New hydrodynamic modes in non equilibrium fluids

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New hydrodynamic modes in non equilibrium fluids

J. Coste, J. Peyraud

To cite this version:

J. Coste, J. Peyraud. New hydrodynamic modes in non equilibrium fluids. Journal de Physique, 1975,

36 (9), pp.751-758. �10.1051/jphys:01975003609075100�. �jpa-00208311�

LE JOURNAL DE PHYSIQUE

NEW HYDRODYNAMIC MODES IN NON EQUILIBRIUM ^FLUIDS

J. COSTE and J. PEYRAUD

Laboratoire de Physique de la Matière Condensée (*),

Parc Valrose, ^{06034 Nice} Cedex, ^France

(Reçu ^{le 22 novembre} 1974, ^{révisé le} 3 janvier 1975, ^{révisé le} 19 février 1975, accepté ^{révisé le 8 avril} 1975)

Résumé.

Nous étudions l’hydrodynamique d’un fluide traversé par ^un bruit sonore intense et anisotrope. Nous obtenons qu’un ^mode hydrodynamique, ^à polarisation partiellement transverse, peut ^se propager dans ^un tel fluide, à condition que l’intensité ^et l’anisotropie ^{du bruit sonore soient}

assez fortes.

Abstract.

We study ^the hydrodynamics of a fluid supporting ^an intense and anisotropic ^acoustic

noise. We show that new hydrodynamic modes, ^with partly transverse polarisation, may exist in such a fluid provided ^{that the} intensity ^and anisotropy ^{of the acoustic noise are} sufficiently high.

Classification

Physics ^Abstracts

6.305

1. Introduction.

Let us consider the propagation

of an acoustic wave packet (or ^more generally ^of

directive acoustic noise) through ^an homogeneous

fluid. Assuming first that linearized hydrqdynamics

is available, ^and considering ^short enough propaga- tion length (i.e. ^short compared ^{to the} damping length

of the sound modes), the acoustic noise is stationary.

The fluid supporting this noise _may therefore be described by adding ^{to the usual} equilibrium ^variables

the _energy density ^{e and the} average unit director

(vector) 0 - k k of the ^II k ^{1 wave packet. Our purpose} in this _paper is to investigate ^{what is} going ^{on when}

one perturbes the above stationary ^{state. We show}

that the perturbation may be described in general by

a state vector B’ whose components ^are macroscopic

variables { e’, P’, p’, J’} (p’ ^{and J’} being ^{the mass}

and momentum density associated with the pertur- bation) ^which obey ^{a set of} coupled ^kinetic equations.

The evolution of the acoustic wave packet ^therefore

does not proceed ^{as in standard} geometrical ^acoustics

where the propagation ^takes place ^{into a} given inhomogeneous fluid : the perturbation ^{of the enve-} loppe (e, fol) modifies the medium’s parameters (p, J) ;

in tum the perturbation ^of (p, J) modifies the _pro-

pagation ^{of the wave} packet. ^Another way of looking

at this problem ^{is to} say that we study the linear

hydrodynamics ^{of a} fluid whose stationary ^{state is}

(*) Laboratoire associé au CNRS no 190.

not the thermodynamic equilibrium, but contains an

anisotropic spectrum of acoustic modes.

In such a fluid the transport properties ^are supported by large ^scale hydrodynamic fluctuations (here mainly

the non equilibrium ^acoustic modes). There results

physical effects which _may be much more important

than in generalized hydrodynamics ^{of the} equilibrium

fluid where the _energy contained in the large ^scale hydrodynamic domain is _very small [cf. ^for example (1)] (except, ^{of course, near} the critical

point).

