Boon, J.P.

Lekkerkerker, H.N.W.

Utrecht University Repository

PRETRANSITIONAL EFFECTS IN THE RAYLEIGH-BENARD INSTABILITYH.N.W. LekkerkerkerFaculteit van de WetenschappenVrije Universiteit Brussel1050 Brussel, BelgiumJean-Pierre BoonFaculté des SciencesUniversité Libre de Bruxelles1050 Bruxelles, BelgiumCONTENTSI. IntroductionII. Hydrodynamic fluctuations near the instability pointIII. Analogies with pretransitional effects in second-order phasetransitionsIV. Experimental possibilities of observationI. INTRODUCTIONRecently there has been a growing interest in pretransitional2effects, in hydrodynamic regime transitions . Particular emphasisis put on the interesting resemblance that these pretransitionaleffects bear to those occurring in equilibrium phase transitions.One of the most striking analogies is the amplification of thermalfluctuations near the transition point.205206 H.N.W. LEKKERKERKER AND J.P. BOONThe simplest example of a hydrodynamic instability is theRayleigh-Bénard or convective instability which occurs in a hori-zontal fluid layer heated from below. When the temperature gradientreaches a certain critical value stationary convection sets onspontaneously. The fundamental physical process which causes thisinstability is the conversion of the energy released by thebuoyancy force into the kinetic energy of the convective motion.Stationary convection sets on when the rate of energy transferfrom the gravitational field to the convective motion balances therate of viscous dissipation of energy of the convective motion.Following essentially the stochastic treatment of Zaitsev andShliomis we first analyze the hydrodynamic fluctuations in the pre-transitional state. Next we discuss the analogies between pretran-sitional effects in second-order phase transitions and in hydrody-namic regime transitions. Finally we suggest the use of lightscattering and Brownian motion to observe experimentally the anoma-lous fluctuations near the instability point.II. HYDRODYNAMIC FLUCTUATIONS NEAR THE INSTABILITY POINT4The basic equations describing the convective instability are<2 = /3-et ' (2.2)Here W is the vertical component of the velocity, & is theperturbation in the temperature, — ß is the (vertical) tempera-ture gradient and CX , ^C and 'O are the thermal expansionPRETRANSITIONAL EFFECTS IN RAYLEIGH-BENARD INSTABILITY 207coefficient , thermal diffusivity and kinematic viscosity, respecti-vely. For the sake of simplicity we consider a fluid layer withfree boundaries, in which case W and Q can be written asn-i,.-»•-*. . ƒ , , . (2.3)L %.Y ) sin (nrLz/oL) .Here 1C - (^ x > ^ u ) is the horizontal wavevector, X — fXjU)is the horizontal position vector and d is the thickness of thefluid layer. Substituting Eq. (2.3) in Eqs. (2.1) and (2.2) oneobtains the following equations for the Fourier amplitudeso - » . , £ _» /£L wfn/JCjt) = - ;Jk w (n, K. t) + <x^ ^,, , > ,,,,, ^ (2.5)-*- Ffo)(n,£,t) ,where K '- ^  + (HK/cij. in order to study the hydrodynamicfluctuations we have introduced fluctuating forces in the hydrody-namic equations. As usual we assume that these fluctuating forcesare gaussian markoffian 0 -correlated random processes and for thefluctuation intensities of these forces we use the equilibriumexpressions.208 H.N.W. LEKKERKERKER AND J.P. BOON(2.6)/ -r2' kB l *u2iizd f OpThe instability sets in as soon as one of the eigenvalues ofthe two coupled linear equations (2.4) and (2.5) goes to zero, i.e.as soon as the determinant of the coefficient matrix becomes equalto zero(2.7)Introducing the Rayleigh number -öó3^C* sOX. t it follows£ i A. — Pthat Eq. (2.7) is satisfied for R^kcL K. .The lowest value of Rfor which this is realized is KC - 2.7 n ƒ4 , obtained for n = 1and 1C - Kc = T^The two coupled equations (2.4) and (2.5) have two eigenva-• ^  \lues for each wavevector (K;f?H/c£) and correspondingly there aretwo