DYNAMICS OF FLUCTUATIONS IN A FLUID BELOW THE ONSET OF RAYLEIGH-BÉNARD CONVECTION.pdf

8/11/2019 DYNAMICS OF FLUCTUATIONS IN A FLUID BELOW THE ONSET OF RAYLEIGH-BNARD CONVECTION.pdf

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Dynamics of fluctuations in a fluid below the onset of Rayleigh-Benard convection

Jaechul Oh,1 Jose M. Ortiz de Zarate,2 Jan V. Sengers,3 and Guenter Ahlers1

1Department of Physics and iQUEST, University of California, Santa Barbara, California 93106, USA2Departamento de Fsica Aplicada I, Facultad de Fsica, Universidad Complutense, 28040 Madrid, Spain

3Institute for Physical Science and Technology, and Departments of Chemical Engineering and Mechanical Engineering,University of Maryland, College Park, Maryland 20742, USA

(Dated: September 1, 2003)

We present experimental data and their theoretical interpretation for the decay rates of temper-ature fluctuations in a thin layer of a fluid heated from below and confined between parallel hori-zontal plates. The measurements were made with the mean temperature of the layer correspondingto the critical isochore of sulfur hexafluoride above but near the critical point where fluctuationsare exceptionally strong. They cover a wide range of temperature gradients below the onset ofRayleigh-Benard convection, and span wave numbers on both sides of the critical value for this on-set. The decay rates were determined from experimental shadowgraph images of the fluctuations atseveral camera exposure times. We present a theoretical expression for an exposure-time-dependentstructure factor which is needed for the data analysis. As the onset of convection is approached,the data reveal the critical slowing-down associated with the bifurcation. Theoretical predictionsfor the decay rates as a function of the wave number and temperature gradient are presented andcompared with the experimental data. Quantitative agreement is obtained if allowance is madefor some uncertainty in the small spacing between the plates, and when an empirical estimate isemployed for the influence of symmetric deviations from the Oberbeck-Boussinesq approximation

which are to be expected in a fluid with its density at the mean temperature located on the criticalisochore.

I. INTRODUCTION

In this paper we present experimental data for anda theoretical analysis of the decay rates of fluctuationsin a fluid layer between two horizontal plates that aremaintained at two different temperatures. The size Lof the layer in the horizontal directions is much largerthan the distance d between the plates. By now it hasbeen well established that the presence of a uniform andstationary temperature gradient T0 in the fluid layer

induces hydrodynamic fluctuations that are long rangedin space.

An important dimensionless parameter that governsthe nature of the thermal nonequilibrium fluctuations isthe Rayleigh number

R=gd4T0

DT , (1)

where g is the gravitational acceleration and where isthe isobaric thermal expansion coefficient, DT the ther-mal diffusivity, andthe kinematic viscosity of the fluid.The Rayleigh number is commonly taken to be positivewhen the fluid layer is heated from below. States in whichthe fluid layer is heated from above then correspond toR < 0. The fluid layer in the presence of a tempera-ture gradient remains in a quiescent state for all R lessthan a critical value Rc for the onset of Rayleigh-Benard(RB) convection, including all negative values ofR. Thepresent paper is concerned with thermal fluctuations insuch a fluid layer for positive values ofR below Rc.

At sufficiently large wave numbers q the inten-sity of nonequilibrium fluctuations is proportional to(T0)2 q4 both for negative and positive R [14]. For

smaller values ofqit was shown theoretically [5] and con-firmed experimentally [6] that the increase of the fluctu-ation intensity with decreasingqis quenched in the pres-ence of gravity, yielding a constant value in the limit ofsmall q. If the presence of top and bottom boundariesis taken into account, one finds that the intensity of thefluctuations vanishes as q2 [4, 7, 8] at small q. Hence,the nonequilibrium structure factor S(q) is predicted toexhibit a crossover from a q4 dependence for largerqtoa q2 dependence in the limit q 0, leading to a max-

imum at an intermediate wave number qmax which hasa value near /d. As the RB instability is approachedfrom below, linear theory predicts that S(qmax), as wellas the total fluctuation power [i.e., the integral ofS(q)]diverges, in agreement with the asymptotic predictionobtained by Zaitsev and Shliomis [9] and by Swift andHohenberg [10, 11]. The predicted increase as the RB in-stability is approached was verified quantitatively by ex-periments using shadowgraph measurements [12]. As thefluctuations become large it was predicted [10, 11] andconfirmed experimentally [13] that linear theory breaksdown and that the fluctuation amplitudes saturate dueto nonlinear interactions.

The present paper is concerned not with the inten-sity, but with the dynamics of the nonequilibrium fluc-tuations. One experimental technique to probe the timedependence of the fluctuations is provided by dynamicRayleigh light scattering. Dynamic light scattering ex-periments in fluid layers with negative Rayleigh numbershave shown the existence of two modes: a heat modewith a decay rate equal to DTq

2 associated with tem-perature fluctuations, and a viscous mode with a decayrate equal to q2 associated with transverse momentumfluctuations [3, 14]. Thus, for largeqwhere gravity and


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boundary effects are negligible the coupling between theheat mode and the viscous mode causes an enhancementof the amplitude of nonequilibrium fluctuations, but doesnot affect the decay rates, in accordance with the originalpredictions of Kirkpatricket al. [1]. However, for smallerq (corresponding to macroscopic wavelengths), gravityand boundary effects induce a coupling between the de-cay rates of the viscous and heat modes, as originally

suggested by Lekkerkerker and Boon [15]. For certainnegative values of the Rayleigh number the modes caneven become propagating [5, 1618]. On the other hand,for positiveR near the RB instability the nonequilibriumstructure factor is dominated by a very slow mode witha decay rate which vanishes as R Rc [911, 15, 19].