Fourier analyzing ^{B’ we} study ^the eigenmodes ^of

the system { fluid ^{+ acoustic} modes ?. ^{A new mode}

appears unexpectedly in addition to the usual hydro- dynamic ^ones (which ^are only slightly modified). ^The

conditions to be satisfied in order to observe it are

that the intensity ^and anisotropy ^{of the} non-equili-

brium acoustic noise are sufficiently high. ^The pola-

rization of the new mode is partly transverse, depen- ding ^{on the} angle 0 between its wave vector and the average direction of propagation of the acoustic

modes, ^{and its} phase velocity ^{is close to c cos 0.}

The origin ^{of the new mode must} be found in the fact that the transport properties of the fluid are made

anisotropic by ^the presence of the directive sound excitation. We think that an interesting analogy ^is provided by ^{the case of a} plasma immersed into a

strong magnetic ^{field H. If H =} 0, ^the dispersion

relations of the longitudinal (plasma) ^{and transverse}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003609075100

752

(electromagnetic) ^{modes are} respectively ^{m =} _wp ^and

w ⁼ kc, (wp ^{and c being the} plasma frequency ^{and the} velocity ^of light), ^for kc > 0153?. In the presence of H

they ^are replaced by ^{w =} cop ^{cos 0 and w = kc cos 0.}

0 being ^the angle ^{of H with the wave vector. In this}

case, it is the conductivity ^tensor which is made

anisotropic (the particles being constrained to move

along ^the magnetic ^field lines).

Finally, ^{this new} mode is found to be unstable if the direction of its wave vector lies in a finite angular

domain around the direction of propagation ^{of the}

acoustic noise. However, ^the stability problem ^{is still}

open, because it depends ^{on the} dissipation ^{and on the}

non-linear behaviour of the acoustic excitation, ^which

are not taken into account in the present ^article.

The coupled dynamics ^{of the wave} packet’s para- meters and of the hydrodynamic ^variables (p, J) ^will

be analyzed ^{in two} successive steps, considering ^first

a quasi-monochromatic ^wave packet :

i) ^we recall the kinetic equations ^{of the wave} packet (a, P) propagating ^{in a fluid} presenting large

scale inhomogeneities (limit ^of geometrical acoustics) ii) ^we derive the hydrodynamic equations ^{of the}

fluid in the _presence of the perturbed ^wave packet.

From the above coupled linearized equations,

we derive the dispersion relations of the new hydro- dynamic mode. We then consider the general ^{case of}

a fluid excited by ^an arbitrary spectrum of acoustic

noise, ^{and we} discuss the approximations ^{of the} theory, giving ^the experimental conditions to be realized in order to observe the new mode.

2. Perturbation of the stationary propagation ^{of a}

narrow sound wave packet. ^Kinetic équations obeyed by ^{the wave} packet’s envelope.

Calling ^{i3 =} pU, the fluid’s intemal _energy and v the oscillating velocity

in the _wave, the _energy density est associated with the sound wave in the stationary ^{state is} given by

where _po and _wo are the equilibrium ^mass density ^and enthalpy per particle, Pst and bst ^{are the} average ^mass and internal _energy densities in the _presence of the

wave packet. As for the _energy flux density q ^asso- ciated with the wave packet, ^{it is} given by

C’ being ^{the adiabatic} compressibility ^{of the} equili-

brium fluid and Po ⁼ k k is ^{1 k 1 the average wave vector}

of the wave packet. ^The stationary ^{state of the} system

{ ^{fluid + wave} packet} may therefore be defined

by ^the state vector Bst ^whose components ^are

The perturbation ^{of this} stationary ^{state will be}

formally ^obtained by making ^{the above state variables}

space and time dependent, ^and by adding ^{to them the}

current density J which is associated with the time variation of the mass density. These variables have

a macroscopic character, ^{in the sense} that their time and _space length ^{scales are assumed to be} large compared ^to the inverse frequency ^{and wave number}

the sound wave packet. Physically, ^the perturbed

state B(x, t) ^{can be} produced, ^either by ^{some inner} large scale fluctuation of the fluid, ^or by any weak extemal source. Let B ⁼ BSt ⁺ B’, where, ^{in the} following, ^B’ designates the deviation from the

stationary ^{state value.}

Let us first study the evolution of the main packet’s

parameters (e, B). ^{It is} govemed by ^the equation ^of geometrical ^acoustics (this ^{is due to the above} large

scale assumption ^on the variation of the macroscopic variables) which reads

cok(x, t) is the local dispersion relation, ^which may be written as

k.J’(x, t) ^is obviously ^the Doppler shift due to the

macroscopic velocity ^{of the fluid at} point (x, t).