eigenmodes. We shall only consider the eigenmodes correspon-ding to the smallest root and with wavevectors close to the criti-cal one, i.e. with n = I and % in the vicinity of /K-i. .Indeed those are the modes that develop anomalous fluctuations nearthe instability point. For these modes we can write the equation1 y/f/,*, t) = - } (/,> K} y (iji, t) + F ( *>(i, *,, t} ,(2.8)where A (ly'lb) is the root of the coupled equations (2.4) and(2.5) that goes to zero for R — K c and 'At = 7CC . The corres-PRETRANSITIONAL EFFECTS IN RAYLEIGH-BENARD INSTABILITY 209ponding mode is a linear combination of ^(/^'TCjt) andOCh^jt) . For K = RC and K - 7CC one can write5/3 (2.9)/ IL ISimilarly the intensity Ca^  of the random force Fis a linear combinatie(2 .9) it follows thatn binat on of Q and Q . From Eq .Qas for typical liquids 0<^T/Cfß ~ IOFrom (2.8) one obtains(2.10)-6( 2 . 1 1 )This result clearly shows that for R close to R cand for % close to ?C/C. , i.e. where *(!•>&•} —*• 0 >the fluctuations in ^ (/.> 'î^j £) decay very slowly(critical slowing down) and have a very large intensity("critical opalescence"). Expressing VS ( I, '70_,tJ and..A. ^0(i>Kjt) in terms of Ij, ( tj *k, 3 C ) onefinds, close to the instability point,210 H.N.W. LEKKERKERKER AND J.P. BOON( 2 . 1 2 )J A('JK) ( 2 .13 )III. ANALOGIES WITH PRETRANSITIONAL EFFECTS IN SECOND ORDER PHASETRANSITIONSIn second-order phase transitions there are two well-knownpretransitional effects : near the critical point the fluctua-tions in the order parameter become very large and they decayvery slowly. These effects can be explained from the behaviorof the suceptibility associated with the order parameter near thetransition point. Starting from the Landau expression for thefree energywhere -"J7 is the order parameter and £ = T/TC ~l one obtainsfor the K - th. spatial Fourier component of the susceptibilityPRETRANSITIONAL EFFECTS IN RAYLEIGH-BENARD INSTABILITY 211associated with 'Ma. e + b k* (3 .1)which diverges for T-+TC and K -> O . From the well-knownconnection between fluctuations and susceptibility( 3-2 )it follows that the fluctuations in the order parameter become verylarge for 7"—* Tc . Further according to irreversible thermody-rtamics the decay rate of small deviations from equilibrium can bewritten asA- i- ,X.where L is an Onsager kinetic coefficient and % is the suscepti-bility associated with the considered variable. According to theconventional theory of critical slowing down^the kinetic coefficientassociated with the order parameter behaves regularly near Tand since % becomes large near T t ^ becomes small> = L = (Zn)* L (a.6. t bk&) .(3'3)As seen in the previous section, near the convective insta-bility point, the fluctuations in the eigenmodes that becomeunstable, are very large and decay very slowly. In view of theanalogy with the order parameter behavior in a second-order phase212 H.N.W. LEKKERKERKER AND J.P. BOONtransition , we will call IL/ the order parameter for the hydrody-namic regime transition. The analogy in the pretransitionalbehavior of Tt and y becomes even more explicit upon analyzingthe behavior of A ( / } f o ) . For R just below R and for '~K/close to K>c one can write(3. A)where in this case £ = / — °l^c , and the expansion coefficientsare a.' = [# 4/(% + VJ] (fc ƒ> tl % 2 j and b ' = ^ 'Z *>/ (*. f V) •Substituting Eq. (3.4) in ( 2 . 