A spectral analysis is presented of the fluctuations in a horizontal fluid layer subject to a downward directed temperature gradient, which, for a critical value, drives the system in a convective instability state. It is found that the external
force resulting from the combination of the temperature gradient and the gravitation force gives rise to a coupling between the heat diffusion mode and a shear mode. As a result of this mode coupling the damping constant of the heat diffusion mode goes to zero when the temperature gradient increases towards its critical value, i.e. the heat diffusion mode behaves like a "soft mode". The implications of the mode coupling and of the ensuing softening of the heat diffusion mode on the light scattering spectrum are discussed

Boon, J.P.

Lekkerkerker, H.N.W.

A spectral analysis is presented of the fluctuations in a horizontal fluid layer subject to a downward directed temperature gradient, which, for a critical value, drives the system in a convective instability state. It is found that the external

force resulting from the combination of the temperature gradient and the gravitation force gives rise to a coupling between the heat diffusion mode and a shear mode. As a result of this mode coupling the damping constant of the heat diffusion mode goes to zero when the temperature gradient increases towards its critical value, i.e. the heat diffusion mode behaves like a "soft mode". The implications of the mode coupling and of the ensuing softening of the heat diffusion mode on the light scattering spectrum are discussed

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Analysis of the approach to the convection instability point

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ANALYSIS OP THE APPROACH TO THE CONVECTIVE INSTABILITY POINTJ.P. Boon (l) and H.N.W. LekkerkerkerFaculté des Sciences, Université Libre Bruxelles,1050 Brussels and Fakulteit van de Wetenschappen,Vrije Universiteit Brussel, 1050 Brussels, BelgiumABSTRACTA spectral analysis is presented of the fluctuations in ahorizontal fluid layer subject to a downward directed temperaturegradient, which, for a critical value, drives the system in aconvective instability state. It is found that the externalforce resulting from the combination of the temperature gradientand the gravitation force gives rise to a coupling between theheat diffusion mode and a shear mode. As a result of this modecoupling the damping constant of the heat diffusion mode goes tozero when the temperature gradient increases towards its criticalvalue, i.e. the heat diffusion mode behaves like a "soft mode".The implications of the mode coupling and of the ensuing softeningof the heat diffusion mode on the light scattering spectrum arediscussed.INTRODUCTIONIn a horizontal fluid layer heated from below a stationaryconvection mode appears spontaneously for a critical value of thedownward directed temperature gradient. This is one of thesimplest examples of a hydrodynamic instability, (well known inclassical physics as the Benard problem (2)).One of the prime objects of investigations in hydrodynamicstability has been the determination of the values of the stateparameters at which the transition from the stability regime tothe instability domain occurs. So far little attention has beengiven to the process of initiation of non-equilibrium instabil!-398ties (see however references 3 and 4). The reason for this maybe the difficulty to observe pretransitional phenomena in classi-cal hydrodynamic experiments. However, it appears that modernexperimental methods allow the investigation of the pretransi-tional states. Indeed, since instabilities are triggered bythermal fluctuations, light scattering spectroscopy which probesthese fluctuations, appears as anappropriate tool to investigatethe pretransitional fluctuations (5,6).The Benard problem, because of its simplicity, was chosenhere to investigate pretransitional phenomena. Using hydrodynamicfluctuation theory, we show how the spectral features of the lightscattering spectrum are modified when the fluid departs from equi-librium towards the instability critical point. The spectralchanges are found to be most important for the central line. Anadditional Lorentzian component arises because the entropy fluctu-ations, which in an equilibrium one-component fluid decay in asingle heat diffusion mode, decay into two non-propagating modesin the presence of the external force. When the system approachesthe instability critical point the damping of one of these modesgoes to zero indicating a "soft mode" type of behavior whichsuggests an analogy between hydrodynamic instability phenomenaand structural phase transitions (7).SPECTRAL ANALYSIS OF THE FLUCTUATIONSAs is well known, the spectral intensity distribution of thepolarized component of scattered light is proportional to Se(k,uu)the spectral density of the kth spatial Fourier component of thefluctuation 6e in the optical dielectric constant. Here k isthe change in wave vector and uo the change in frequency uponscattering. The dielectric constant can be considered as afunction of the thermodynamic state of the system, so that oecan be expressed in terms of the fluctuations a^_ in the thermo-dynamic state variables A^. Consequentlyse (k.»)-s (f )(f )sa>a- (k,.). (i)i > J J i JThe spectral densities of the fluctuations in the thermodynamicstate variables occuring on the right hand side of equation (1)will be calculated here by assuming that also in non-equilibriumsystems do the fluctuations on the average decay according to theappropriate hydrodynamic equations, i.e. the linearized equationsfor the fluctuations in the steady state variables. In the Benardproblem the steady state can be described as follows : an adverselinear temperature gradient is maintained steadily; there is noconvective motion; and the gravitational force is balanced by399the pressure gradient. Labeling the steady state variables withthe superscript s one hasTS(z) v3 = O (z) = -i (2)where the subscript o denotes the value of a variable at thereference position, which for the sake of convenience is taken atthe lower boundary (see figure 1). For the purpose of obtainingthe linearized equations for=the fluctuations in the steady statez]variables the equation for pSand p~(z) needonot be solvedexplicitly as long as |p°(z) - p | « P and |p (z) - PQ| « pQwhich conditions are usually satisfied for normal fluids and foractual experimental conditions (layer tickness ~ 0.1 - 1 cm).In the present case, we choose to write the thermodynamicequations for the following set of variables:53 = s - s op = p - p ; = ô v +x x o vy y v ; (^ v) .