c’ which is the perturbation ^{of the} stationary ^sound velocity ^{is a} function of p’ ^{and e’} through ^{the local} equation ^{of state} of the fluid. In tum, p’ ^{and J’} obey hydrodynamic equations ⁱⁿ which the stress tensor

depends ^{ont and} B’. We therefore have to deal with a self consistent problem. ^{Let us} first derive the equations obeyed by ^{e’ and} B’ ^{in a} given ^non- equilibrium ^fluid [that ^{is for a given function A’(x,} t)].

Eq. (4) reads, ^to first order with respect ^{to the} per-

turbation,

(We ^{recall that} Bo is the unit vector of the _propa-

gation ^{in the} stationary system.) Now, ^{we have from}

eq. (6)

Another useful relation is a consequence of _eqs. (3,4), namely :

with the help of relations (9,10), eq. (8) may be written

as :

Finally, ^the perturbed ^{form of} energy eq. (5) ^is

3. Hydrodynamics ^{of the fluid in the} présence of the

wave packet.

p’ ^{and J’} obey the linearized conser-

vation equations :

where n’j ⁺ bij p’ is the total stress tensor. Following

the same point ^{of view as in} generalized hydrodyna-

mics [cf. ^for example (1)], ^{we assume that our fluid,}

even while supporting the acoustic noise, ^{is at} any

point ^{close to local} thermodynamic equilibrium. ^Then

’ ⁱⁿ eq. ^{q( )} (14), ^{as c’} - Cok ( k ) ^{in eq. q ) (17), are the er- p}

turbed thermodynamic ^state functions of the fluid in the presence of the acoustic noise. ^The non-diagonal part 7rj ^{of the} stress tensor is cleârly ^defined by

and has the meaning ^{of the} perturbed correlation tensor of the fluctuating ^currents in the fluid. 5J is the sum of short scale and large ^scale (hydrodynamic)

fluctuations [cf. (1)]. ^{The first ones are} responsible

for the usual viscous terms in _eq. (14) ^{and are} neglected

in the present theory (this approximation ^{will be}

discussed later). ^{It is worth} noting that, ^{in the same} approximation, ^{we have} neglected ^{in the} energy eq. (5)

the source of acoustic fluctuations in the fluid. Among

the large ^scale fluctuations, ^we find those of the

equilibrium fluid and also those which are associated with the superimposed acoustic noise. We assume that the intensity of this noise is sufficiently large ^{that the}

contribution of the acoustic modes to the stress tensor will be dominant. In the case of a narrow

sound wave packet, ^{the average current and mass}

densities fluctuations are given by

Now P and c, which depend ^{on the local} thermody-

namic state, have ^a non-linear dependence ^on p and 13.

We assume that their _average values in the perturbed

state may be obtained through ^a Taylor expansion

of the equation ^{of state} around the equilibrium

values (po, i3o). ^This expansion ^{reads :}

where A ⁼ P or c, and Ap ⁼ p’ ⁺ bp, bp being ^the fluctuating ^mass density associated with the acoustic modes. After some straightforward thermodynamic algebra (see Appendix), ^we obtain the expression ^of A(p, 13), ^whose perturbed part ^{reads :}

where c5V2)’ and c5p2)’ may be respectively expressed ^{in terms of e’} through eq. (1, 17).

Eq. (1, 6, 7, 11-17) ^and eq. (19) written for A’ = [c’, p 1 ^{form a} closed set, which allows ^{us to} describe completely ^the kinetics of the components of the perturbed ^{state B’.}

4. Dispersion ^{relation of the new} hydrodynamic

mode.