The phenomenon of critical slowing down of the fluctu-ations as the RB instability is approached from below hasbeen observed experimentally, but the results obtainedso far are qualitative. Using forced Rayleigh scatteringAllainet al. [20] measured the decay of imposed horizon-tal spatially periodic temperature profiles with variouswave numbers q. The decay time of these imposed de-

viations from the steady state increased as the temper-ature gradient approached the value associated with Rc.Furthermore, the decay times became larger for periodictemperature profiles with qcloser toqc. Critical slowingdown of nonequilibrium fluctuations has also been ob-served by Sawada with an acoustic method [21]. In thisexperiment, first a deterministic RB pattern was estab-lished and then the cell was turned over so as to changethe sign ofR. The experimental observations were inter-preted with an amplitude equation that did not includeany wave-number dependence. The single decay time ex-tracted from the data did indeed increase as the onsetof convection was approached, but numerical agreementwith the amplitude equation was poor. Using neutronscattering, Riste and coworkers [22, 23] observed the crit-ical slowing down of nonequilibrium fluctuations close tothe onset of convection in a liquid crystal. Quantitativeexperimental studies have been performed showing theslowing down of the dynamics of deterministic patternsas the RB instability is approached from above [24, 25].

In the present paper we present a quantitative theoret-ical and experimental study of the decay rates of fluctu-ations for positiveR but below the RB instability. Theresults of both experiment and theory reveal the criticalslowing down as the RB instability is approached frombelow. Theoretically, the decay rates can be determined,in principle, from a linear stability analysis of the de-terministic Oberbeck-Boussinesq (OB) equations againstperturbations, as has been done traditionally [15, 2630]. However, here we derive these decay rates by solvingthe linearized stochastic OB equations, obtained by sup-plementing the deterministic OB equations with randomdissipative fluxes [9, 10, 31, 32]. Because of Onsagers re-gression hypothesis, the two procedures should yield thesame result, and indeed they do. However, it is impor-tant to note that a solution of the stochastic OB equa-tions is required to obtain the correct amplitudes of the

nonequilibrium fluctuations. For instance, it turns outthat the wave number qmax corresponding to the maxi-mum enhancement of the intensity of the fluctuations asa function ofqdiffers, at least forR < Rc, from the valueofqat which the decay rate has a minimum [4].

We present in this paper the results of experimentalshadowgraph measurements [33] of the fluctuations in athin horizontal layer of sulfur hexafluoride heated from

below but with R < Rc. We show that the decay ratesof the fluctuations can be obtained from an analysis ofimages acquired at various exposure times, and comparethe experimental results with the theoretical predictions.The data as well as the theory reveal the critical slow-ing down of the fluctuations as the onset of convectionis approached. To our knowledge, previous shadowgraphexperiments were used primarily to obtain the fluctuationintensities [12, 34]. Exceptions are studies of the dynam-ics of deterministic patterns above the RB instability thatused image sequences at constant time intervals [35, 36].Measurements of the structure factor with the shadow-graph method can probe fluctuations at the small wave

numbers which reveal the influence of the finite geome-try. These wave numbers generally are not accessible tolight scattering.

We shall proceed as follows. In Sect. II we derive thedecay rates of the nonequilibrium fluctuations in a fluidlayer between two horizontal rigid boundaries. To obtainanalytic expressions we determine the dynamic structurefactor in a first-order Galerkin approximation. This sec-tion is a sequel to a previous publication [4], in whichthe static nonequilibrium structure factor was consid-ered. In Sect. III we then use the dynamic structurefactor to calculate the dependence of shadowgraph sig-nals on the exposure-time interval . In Sect. IV wedescribe the experimental conditions and procedures, in-

cluding the collection and analysis of the shadowgraphimages. Specifically, we show how the decay rate (q)of the fluctuations below the RB instability can be de-termined experimentally from shadowgraph images ob-tained with two different exposure times. A comparisonof the experimental results obtained for (q) with ourtheoretical predictions is presented in Sect. V. Finally,our conclusions are summarized in Sect. VI.

II. DYNAMIC STRUCTURE FACTOR AND

DECAY RATES OF FLUCTUATIONS

As was done previously [4, 7, 8], we determine thenonequilibrium structure factor by solving the linearizedstochastic OB equations for the temperature and veloc-ity fluctuations. It should be noted that the use of thelinearized OB equations implies that the critical slow-ing down of the fluctuations near the RB instability istreated in a mean-field approximation in which the de-cay rate will vanish at R = Rc when the wave number qequals a critical wave numberqc[9]. It is known from the-ory [10, 37] and experiment [13] that nonlinear effects will


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cause a saturation of the fluctuating amplitudes whenthey become very large. The effect of nonlinear terms onthe decay rates depends on the nature of the bifurcationin the presence of fluctuations. In the RB case the fluc-tuations change the bifurcation, which is supercritical inthe absence of fluctuations [38], to subcritical [10, 13].In that case one would expect the decay rates to remainfinite at the bifurcation; however, this issue is beyond the

scope of the present paper.For the case of a fluid layer between two plates withstress-free (i.e., slip) boundary conditions the intensityand the decay rates of the nonequilibrium fluctuationshave been obtained in a previous publication [8]. Theadvantage of stress-free boundary conditions is that theypermit an exact analytic solution of the problem. Here,instead, we consider the more realistic case of a fluidlayer between two rigid boundaries, corresponding totwo perfectly conducting walls for the temperature andwith no-slip boundary conditions for the local fluid ve-locity. While this case does not permit an exact an-alytic solution, a good approximation can be obtainedby representing the dependence of the temperature andvelocity fluctuations upon the vertical z coordinate byGalerkin polynomials [30]. Specifically, two of us havedetermined the static nonequilibrium structure factor ina first-order Galerkin approximation [4]. A first-orderGalerkin-polynomial solution appears to provide a goodapproximation below the convection threshold for thetotal intensity of the fluctuations [4, 39]. We thus ex-tend here the first-order Galerkin treatment consideredin Ref. [4], in the expectation that it will also provide agood approximation for the decay rates and the actualamplitudes of the two coupled hydrodynamic modes.

The decay rates of the hydrodynamic modes can bereadily obtained by finding the roots in of the determi-

nant of the matrix H(, q), defined by Eq. (15) in Ref. [4].There are two coupled hydrodynamic modes whose decayrates (q) are given by

(q) =DT2d2

(q2 + 10)

(q4 + 24q2 + 504)

(q2 + 10)(q2 + 12) + 1 (2)

(q4 + 24q2 + 504)

(q2 + 10)(q2 + 12) 1

2+

27Rq2

7(q2 + 12)(q2 + 10)2

,

where / is the Prandtl number, and q = qd isthe dimensionless horizontal wave number of the fluctu-ations. We note that in the previous publication [4] the

symbol q was used to emphasize that it represents themagnitude of a two-dimensional wave vector qin the hor-izontalX Y-plane. Here we drop the subscript , since inthe present paper the wave vectors will always be two-dimensional in the horizontal plane. In Eq. (2), (q)

represents the decay rate of a slower heat-like mode whichapproachesDTq

2 for large values ofq, while +(q) repre-sents the decay rate of a faster viscous mode approachingq2 for large q. The advantage of the Galerkin approx-imation is that one can specify the decay rates (q)explicitly as a function of q. We note that in the first-order Galerkin approximation considered here, we findonly two decay rates, instead of a series of decay rates

that would be obtained if higher orders were consideredin the Galerkin expansion. For studying the situation be-low the RB instability, where fluctuations decay to zero,consideration of the two primary decay rates will be ade-quate. As discussed more in detail later, for these Eq. (2)is a good approximation.

