>,- Tj;

As it is seen from these equations the difference between obtainet results is very imponant.

Ia the first chapter of the fourth section the shear parameter of accretion flows onto compact object (Kerr black hole) is studied. Its behaviour for difference distance from the black hole is investigated.

Ia thi second subsection of the fourth section the general scheme for the construction of the general-relativistic model of the magnetically driven jet is suggested. The equations for basic physical varisbtes is obtained.

fa the isa section the main results of the work is outlined.

Rrfntts: The work is devoted to the investigation of the shear induced phenomena:

1. Tfee linear evolution of acoustic waves in a fluid flow with uniform mean density and uniform shear of velocity is investigated. The process of the mean flow energy extraction by the three-dimensional acoaatc waves, stimulated by the non-normal character of the linear dynamics in the shear flow, is snifyssd. The thorough examination of the dynamics of different physical variables characterising the wave evobtion is presented. The physics of gaining of the shear energy by acoustic waves is described;

2. Iscompressible fluctuation background ia smooth laminar shear flows is studied in three and two (feseacioial cases. Calcsdations are performed for parallel Couette flow in the framework of nonmodal approach of perturbations linear dynamics. This approach allows to grasp phenomena that has been overtoeked ia earty investigstieas and, thereby, puts fluctuation background in absolutely new light — cpBCisl fpectral density cf ihe fluctuation background energy is anisotropic in incompressible shear flows, ¡rfflreavOT its level by far exceeds level of thermal noise in the certain region of wave-number space. The posdhls EppearsEces of fluctuation background are described;

3. T!:e reccr.d way of energy transfer tato heat is discussed. It is shown that the ru-w, indirect way cf tassa Etow energy coaversioa to hext in the nonequilibrium system in stationary state is revealed —there is peraaisect traasfonnstioa cf mean flow energy to the spatial Fourier harmonics of vortex perturbation» ta<3 at last — to thermal energy;

4. Broivnisn particles suspended in unlimited laminar Couctte flow with linear shear is considered. The results of a study of the mean-square displacement of Brownian particles in Couette flow is presented. In contrast with earlier authors, this problem is studied using the time-dependent linearized incompressible Navier-Stokes equation in nonmodal approach. It is shown that mean-square displacement is strongly influenced by fluctuation background in shear flow (discussed in the previous sections), namely in the Couette flow,

5. Tlie shear parameter of accreting flow onto Kerr black hole is investigated;

6. The general schcme for the construction of the general-relativistic model of the magnetically driven jet is suggested.

Publications

1. G.D. Chagelishvili, G.R. Khujadze, J G. Lominadze and A.D. Rogava: "Acoustic waves in unbounded shear flows", ICTP, IC/96/50, Miramare, Trieste, 1996;

2. G.D. Chagelishvili, G.R. Kliujadze, J.G. Lominadze and A.D. Rogava, "Acoustic waves in unbounded shear flows", Phys. Fluids, 9, 1997;

3. A.D. Rogava and G.R. Kliujadze, "General - Relativistic model of magnetically driven jet". General Relativity and Gravitation, Vol. 29,3, 1997;

4. G.D. Chagelishvili and G.R. Khujadze, "Fluctuation background due to incompressible disturbances in laminar shear flows" JETP, 85, 1997;

5. G.D. Chagelishvili, G.R. Khujadze, A.D. Rogava: "Linear dynamics of acoustic waves- in unbounded shear flows", in the proceedings of the EUROMECH Colloquium 352 "Mean Flow Effectv in Acoustics" University of Keele, Department of Mathematics, Keele, UK, 9-12 July 1996;

6. G.D. Chagelishvili, O.G. Chklictiani, G.R. Khujadze, A.D. Rogava and A.G. Tevzadze, "Mutual transformation of waves m smooth shear flows", in the proceedings of the conference'. Wind-over-wave couplings: perspectives and prospects, University of Salford, UK, (held by THE INSTITUTE OF MATHEMATICS AND ITS APPLICATIONS (IMA)) 1997;

CPtfrt Lj?CTPfm

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3. S. C. Reddy, and D. S. Hermingjon, "Energy growth in viscous channel flows," J. Fluid Med). 252, 209 (1993):

4. Kato, Perturbation Theory for Linear Operators (Springer-Verla3, New York 1976);

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8. P. Goldrddi and S. Tremaine, "Tha excitation and evolution of density waves," Astrophys. J. 222, 850 (1978);

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10. G. D. Chagdishvili, R. G. Chanishvili, J. G. Lominadze, and I. N. Segal, Proceedings of tht fourth International Conference on Plasma Fhysics and Controlled Nuclear Fusion, held in Told, Japan 17-20 November 1992 (ESA SP—351, 1993);

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