We now proceed ^to study ^the eigenmodes

of the non-stationary fluid. We are therefore led to Fourier analyse ^{B’ with} respect ^to space and time _or,

equivalently, ^{to state} in the above equations ^{that B’}

varies as ei(ktlh.x-wlt) (with k 1 « k in the frame of geometrical acoustics). ^{Let us} introduce :

and

Eliminating (Pi.P) ^between eq. (22) ^and (23) ^we obtain

t ¹ À , ^/

754

Eq. (22, ^23, 24) exhibit the pole kW1 ^{1 Co}

^{cos 0, which}

is the origin ^{of the} hydrodynamic ^{mode. The existence}

of this pole is related to the fact that the evolution of the wave packet’s energy, ^as well as those of its direction of propagation, ^are determined by ^the gradient ^{of the} hydrodynamic field in the direction of the wave packet’s group velocity. ^{Since the} perturbed

stress tensor { n’j ⁺ b ij p’} ^{also depends} linearly

ont through ^relations (16) ^and (19), it exhibits the

same resonance. a’ being itself related to J’ through

eq. (24), ^the stress tensor takes the form (Xij Jj, ^where

the non-equilibrium transport coefficients (Xij ^are

proportional ^{to E’.} Therefore, ^the _aij ^{’s are themselves}

resonant and we conclude that the existence of the new

hydrodynamic ^{mode is a} manifestation of the strong

anisotropy of the fluid’s transport coefficients in the presence of the acoustic noise.

Looking ^{for a} solution of the kinetic equations ^such

that  - cos 0 ~ 0, ^{we are led to} neglect ^{the second} term in _eq. (24) compared ^{to the first one. But this}

second term was produced by ^the perturbation ^{of the}

local phase velocity in the flux term of the _energy eq. (5). ^{It is} easily ^seen that, ^{in the same} approximation

we may replace ffl ^{by co in} expression (17) for f>p2 >’.

As a consequence the expression

entering eq. (19) ^{vanishes, and we are} left with :

Going ^{back to the} (x, t) space, and summarizing ^the

above results, the linearized hydrodynamic equations

of the non-equilibrium fluid, leading ^{to a convenient} description ^{of the new} hydrodynamic mode read :

Let us now put :

Using eq. (26c) ^and (26d) which define the kinetics of the wave packet, ^{the momentum} equation may be put in the form :

(where ^{we retain} only the dominant contribution with respect ^{to the resonant factor} (À - ^cos 0)-’). Eq. (29)

shows that the momentum density ^{J’ of the macros-} copic perturbation lies in the (Pô. B’) plane. ^{We shall}

therefore use the following geometry (Fig. 1).

r

FIG. 1.

Geometry of the interaction.

Working ^now in Fourier _space (with respect ^{to x} and t variables), eq. (26c) ^and (26d) ^{lead to the} already given eq. (24), which reads in terms of Q’ and q (and retaining the first dominant term) :

with

from which

Now, ^the x, y projections ^of eq. (29) ^{read :}

Anticipating ^{the final result, we shall} replace ^À by

cos 00. ^{except in the resonant} factors. Then _eq. (31),

(32), yield ^the polarisation ^{of the} hydrodynamic ^{mode :}

Puttine ^{(2 - cos} 0)’ ⁼ aq, eq. (30) ^{reads :}

Replacing (po A’) by ^{the above} expression ⁱⁿ eq. (32),

we obtain the dispersion ^{relation :}

or

It is important ^{to note that} eq. (35) ^{is the} dispersion

relation of the new hydrodynamic ^mode only. Indeed,

eq. (30, 31, 32) are ^{also satisfied} by Â’ ^{= 1 + 0(q)}

(ordinary sound modes with quasi longitudinal polari- zation). Eq. (35) has been obtained by picking ^out

the pole, ^{À N cos 0.}

5. The general ^{case of a fluid.} supporting ^an arbitrary spectrum ^{of acoustic modes.}

The acoustic spectrum will be defined by

The kinetic equations ^{are obtained} straightforwardly

from the equations ^{of the} single ^{mode case. We must} replace Q’ by

where

q(k) being ^the intensity ^{of mode} k, and 0 the initial

propagation’s direction of this mode. Eq. (27) ^{reads :}

,, , .