We are interested in the time correlation function forthe density fluctuations at constant pressure which isdirectly related to the time-dependent autocorrelationfunction T(q, z , t) T(q, z, t) of the temperaturefluctuations by [4]

T(q, z , t) T(q, z, t) =

F(q , z , z, t) (2)2

22(q q). (3)

Here t= |t t|and is the average fluid density. Thetime correlation functionF(q , z , z, t) is just the inversefrequency Fourier transform of the dynamic structure fac-torS(, q, z, z ) defined by Eq. (9) in Ref. [4]. Followingthe steps described in Ref. [4] we find after some longalgebraic calculations that the time correlation functionF(q, z, z , t) can be expressed as the sum of two expo-nentials:

F(q , z , z, t) =SE

dA+(q)exp[+(q) t] (4)

+ A(q)exp[(q) t]z

d

z2

d2

z

d

z 2

d2

.

Here the coefficient SE represents the intensity of thefluctuations of the fluid in a local thermodynamic equi-librium state at the average temperature T = T (seeEq. (2) in Ref. [4]). The Galerkin polynomials appearin Eq. (4) because they have been used to represent thedependence of the temperature fluctuations on the verti-cal coordinate z. We introduce the dimensionless decayrates

(q) = tv (q) (5)

wheretv d2/DT is the vertical thermal relaxation time.

The dimensionless amplitudes A(q) in Eq. (4) are thengiven by


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A(q) = 30

(q2 + 10)

2(q)

2

q4 + 24q2 + 504

q2 + 12

2

27(q4 + 24q2 + 504)q2

28(q2 + 12)2 (S0NE R)

(q) [2+(q)

2(q)]

. (6)

In Eq. (6), S0NEdenotes the strength of the enhancementof the static nonequilibrium structure factor of the fluidin the absence of any boundary conditions:

S0NE= R +(cP/T)d4

D2T(T0)

2. (7)

HerecPis the isobaric specific heat per unit mass [4]. InEq. (7), as everywhere else in the present paper, all ther-mophysical properties are to be evaluated at the averagetemperatureT = T. We note that the amplitudes A(q)depend on the Prandtl number and on the Rayleighnumber R not only explicitly in accordance with Eq. (6),

but also implicitly through expression (2) for the decayrates (q).As noted in the Introduction, the decay rates of the

fluctuations can also be obtained from an analysis of de-viations from steady state on the basis of the determinis-tic OB equations, i.e., the OB equations without randomnoise terms. Hence, Eq. (2) for (q) is implicit in thestandard calculation of the convection threshold withinthe same Galerkin approximation employed here [30].

However, to obtain the correct amplitudes A(q), it isnecessary to solve the stochastic OB equations for thefluctuating fields.

Expression (2) for the decay rates, as well as Eq. (6) for

the amplitudes, are rather complicated. They simplifyconsiderably for . In that limit the decay rate ofthe slower mode (q) reduces to

(q) (q2 + 10)

27q2R

28(q4 + 24q2 + 504), (8)

while the decay rate +(q) of the faster mode becomesproportional toand is so large that the first exponentialterm in Eq. (4) can be neglected. The amplitude A(q)of the remaining exponential term reduces, in the same limit, to a simpler expression to be used later, inEq. (16). It is interesting to note that the limit for largePrandtl numbers is approached rather fast. For instance,at = 15, the difference between the actual (q) givenby Eq. (2) and the value deduced from the asymptoticexpression (8) is always smaller than 3% for any value ofq, with the larger deviations of about 3% at qvalues closeto qc. From Eq. (8) we find that (q) approaches 10when q 0, independently of the values of the Rayleighor Prandtl number. The value 10 obtained from our first-order Galerkin approximation has to be compared with2 9.87, obtained from the exact theory for the limitingvalue at q 0 of this decay rate [30].

The analysis in the previous paper [4] was specifi-cally devoted to the static structure factor, S(q , z , z),which may be obtained by setting t = 0 in Eq. (4)or, equivalently, by integrating the dynamic structurefactor S(, q, z, z ) over the entire range of frequencies[4]. However, as discussed in more detail in Ref. [4], thestructure factorS(q) that is actually measured in small-angle light-scattering or in zero-collecting-time shadow-graph experiments is the one obtained after integrationof the full static structure factor over z and z [18], sothat:

S(q) =1

d d

0

dz d

0

dzF(q , z , z, 0) (9)

=SE36

A+(q) + A(q)

=SE

5

6+ S0NE

R0(q)

,

where R0(q) represents the normalized enhancement ofthe intensity of nonequilibrium fluctuations in the first-order Galerkin approximation, as given by Eq. (20) inRef. [4]. We note that, in accordance with Eq. (9), thestatic structure factorS(q) is expressed as the sum of anequilibrium and a nonequilibrium contribution. However,the dynamic structure factor and its equivalent, the timedependent correlation function given by Eq. (4), can nolonger be written as a sum of equilibrium and nonequi-

librium contributions. This difference between the staticand the dynamic structure factor results from the cou-pling between hydrodynamic modes due to the presenceof gravity and boundaries. For the same reason, thenonequilibrium intensity enhancement S0NEno longer ap-pears as a simple multiplicative factor in the expressionfor the dynamic structure factor. The observation thatthe nonequilibrium dynamic structure factor is no longerthe sum of an equilibrium and a nonequilibrium contri-bution already pertains to the dynamic structure factorof the bulk fluid (i.e., without considering boundaryconditions), where mixing of the modes is still caused bygravity effects [5].