1 1 ) and taking t = 0 one obtains<l f — i f f - •» y*v) (3.5)Notice the striking analogy between Eqs. (3.2) and (3.5) i.e.for the order parameter fluctuations in a second-order phasetransition and in a hydrodynamic regime transition, respectively.IV. EXPERIMENTAL POSSIBILITIES OF OBSERVATIONIn this section we briefly consider the possibilities toobserve experimentally the anomalous pretransitional fluctuationsnear the convective instability. In view of the great success oflight scattering studies of critical phenomena near second-orderphase transitions and because hydrodynamic instabilities aretriggered by thermal fluctuations which can be probed by lightscattering,it seems logical to f irst consider this tool for theinvestigation of pretransitional effects near hydrodynamic insta-bility points. Since this topic has been reviewed recently by oneof us ( J .P .B. ) we shall restrict ourselves to a few comments.PRETRANSITIONAL EFFECTS IN RAYLEIGH-BENARD INSTABILITY 213The fluctuations in the order parameter -TL ( / , - î b j ^ )are only strongly amplified for % close to '&c. - T~- / tf? dSince the thickness of the fluid layer in which the instabilitytakes place is typically of the order of 0.1 - 1 cm it follows thatin order to observe the large fluctuations in the order parameter-3 -4one has to use scattering angles as small as 10 -10 radian.This certainly represents a non trivial experimental problem andso far no experimental evidence for anomalous fluctuations nearthe convective instability point has been produced.Fluid fluctuations can also be probed through the Brownianmotion of suspended particles. Using Faxen's theorem it can beshown that the diffusion coefficient of a spherical Brownian parti-Qcle can be written asJ> =where IT (à) is the average of V, ( f j t) j the fluctua-d dting fluid velocity field, over the surface of the Brownian parti-cle. Using Eq. (<2./2) for the fluctuating fluid velocity field itcan be shown that the diffusion coefficient in addition to itsregular Stokes-Einstein part contains a critical part such that(4.2)i>r\crit A'«VAs can be seen from Eq. (4.2) the ratio of D to U J/•• •• 3^ i? / ^is determined by c OtjCL . Consider £L 'N/ //-*'r' and_2d ~ / mm. ; then at C = 10 , D becomes about 50%larger than its regular value. Given that the instability occursat a temperature difference of about 10°C between the lower andc- -4upper boundary, values of c down to 10 are in experimental214 H.N.W. LEKKERKERKER AND J.P. BOONreach without great difficulty. Photon correlation spectroscopyis an appropriate method to probe the dynamics of Brownian motionand by this method, diffusion coefficients can be determined veryaccurately. Consequently there should be no problem to observe theincrease in the diffusion coefficient. With sufficient experimen-tal accuracy it might even be possible to extract the value of thecritical exponent.REFERENCES AND FOOTNOTES1. Chercheur Qualifié au Fonds National de la Recherche Scientifi-que (F.N.R.S.) Belgium.2. R. Graham, Macroscopic Theory of Fluctuations and Instabilitiesin Optics and Hydrodynamics, and references contained therein,in this volume.3. V.M. Zaitsev and M.I. Shliomis, Zh. Eksp. Teor. Fiz. 59, 1583(1970), (Sov. Phys. JETP _32, 866 (1971)).4. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability,(Clarendon, Oxford, 1961) Eqs. (74) and (76), p. 21.5. J.P. Boon, Light Scattering from Non-Equilibrium Fluid Systems,in Adv. of Chem. Phys., Vol. 30, I. Prigogine and S.A. Rice,eds., (Interscience, New York, 1975).6. It should be mentioned that in certain electrohydrodynamicalinstabilities in nematic liquid crystals the wavelength of theunstable mode is relatively short and an increase in theintensity of the scattered near the instability has recentlybeen observed ; I.W. Smith, Y. Galerne, S.T. Lagerwall, E. Dubois-Violette, and G. Durand, preprint entitled "Dynamics of Electro-hydrodynamic Instabilities in Nematic Liquid Crystals" (1974).7. R. Zwanzig, J. Res. Nat. Bur. Stand. 68B, 143 (1964).8. H.N.W. Lekkerkerker, preprint entitled "Brownian Motion andHydrodynamic Fluctuations near the Convective Instability Point"(submitted to Physica).

Pretransitional effects in the Rayleigh-Bénard instability

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Pretransitional effects in the Rayleigh-Bénard instability

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