Let us consider a sound spectrum with small angular dispersion (Bk~‘ Pô)’ ^We may obviously approximate Pk by Po except ^{in the resonant factors} [2 - ^cos 0] - ^2.

We shall therefore write :

with

and retain the kinetic equations ^{of the} previous section, except ^{for the} change ^{a - 6. The} dispersion ^relation

will be obtained from _eq. (34) which reads :

Passing ^{to a} continuous description ^{of the} spectrum, let us put :

we have

where ? is a convergence ^tenn introduced by ^the causality requirement. Now, ^the angular spectrum distribution q(z) will be assumed to take a gaussian

form

Putting

the dispersion relation takes the form :

where

The form of dispersion ^relation (43) ^is quite analogous

to those of longitudinal ^{waves in} plasma physics.

The existence of a propagative ^root is determined by

the comparison off with the width à of the angular

distribution.

i) If A « A, ^{the roots} of eq. (43), if any, ^are strongly damped.

ii) ^If A > A, ^we may write

756

The imaginary ^term corresponds ^{to a small} damping (the equivalent ^{of the Landau} damping ^of plasma waves).

If A > 0, the solution reads

If A 0, ^the solution is purely imaginary ^{and we} may

ignore the Landau damping ^{term :}

therefore, there exists an unstable mode.

Concluding, ^we may say that if 4 A, ^{we recover} the dispersion ^relation (35) ^{of the} single ^{mode case} (except ^{for a} very small damping). ^{If 11} > 1, ^the angular spreading of the acoustic spectrum ^{kills the}

new hydrodynamic ^mode.

6. Discussion and comments.

Let us now give

some comments on the properties ^{of the new} hydrody-

namic mode and on the conditions needed for its existence.

6.1 A first striking feature of the dispersion

relation (35) ^{is that Re} (co) ^is nearly independent ^of

the intensity of the acoustic _pump. This intensity only plays ^{a role} together ^{with the} anisotropy ^{as a condition}

of existence of the hydrodynamic mode. This condition is that the phase mixing ^{due to the} angular spread ^of

the pump does ^not kill the resonant effect which generates ^the mode, ^{and it} reads A « A ^or

A quite similar situation occurs in plasma physics,

as was already said, concerning ^the propagation ^of longitudinal ^or transverse waves in the présence of

a strong magnetic ^{field H. New} dispersion ^relation

also appear, which ^{are H} independent, ^{but these}

modes only exist if H satisfies appropriate ^threshold

conditions.

6.2 In contradistinction to the plasma case, the

new mode must propagate in the direction of the _pump average ^{wave vector} (i.e. ^the projection ^{of the} phase velocity ^on this direction is always positive). ^Such

a feature obviously ^{does not} appear for the ^wave

propagation ⁱⁿ magnetized plasmas ^{because H does}

not modify ^the longitudinal particle’s ^motion.

6.3 The polarization ^is partly transverse. This

unexpected feature in fluid dynamics ^{is due to the}

fact that the gradient ^{of the} stress tensor

is along the pump ^{wave vector.} The polarization angle 0 is finite for finite 0 and in the limiting ^cases

0 --+ 0 and 0 --> n/2 ^{we obtain} respectively, quasi- longitudinal ^and quasi-transverse polarizations.

6.4 The modes are found unstable in a finite

angular 0 domain around 0 = 0, corresponding ^to

A 0, ^and depending ^on the values of cT/c ^and c!/c.

The growth ^{rate is then} y = 1 fjA 11/2 k’ ^{c and it is}

maximum for 0=0. On the contrary, ^{the modes} whose transverse polarization ^is important ^{are found}

stable.

However, the above results are not sufficient in order to treat the stability problem, ^{and we must first}

discuss the two approximations ^{of the} theory :

6.4.1 We have neglected ^the dissipation ^{of the}

acoustic modes. As stated above, ^this assumption

is tenable if we consider time intervals short com-

pared ^to the lifetime of the _pump modes. More- over, it is _easy to obtain the modification of the _disper-

sion relation due to the damping of the waves ys k2.