III. APPLICATION TO THE DEPENDENCE OF

SHADOWGRAPH SIGNALS ON EXPOSURE

TIME

The shadowgraph method provides a powerful toolfor visualizing flow patterns in Rayleigh-Benard convec-tion [33, 36, 40]. Of special interest for the present pa-per is that the shadowgraph method can also be used to


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measure fluctuations in quiescent nonequilibrium fluidsat very small wave numbers [6, 12, 34, 40, 41].

In shadowgraph experiments an extended uniformmonochromatic light source is employed to illuminate thefluid layer. Many shadowgraph images of a plane perpen-dicular to the temperature gradient are obtained with acharge-coupled-device (CCD) detector that registers thespatial intensity distribution I(x, ) as a function of the

two-dimensional position in the imaging plane xand thenonzero exposure time used by the detector to averagephotons for a single picture. An effective shadowgraphsignal I(x, ) is then defined as:

I(x, ) = I(x, ) I0(x, )

I0(x, ) , (10)

whereI0(x, ) is a blank intensity distribution in the ab-sence of any thermally excited fluctuations. In practice,I0(x, ) is calculated as an average over many originalshadowgraph images, so that fluctuation effects cancel.

From a series of experimental shadowgraph signalsI(x, ), the experimental shadowgraph structure factor

Ss(q, ) is defined as the modulus squared of the two-dimensional Fourier transform of the shadowgraph sig-nals, averaged over all the signals in the series: Ss(q, ) |I(q, )|2. The physical meaning of the shadowgraphstructure factor Ss(q, ) in the past has been based onthe assumption that the shadowgraph images are takeninstantaneously. In the limit 0 Ss(q, ) has beenrelated to the static structure factor of the fluid S(q) bya relation of the form [4, 12, 36, 40, 41]

Ss(q, 0) = T(q) S(q). (11)

In Eq. (11), T(q) is an optical transfer function that con-tains various properties, such as the wave number of the

incident light, the temperature derivative of the refrac-tive index of the fluid, and details of the experimentaloptical arrangement. For the present work a specifica-tion ofT(q) is not needed since it will be eliminated inthe treatment of the experimental data in Section IV.The quantityS(q) in Eq. (11) equals the static structurefactor of the nonequilibrium fluid, as defined in Eq. (9).

To determine the dependence of the experimentalshadowgraph structure factor Ss(q, ) on the exposuretime we need to extend the physical optics treatmentperformed by Trainoff and Cannell [40]. We then findthat Eq. (11) is to be generalized to:

Ss(q, ) = T(q) S(q, ), (12)

where T(q) is the same optical transfer function as inEq. (11) and where S(q, ) is a new exposure-time-dependent structure factor, which is related to the au-tocorrelation function of temperature fluctuations by

(2)2

22 S(q, ) (q q) =

1

d2 (13)

0

dt

0

dt d0

dz

d0

dz T(q, z , t) T(q, z, t).

0 2 4 60.0

0.2

0.4

0.6

q

104

S(q,)/SESNE

~

~

~

0

FIG. 1: Difference S(q, )/SES0

NE (see Eq. 16 below) be-tween the theoretical structure factors atR= 1371 (Eq. (14)),and at R = 0 (Eq. (15)) as a function of q, for three differentcollecting times. The solid curve corresponds to= 0 ms, thedashed curve to = 200 ms, and the dotted curve to = 500ms. The Prandtl number is = 34 and tv= 0.74 s.

In principle, S(q, ) can be evaluated by substitutingEq. (3) with Eq. (4) for the correlation function of thedensity fluctuations into the right-hand side of Eq. (13).However, for the experimental results to be presented,it is sufficient to evaluate S(q, ) for large values of thePrandtl number , in which case the exponential contri-bution with decay rate +(q) in Eq. (4) can be neglected,as discussed before. Retaining only the contribution fromthe slower mode with amplitude A(q) in Eq. (4) andperforming the integrations, we deduce from Eq. (13)

S(q, ) =SEA(q) (q) 1 + exp[ (q)]

182 2(q), (14)

where is a dimensionless exposure time, defined by = /tv with tv the vertical relaxation time introducedin Eq. (5). In the limit 0, Eq. (14) reduces to

S(q) = SEA(q)/36, so that it equals the static struc-ture factor measured in small-angle light-scattering or inshadowgraph experiments in the zero collecting time ap-proximation, as given by Eq. (9). It is interesting to note

that, due to the nonzero collecting time, , even in ther-mal equilibrium (R= 0) the shadowgraph measurementspresent some structure. From Eq. (14), we find thatthis equilibrium structure, as a function of the dimen-sionless collecting time, , is given by

SE(q, ) = SE5

32(q2 + 10) 1 + exp[(q2 + 10)]

(q2 + 10)2 .

(15)It can be readily checked that, in the limit 0,Eq. (15) reduces to the structureless constant (5/6)SE, in


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agreement with Eq. (9). This value is 17% lower than theactual value SE, due to the use of a first-order Galerkinapproximation [4].

To gain insight into the effect of the exposure time onshadowgraph measurements, we show in Fig. 1 the dif-ference between the nonequilibrium structure factor asobtained from Eq. (14) and the equilibrium structure fac-tor (R = 0) as obtained from Eq. (15), as a function of

q, for three different collecting times. Corresponding tosome of the experimental results to be discussed below,we usedR = 1371, a vertical relaxation time tv = d2/DTof 0.74 s, and a Prandtl number equal to 34. We eval-uated the difference for = 0, = 200, and = 500ms. The structure factorsS(q, ) andSE(q, ) have been

normalized by dividing them by the productSES0NE.

From Fig. 1, we arrive at the following conclusions: (i)The main effect of a nonzero collecting time is to lowerthe height of the measuredS(q, ); this is expected sincefluctuations cancel out when larger exposure times areused. (ii) An additional effect of a nonzero collectingtime, which can be observed in Fig. 1, is a displacement

of the maximum in S(q, ) to lower q values. This effectis mainly due to the subtraction of the equilibrium struc-ture given by Eq. (15), which decreases with increasingq. (iii) Another interesting feature, we infer from Fig. 1,is that the dependence on q2 for low qis preserved, whilethe dependence on q4 for large q is destroyed for theeffective structure factor. Actually, at large q, there ex-ists a crossover from a q4 dependence to a q6 depen-dence, for nonzero values of, as will be shown below inEq. (17).