& •'•>The above description of the velocity field is a consequence ofthe symmetry of the system under consideration. It can be shownthat cp and vz have the same transformation properties as 6sand 6p under the symmetry operations of the system, whereas(Vxv) does not and will therefore not couple to 6s and 6p."" ZTI9P' ATFigure 1. Representation of the geometry of a fluid layer subjectto an adverse temperature gradient.400One then obtains the following set of coupled linearized equations(ot - 0^ )68 - (a/p)DTV26p - (Cp/T)ßvz = 0,( à - (y-l)D V 2 )6P - (p/a)(y-l)D V2 6Su i i( a _ ( v f v ' ) V 2 + v 'o 2 )cp - v ' ( v 2 - o 2 ) ö v+ p-1(v3-ôpôP = o,(ö t -vV 2 -v l £P)v z - v'ôzcp + (gXg + p"1ô z)ôP- (s*T/Cp)6s = O. (4)Here is ot the thermal expansitivity, Xs "the adiabatic compressi-bility, y the ratio of the specific heats at constant pressure(Cp) and at constant volume (Cv), Dip the thermal diffusivity, vthe kinematic viscosity and v' = ( § + V3)/P where T) is theshear viscosity and § the bulk viscosity.As the system is supposed to be infinite in the x and ydirections (i.e. the actual geometry of the system is such thatthe thickness of the fluid layer is much smaller than its dimen-sions in the x,y plane) a spatial Fourier transformation can beperformed in these coordinates. For the sake of mathematicalsimplicity we consider hypothetical boundary conditions at thelower and upper plates such that the variables 6s, 6P, 9 andvz can also be Fourier transformed in the z direction. Furtherthe set of equations (4) is Laplace transformed in the timevariable. Considering the terms due to the presence of thetemperature gradient and of the gravitation force as small, onecan treat the dispersion equation of the Fourier-Laplace transfor-med coupled equations (4) by perturbation theory. To zeroth orderone retrieves the usual modes for the system at equilibrium.From the higher order terms it is found that the non-propagatingmodes are weakly affected by it. Therefore the latter can safelybe neglected here and one then obtains for the damping factors ofthe non-propagating modess, = - f- (vfDj ± f- [(v-Dj3 + 4vDm ^ VT F, (5)i — P *"*'m/ O L x T T Rwith401k2 - k2Here R is the Rayleigh number and Rc(k) its critical value for themode with wave vector k. Indeed, note that R/Rc(k) depends stronglyon the wavelength of the fluctuation under consideration. Thisfact has important consequences for probing pretransitional pheno-mena with light scattering spectroscopy and will be discussed inthe next section. Por most liquids v > Drp and it is easily seenfrom equation (5) that in that case the external force induces acoupling such that the heat diffusion mode (S ) behaves like a"soft mode" simultaneously with the broadening of the shear mode(S ). When considering only the non-propagating modes, the setof equations (4) reduces to two coupled equations. These caneasily be solved for 6s(k,s)(the Fourier-Laplace transform of 6s),from which the appropriate correlation function is constructed toyield the spectral distribution of scattered light. One obtains...central / oe \ 2 /. / , \ i ? \ / '+' \ -\ . .I œ ( < ös(k) >(A- + A- , (7)'with(vk2 - s J2A+ = 1 - A =(vk2 - s+)2 + vDTk4R/Rc(k)One thus observes that the central peak for a system subject to anadverse temperature gradient consists of two Lorentzians (Notethat in the limit R -• 0, i.e. ß -• 0 one retrieves the single heatdiffusion line as expected for a fluid at equilibrium). Thespectrum as given by equation (7) is illustrated in figure 2.DISCUSSIONThe most interesting result is undoubtedly the fact that thedamping factor of one of the non-propagating modes goes to zerowhen the instability critical point is approached. Prom theexpression of R/R (k) given by equation (6), it is clear thatfor a given temperature gradient the effect of the external forcewill be most important for small values of k. This means that themost dramatic spectral changes are to be observed at smallscattering angles. Indeed, the wavelength of the mode which isthe first to become unstable is of the order of the verticaldimension of the system.402Heat Diffusion ModeShear ModeFigure 2. Representation of the central spectral components of afluid subject to an adverse temperature gradient. Thearrows indicate the modification of the width of thespectral components when the temperature gradientincreases towards its critical value.Presently available techniques used in light scattering spec-troscopy should allow to probe modes which are expected to beaffected in a fluid subject to an external force.ACKN OWLEDGEMENTSThe award of a NATO Science Fellowship by the NetherlandsOrganisation for the Advancement of Pure Research for the academicyear 1971-1972, when this work was initiated, is acknowledged byHNWL. This research has been sponsored in part by the Internatio-nale Instituten voor Fysica en Chemie, gesticht door E. Solvay.REFERENCES AND FOOTNOTES(1) Chercheur qualifié au Fonds National de la Recherche Scienti-fique (FNRS), Belgium.(2) S. Chandrasekhar, Hydrodynamics and Hydromagnetic Stability,(Oxford, Clarendon Press, 196l), Chapter 2.403(3) V.M. Zaitsev and M.I. Shliomis, Soviet Physics JETP £2(1971).(4) J.P. Boon, Phys. Chem. Liquids 2> 157 (1972)-(5) J.P. Boon and Ph. Deguent, Phys. Letters 39A, 315 (1972).(6) J.M. Deutch, S. Hudson, P.J. Ortoleva and J. Ross, J. Chem.Phys. 51, 4327 (1972).(7) H. Thomas, I.E.E.E. Trans. Magn. 5, 874 (1969).

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Analysis of the approach to the convection instability point

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