Indeed, ^{we have to} replace ^û/ by w’ - iys ^{k2 in the}

resonant factor (Â - ^cos 0) - 1, ^{and therefore in the} dispersion relation. We therefore conclude that the

instability ^criterium (for ^A 0) ^{due to} dissipation

reads :

or

6.4.2 we have neglected the non-linear evolution of the pump ^wave packet (building up of a shock

wave). The characteristic time for the apparition ^of

non-linear effects is of the order of

This time must be large compared ^{to the} frequency

k’ c cos 0 of our new mode. This condition, ^which reads

is not very stringent ^{in most} experimental situations.

Eq. (49) is, ^like eq. (48), ^a necessary condition for the existence of the new mode. The existence of unstable modes requires ^{that the} non-linear time be larger ^than

their growth ^{rate. Such a condition} implies ^that

k k’, and therefore cannot be fulfilled in the limit of geometrical ^{acoustics which we} consider here.

The conclusion concerning the existence of the insta-

bility ^can only ^{be obtained} by considering ^{the non-}

linear behaviour of the pump, which is outside the scope of the present paper.

A last comment will concem the comparison

between the present theory ^and hydrodynamic theory

of the quasi-equilibrium ^fluid [1] ] (quasi-equilibrium meaning ^{that the} unperturbed ^{state is not the} quasi- stationary ^state of the fluid supporting ^{an external}

acoustic noise, ^{but the true} equilibrium state).

i) ^The present theory proceeds along ^{a more}

concise _way, thanks to the use of geometrical optics.

It may be shown that our previous ^formalism [1] ^is equivalent ^{to the} present ^one (including the appearance of the resonant factor (Â - ^cos 9) -1 ), ^{provided that}

the sound propagators in Fourier _space are expanded

up ^to the second order with respect ^{to k’.}

ii) ^The hydrodynamic equations ^{of the} non-equili-

brium fluid contain o2cjop2 (through ^the expression

for A’), which does not appear in the previous theory.

This term enters the kinetic equation ^{of the} perturbed

correlation tensor, and yields ^{a finite} contribution to the dispersion ^{relation. It} is associated with a third order expansion of the local thermodynamic pressure, and would generate ^{in the} frame of the previous theory

a contribution to the kinetic équation ^{for the corre-}

lation tensor proportionnal ^{to the} fourth order correlation, ^{which is} neglected in standard bilinear

hydrodynamics. ^{In the} present theory ^the o2cjop2

term proves ^to be one of the dominant terms which

are proportional ^to q 2. ^{(/ 2013 cos 0) It may be reasonably}

conjectured ^that higher ^order expansion ^{of the non-}

linear local state functions would lead to higher ^order

terms with respect ^{to the small} coupling parameter ^q.

7. Conclusion.

As stated earlier, the existence of the new hydrodynamic ^{mode is a} manifestation of the anisotropy created in the fluid by ^{the sound} excitation. An interesting situation which could be

investigated ^{is the case of a} fluid in which the sound

velocity ^is anisotropic in the absence of any excitation :

a new resonant effect could occur if the dispersion

relation of the new mode (in ^the non-equilibrium fluid) already belongs ^{to the} original ^fluid (1). ^{In the}

present ^{case of an} isotropic fluid, ^some further studies

(either theoretical or expérimental) ^{are needed in}

order to analyse ^the stability problem.

Appendix.

Averaging ^the Taylor expansion (18), ^{we obtain :}

Using the relations

and dropping ^the terms bp ^bs y and ^Ôs’ y (the acoustic fluctuations are assumed nearly isentropic), ^we

obtain

with the help ^of thermodynamic ^relations

758

and expressing (i3’ - wo p J ^{in terms of E’} and ^bv2 )’ by :

we finally put A (p, i3) ^{in the form :}

References

[1] COSTE, J. and PEYRAUD, J. To be published.

[2] LANDAU, ^{L. D. and} LIFSHITZ, E. M., Mécanique ^des fluides

(Editions ^de Moscou) ^1971.