A simpler formula for S(q, ) can be obtained by in-

troducing into Eq. (14) the limiting value ofA(q) forlarge Prandtl numbers. We then obtain

S(q, )

SES0NE=

S(q, ) SE(q, )

SES0NE

= 5

3

(q) 1 + exp[(q)]

2 2(q)

27q2

28(q2 + 10)(q4 + 24q2 + 504) 27Rq2, (16)

where for (q) Eq. (8) for large should be used. Asdiscussed after Eq. (8), for Prandtl numbers near 15the asymptotic limit (16) is already closely approached.Equation (16) reduces, in the limit 0, to the expres-sions obtained in Ref. [4]. It is interesting to note that,in the limit q , Eq. (16) reduces to:

S(q, )

SES0NE

q

45

28

1

q6, (17)

showing, as mentioned above, that a nonzero collectingtime changes the asymptotic behavior from q4 to q6.Note that Eq. (17) is evidently not valid when = 0 inwhich case one needs the asymptotic expression given byEq. (30) in Ref. [4].

IV. EXPERIMENTS

A. Apparatus

To visualize the thermal fluctuations with the shadow-graph method one needs to perform the experiments witha fluid in which the thermal noise will be large [11, 42].This goal can be accomplished by selecting a fluid in the

vicinity of its critical point [12, 13, 33]. The measure-ments reported here were obtained for sulfur hexafluo-ride. The apparatus and experimental procedures havebeen described in detail elsewhere [33]. Here we describeonly those aspects which are specific to the experimentwith a fluid near its critical point.

For the details of the cell construction we refer to Fig.9 of Ref. [33]. Initially we used a diamond-machinedaluminum bottom plate which could be positioned withpiezo-electric elements. The bottom-plate thermistorswere embedded approximately 0.64 cm below its top sur-face. Even though to the naked eye this plate had a near-perfect mirror finish, the tool marks from the diamond

machining imposed a preferred direction on the fluctua-tions below the onset of convection. Thus we placed anoptically flat sapphire of thickness 0.318 cm on top of thealuminum plate. A thin silver film was evaporated on thetop surface of this sapphire to provide a mirror for theshadowgraphy.

Initially we used a sapphire of thickness 0.952 cm forthe cell top. In addition a sapphire of thickness 1.90 cmwas in the optical path of the shadowgraphy and providedthe top window of the pressure vessel. In combinationwith the very small cell thickness used in these experi-ments, it turned out that the optical anisotropy of theserandomly oriented sapphires introduced an anisotropy of

the shadowgraph images which obscured the rotationalsymmetry of the fluctuations. In order to minimize thiseffect, we replaced the pressure window by a fused-quartzwindow and used an optically flat sapphire of 0.318 cmthickness for the cell top. Under these conditions wefound that the fluctuations below the onset of convectionyielded a structure factor which was nearly invariant un-der rotation, as can be seen from Fig. 5 below.

The cell spacing was fixed by a porous paper sidewallwith an inner (outer) diameter of 2.5 (3.5) cm. Since thetop sapphire was supported along its perimeter whichhad a diameter of 10 cm (i.e., considerably larger thanthe cell wall), the force exerted on the cell top by thebottom plate and the wall caused a slight bowing of theinitially flat top. Over the entire sample diameter thisyielded a radial cell-spacing variation corresponding toabout one circular fringe when illuminated with an ex-panded parallel He-Ne laser beam. This variation of thethickness by about 0.3m assured that convection wouldstart in the cell center, rather than being nucleated in-homogeneously near the cell wall. Assuming a parabolicradial profile for the cell spacing, we estimate that thespacing was uniform to much better than 0.1% over the1.31.3 cm2 area near the cell center which was actu-


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47.0

46.5

46.0

45.5

45.0

T

[C]

0.90.80.70.6

, g/cm3

FIG. 2: The temperature-density plane near the critical pointof SF6. The dashed line indicates the coexistence curve sep-arating liquid and vapor. The vertical dotted line is the

critical isochore. The solid circle is the critical point withTc = 45.567

C, Pc = 37.545 bar, c = 0.742 g/cm3. The

solid line represents the isobar P = 38.325 bar used in ourmeasurements, and the heavy section of this line shows thetemperature-density range on the isobar which is spanned bythe sample with thickness d = 34.3 m when T = 0.22C.

ally used for the shadowgraph images. The actual sam-ple thickness was measured interferometrically [33] andfound to be 34.3m.

B. Properties of SF6 near the critical point

The thermodynamic properties of sulfur hexafluoridein the critical region can be calculated from a equationdeveloped by Wyczalkowska and Sengers [43]. For theviscosity we used a fit of data from Refs. [44] and [45]to a smooth function. This approach neglects a smallanomaly of the viscosity at the critical point. We usedfits of smooth functions to the conductivity data fromRefs. [4651].

The measurements were made at constant pressurePand at constant mean sample temperature T. The meantemperature was adjusted so that the density (T) wasthe critical density c = 742 kg m3. The imposed tem-perature difference Tcaused a density variation of thesample along an isobar. This is illustrated in Fig. 2for the conditions of the present experiments, namelyfor P = 38.325 bar and T = 46.50C. At = c,we have = 34.0, DT = 1.59 10

5 cm2/s, andtv d2/DT = 0.738 s.

46.2 46.4 46.6 46.80.0

0.5

1.0

1.5

T ( oC )

(

1/K

)

FIG. 3: The thermal expansion coefficient along the isobarP = 38.325 bars. The heavy part of the line terminated bytwo open circles indicates the temperature range spanned bythe sample with d = 34.3 m and T= Tc = 0.44

C.

C. Symmetric departures from the

Oberbeck-Boussinesq approximation

Much of the theoretical work on RBC has been done inthe OB approximation [26, 27], which assumes that thefluid properties do not vary over the imposed tempera-ture interval, except for the density where it provides thebuoyant force [28]. Non-OB effects have been consideredby a number of investigators, and most systematicallyby Busse [52]. At leading order, they break the reflection

symmetry of the system about the horizontal midplane,and at the onset of convection they yield a transcriti-cal bifurcation to a hexagonal pattern [36], instead ofthe roll pattern of pure OB convection. When the meantemperature corresponds to the critical isochore, = c,this effect is of modest size. Although several proper-ties contribute, we illustrate this by showing the isobaricthermal expansion coefficient in Fig. 3 along the isobarof Fig. 2 as an example. One can approximate as asum of two contributions, one of which is anti-symmetricand the other one symmetric about the mean tempera-ture (and thus approximately also about the horizontalmidplane) of the sample. Only the anti-symmetric partis considered in the theory [52]. Its smallness near theonset of convection is seen from the similar values ofat the top and bottom of the cell (open circles in the fig-ure). A quantitative calculation of the parameterP in-troduced by Busse [52] (see Eqs. (13) in Ref. [36]) yieldsP= 0.23. This value indicates that, within our exper-imental resolution, only rolls should be seen near onset.This is indeed the case [13]. However, the variation ofwhich is symmetric about the midplane is quite large. Itdoes not break the reflection symmetry about that planeand thus permits the existence ofrolls at onset rather


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46.3 46.4 46.5 46.6 46.70

1000

2000

3000

T ( oC )

R

FIG. 4: The Rayleigh numberR(see Eq. (1)) along the isobarP = 38.325 bar for d = 34.3 m. From top to bottom, thecurves are for T/Tc = 1.000, 0.860, 0.699, 0.430, and 0.269

with Tc= 0.44

C.

than requiring a hexagonal plan form. Behavior simi-lar to that of is found for the specific heat cP and forthe Rayleigh number. At present there is no theoreticaltreatment of these higher-order non-OB effects. Thus weproceed empirically by examining the variation ofR asa function of vertical position or local sample tempera-ture. If we neglect the temperature dependence of thethermal conductivity and assume that the local temper-ature in the fluid layer still varies linearly as a functionof z, we obtain the Rayleigh number profiles shown inFig. 5. The curve at the top is for the experimentalvalue T = Tc = 0.44C corresponding to the onsetof convection. The other curves, from top to bottom,correspond to T /Tc =0.860, 0.699, 0.430, and 0.269.For T = 0.44C one sees that the local R(c) 3190far exceeds the value Rc = 1708 for the uniform sys-tem. On the other hand, near the top and bottom of thesample the local R is well below the onset of convectionfor the homogeneous system. The data in Fig. 5 showthat, though we find rolls above threshold, non-OB ef-fects cannot be completely neglected in our experiments.We shall return to the influence of non-OB effects on thecomparison between experiment and theory in Sect. V.

D. Sample temperature

The bath temperature Tbathand the bottom-plate tem-peratureTBP were adjusted so as to hold the mean sam-ple temperature constant. This temperature was chosenso that the mean density corresponded to the criticaldensity c. Because of the small sample thickness thethermal resistance of the sample was comparable to thatof the top and bottom confining plates. This fact re-

FIG. 5: Shadowgraph signals (left column) and the mod-

uli squared of their Fourier transforms (right colum) forT = 0.189 (top row) and T = 0.378 K (bottom row).The exposure time was 500 ms.

quired a special procedure to assure that the sample wasindeed at the temperature corresponding to c. Before arun at a given fixed pressure was started, we measuredthe power of shadowgraph images of the fluctuations ata fixed imposed Text = (TBP Tbath) as a function ofthe mean temperature Text = (TBP + Tbath)/2 of the sys-tem consisting of the bottom plates, the sample, and thetop plate as determined by the bath temperature and thebottom-plate thermistor. This temperature difference is

the sum of those across the bottom aluminum plate TAl,across the boundary between the aluminum plate and thebottom sapphire Tb1, across the bottom sapphire Tsb,across the sample T, across the top sapphire Tst, andacross a boundary layer above the top sapphire in the wa-ter bath Tb2. From estimates of the thermal resistancesof these sections we find T /Text= 0.473. Thus, at theonset of convection, we measured Text,c= 0.930C anddeduced Tc= 0.44C.

E. Analysis of shadowgraph images

At each T, three shadowgraph-image sequencesIi(x, j), i = 1,...,N with N = 1024, and j = 0, 1, 2were acquired. For each sequence, a different exposuretime j was used, namely 0 = 0.500 s, 1 = 0.350 s,and 2 = 0.200 s. The time intervalt between the im-ages was typically one or two seconds, which was largeenough for the images to be nearly uncorrelated. Theimages of each sequence were averaged to provide a back-ground image I0(x, j), as discussed in Sect. III. Then,for each image of the sequence, a dimensionless shadow-


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graph signal Ii(x, j) was computed, in accordance withEq. (10). The mean (over x) value of a typical Ii(x, j)was within the range 0.01, indicating adequate stabilityof the light intensity and image-acquisition system. Next,the two-dimensional Fourier transform of each shadow-graph signal in the sequence was computed, and the mod-ulus square calculated, obtaining series of|Ii(q, j)|

2 forfurther analysis. Typical shadowgraph signals and the

squares of the moduli of their Fourier transforms areshown in Fig. 5. The nearly-uniform angular distributionof the transforms illustrates the rotational invariance ofthe Rayleigh-Benard system.

As explained in Sect. III, due to the rotational in-variance of RB convection in the horizontal plane, themodulus squared Fourier transformed shadowgraph sig-nals have rotational symmetry, and for an infinitely ex-tended sample they would depend only on the modulusqof the wave vector q. However, the finite spatial extentof the images leads to random angular fluctuations of|Ii(q, j)|2. To reduce these fluctuations, we performed

azimuthal averages |Ii(q, j)|2 over thin rings in Fourier

space (the angular average is denoted by the overline anddepends only on q and j). Now for each shadowgraphmeasurementIi(x, j) the integral

Pi(j) =

0

2q |Ii(q, j)|2 dq (18)

is the total power and, by Parsevals theorem, has tobe equal to the variance of the original Ii(x, j). Weused Eq. (18) as a check of consistency for the entireprocedure of taking the Fourier transform, of calculatingthe modulus squared and the azimuthal average over thinrings, and of assigning to each ring a q value. Finally,we averaged over the Nindividual |Ii(q, j)|2 of eachj

series, to obtain the experimental shadowgraph structurefactorSs(q, j) |Ii(q, j)|2.For the remainder of this paper we shall use the exper-

imentally determined cell spacing d = 34.3m to scaleq according to q = qd. Two examples of the product2qSs(q, j), both for P = 38.325 bar, T = 46.5C,and T = 0.189C, are shown in Fig. 6. The solidsquares are for2 = 0.200 s, whereas the solid circles arefor 0 = 0.500 s. As discussed in the previous section,the relationship (12) between the experimental shadow-graph structure factor Ss(q, j) and the correspondingfluid structure factorS(q, j) involves the optical transferfunction T(q). Nevertheless, we observe some qualitativeagreement of the experimental results for 2qSs(q, ) inFig. 6 and the theoretical results for S(q, ) shown inFig. 1. The expected maximum near q= qc is present,and the decrease as qvanishes is consistent with the pre-dicted q2 dependence. The longer averaging of the ran-dom fluctuations diminishes 2q Ss(q, ), which was oneof the features noted after Fig. 1 for S(q, ).

The experimental structure factor Ss(q, j , T) de-pends on the Rayleigh number and, hence, on T. ForT= 0, instrumental (mostly camera) noise is expectedto be the dominant contribution, i.e., the contribution

0 2 4 60

1

2

3

4

q

104

2q

Ss

(q,

)

~

~

~

FIG. 6: Experimental shadowgraph structure factor2qSs(q, ) as a function of the dimensionless wave num-ber q. These results are for P = 38.325 bar, T = 46.5C,and T = 0.189C. The solid squares (solid circles) are for = 0.200 s ( = 0.500 s). The open symbols are the corre-sponding background measurements for T = 0.

from the equilibrium fluctuations is negligible and thetheoretical equilibrium structure, given by Eq. (15), isexpected to be unobservable in our experiments. Exam-ples of the shadowgraph structure factors for T = 0are shown by the open symbols in Fig. 6. One sees that2qSs(q, j , 0) is well represented by a straight line, cor-responding to white noise. At large q, 2qSs(q, j, 0)merges smoothly into the data for the 2qSs(q, j , T)with the same j , as one would expect.

A detailed comparison between the experimental andthe theoretical structure factors requires knowledge of theoptical transfer function, T(q), which depends on the de-tails of the experimental optical arrangement. It involves,for instance, the size of the pinhole and the focal length ofthe lens used to make the parallel beam, and the spec-tral width of the light source.[40] In the present workthe spatial structures to be determined (the fluctuationwavelengths) had length scales [typically O(50m)] thatare one or two orders of magnitude smaller than thoseof more conventional RB experiments. For this reason,we found it difficult to obtain T(q) for our instrumentwith sufficient accuracy to avoid significant distortionof S(q). We circumvented the difficulty of the opticaltransfer function by deriving the dynamicproperties ofthe fluctuations from ratios ofSs(q, j) with different val-ues of j . To account for the instrumental white noise,before taking such ratios, we subtracted the measuredSs(q, j , 0) in the absence of a temperature gradient from


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10

0 2 4 6

0.2

0.3

0.4

q

R

(q,j,

0,

T

)

~

~

FIG. 7: The ratio R(q, j , 0,T) (see Eq. (20)) based onj = 0.200 s and 0 = 0.500 s. From top to bottom the dataare for T = 0.118, 0.189, 0.307 and 0.378C. The onset ofconvection occurred at Tc = 0.440

C.

Ss(q, j , T) to yield

Ss(q, j , T) = Ss(q, j , T) Ss(q, j , 0). (19)

After this background subtraction, we formed for j = 1, 2the ratio

R(q, j , 0, T) 2j Ss(q, j , T)

20 Ss(q, 0, T) . (20)

where 0 is the longest exposure time 0 = 0.500 s. Theratios R obtained for j = 0.200 s as a function of qare shown in Fig. 7 for four values of the temperaturedifference T. The shadowgraph transfer function T(q)cancels and is no longer contained in R. In addition,for the ratios R, it is irrelevant whether we use the def-inition of the structure factor considered in Sect. II, orthe shadowgraph definition displayed in Fig. 6, the lat-ter including a factor 2q. Thus, we are allowed to useEq. (14) for Ss, so that

R(q, j , 0, T) = j(q) 1 + exp(j(q))

0(q) 1 + exp(0(q)) . (21)

One sees that R depends only on0, j ,q, and (q, T).At each qand T, only the decay rate is unknown andthus can be determined from the experimental value ofR.

V. EXPERIMENTAL RESULTS ANDCOMPARISON WITH THEORY

In Fig. 8 we show the decay rate as a function ofq. The symbols represent the experimental values de-

0 2 4 61

10

q

(

1/

s

)

~

FIG. 8: Results for (q, T), in units of inverse seconds, ob-tained from the values ofR shown in Fig. 7 (see Eq. (21)).The symbols are the same as in Fig. 7. The top (dash-double-dotted) curve is the prediction for T = 0. The remainingcurves are the theoretical predictions (see Eq. (8)) for thevalues of Tof the data (see Fig. 7).

duced from the data for R displayed in Fig. 7 by solvingEq. (21). From top to bottom, the data sets are forT /Tc = 0.269, 0.430, 0.698, and 0.861. The curvesrepresent the theoretical values calculated from Eq. (8)with tv = 0.738 s (the value derived from the fluid proper-ties at the mean temperature and at the critical density).The topmost (dash-double-dotted) line is for equilibrium:T= 0 (i.e., R= 0). The remaining four curves are forthe values of T /Tc of the data sets, if we adopt theBoussinesq estimate

ROB = (T /Tc)Rc (22)

with Rc = 1730 (the value obtained from the Galerkinapproximation used in this paper [30]) for the Rayleighnumber in Eq. (2). One sees that the predictions, basedon the Boussinesq approximation, do not agree very wellwith the experiment. However, both theory and exper-iment reveal clearly a minimum of near q 3 whichbecomes more pronounced as T approaches Tc. Forthe larger values of T /Tc the experimental values of have a maximum near q= 5. We expect that this isdue to nonlinear effects in the physical system which leadto second-harmonic generation. This phenomenon is notcontained in the theory.

We believe that the major disagreement between themeasurements and the calculation is due to symmetricnon-OB effects discussed in Sect. IV C. Since there isno quantitative theory, we proceeded empirically and ex-plored the possibility that non-OB effects can be accom-


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0 2 4 6

1

10

fq q

tv

~

FIG. 9: Results for tv as a function of fq q, using the val-ues tv = 0.551 s and fq = 0.944 from the least-squares fitdescribed in the text. The data and symbols correspond tothose in Fig. 8. The curves indicate the corresponding theo-retical results obtained from Eq. (8) by using the scale factorsof the Rayleigh number fR,k from the least-squares fit.

modated to a large extent by multiplying the experi-mental T /Tc used to estimate R by an adjustablescale factor. We introduced an adjustable parameterfR,kwhich was allowed to be different for each Tk and used

Rk= fR,kROB (23)

for the Rayleigh number. In addition, we introduced asingle adjustable scale factor for all data sets which ad-

justed the length scale of the experiment so as to yield acorrected wave number

qcorr = fqq. (24)

to be used in the fit of the theory to the data. We expectfq to compensate for experimental errors in the cell spac-ing and in the spacing between the pixels of the images,to be within a few percent of unity, and to be the same forthe runs at all T. Finally, we treated the vertical relax-ation time tv as an adjustable parameter. We note thattv = d2/DT depends on the cell spacing, and thus anyerror in the length scale will lead to an error in the timescaletv. One also might expecttv to depend on T be-cause quadratic non-OB effects would be larger at largerT; but it turned out that a single value for tv for theruns at all Twas sufficient to describe all the data. Wecarried out a simultaneous least-squares fit of Eq. (21) toa group of 11 data sets of , like those in Fig. 8, based

0.0 0.2 0.4 0.6 0.8 1.01.0

1.5

2.0

2.5

T / Tc

fR

FIG. 10: Values of the fit parameterfR,k obtained from theleast-squares fit described in the text. Solid symbols corre-spond to based on exposure times 0 = 0.500 s and j =0.200 s. For the open symbols j was 0.350 s. The solid curveis the polynomialfR = 2.742.87T/Tc+ 1.13(T/Tc)

2

which passes through the point (1,1). The dashed curve rep-resents R/Rc and was calculated from values ofR likethose shown in Fig. 5.

on 0 = 0.500 s and j = 0.200 s. Each data set was fora different Tk, and collectively they spanned the range0.27 Tk/Tc 0.86. A separate fit was done to thesecond group of 9 available data sets based on 0 = 0.500

s and j = 0.350 s. For each group we simultaneouslyadjustedfq, tv, and the eleven or nine fR,k. We obtainedthe same result fq = 0.944 0.010 from both groups.The fit also gave tv = 0.551 0.02 (0.565 0.029) s forthe group based on j = 0.200 (0.350) s. Qualitativelyconsistent with the expected influence of the symmetricnon-OB effects, the fitted value oftv is somewhat smallerthan the value 0.738 s estimated for = c and the ex-perimental value d = 34.3 m. As said above, part ofthis difference is attributable to the error in d indicatedby the result obtained for fq. In Fig. 9 we show the

results for the product = tv for the four examplesdisplayed in Fig. 8 as a function offqq, together with the

corresponding predictions generated by using the valuesoffR,kfrom the fit. Except for the second-harmonic con-tribution at large qand T, the adjusted theory agreesquite well with the data.

The values obtained forfR,kare given in Fig. 10. Thetwo groups (j = 0.200 s and j = 0.350 s) agree verywell with each other, showing that consistent results areobtained with different exposure times. Also shown is afit to the data sets with j = 0.200 s in which the indi-vidualfR,k were replaced by a quadratic function which


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was forced to pass through fR,k= 1 at T = Tc. Thisfit gave

fR(T /Tc) = 2.74 2.87T

Tc+ 1.13

T

Tc

2. (25)

This two-parameter representation of fR fits the dataequally well.

One sees thatfRis largest at the smallest T, indicat-ing that the estimate R = ROB (see Eq. (22)) becomesworse as T decreases. This is to be expected becausethe approximation for fR must approach unity as Tapproaches Tc in order for (qc, Tc) to vanish. As asimplest empirical attempt to include non-OB effects inthe comparison between experiment and theory, we de-fine, in analogy to Eq. (22), a non-Boussinesq Rayleighnumber

RNOB(T) = R()

R()cRc, (26)

where as beforeRc = 1730 and where the angular bracket

indicates an average over the spanned temperature (andthus density) range along the isobar. The averagedcritical Rayleigh number R()c is equal to R() forT = Tc. It turns out that R()c = 1120 for ourexperiment. This approximation corresponds to a re-defined

fR(T) =Tc

T

R()

R()c. (27)

We note that by definition fR(Tc) = 1 and thusR = 1730 as it should be. Equation (27) is plotted inFig. (10) as a dashed line. One sees that this simplestnon-OB model accounts very well for the experimental

data offR. Of course, it would be very helpful to havea proper theory (rather than an empirical model) of thisinteresting effect.

VI. SUMMARY

In this paper we have reported on experimental andtheoretical studies of the dynamics of thermal fluctua-

tions below the onset of Rayleigh-Benard convection ina thin horizontal fluid layer bounded by two rigid wallsand heated from below. Starting from the fluctuatinglinearized Boussinesq equations, we derived theoreticalexpressions for the dynamic structure factor and the de-cay rates and amplitudes of the hydrodynamic modesthat characterize the dynamics of the fluctuations. Thedynamic structure factor is dominated by a slow mode

with a decay rate that vanishes as the Rayleigh numberR becomes equal to its critical value Rc for the onset ofconvection.

We used the shadowgraph method to determine theratioR of shadowgraph structure factors obtained withdifferent camera exposure times. From the theoreticalresults for the dynamic structure factor, we derived arelationship between this ratio and the decay rates of thefluctuations. Using this result and the experimental val-ues ofR, we obtained experimental decay-rate data fora wide range of temperature gradients below the onset ofRayleigh-Benard convection in sulfur hexafluoride nearits critical point. Quantitative agreement between the

experimental decay rates and the theoretical predictioncould be obtained when allowance was made for someexperimental uncertainty in the small spacing betweenthe plates and an empirical estimate was employed forsymmetric deviations from the Oberbeck-Boussinesqapproximation which are expected in a fluid with itsmean density on the critical isochore.

VII. ACKNOWLEDGEMENTS

The research of J. Oh and G. Ahlers was supported byU.S. National Science Foundation Grant DMR02-43336.G. Ahlers and J. M. Ortiz de Zarate acknowledge sup-port through a Grant under the Del Amo Joint Programof the University of California and the Universidad Com-plutense de Madrid. The research at the University ofMaryland was supported by the Chemical Sciences, Geo-sciences and Biosciences Division of the Office of BasicEnergy of the U.S. Department of Energy under GrantNo. DE-FG-02-95ER14509.

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