traveling wave instability in sustained double-diffusive...

Traveling wave instability in sustained double-diffusive convectionA. A. Predtechensky,') W. D. McCormick, J. B. Swift, A. G. Rossberg, andHarry L. Swinneyb)Center for Nonlinear Dynamics and the Department of Physics, The University of Texas, Austin,Texas 78712

(Received 26 May 1994; accepted 23 August 1994)

Experiments on buoyancy-driven double-diffusive convection sustained by imposed verticalconcentration gradients (one stabilizing, the other destabilizing) have been conducted in a thin(Hele-Shaw) isothermal rectangular cell. Novel gel-filled membranes were used to sustain theconcentrations at the boundaries. When the destabilizing solute diffuses more rapidly than thestabilizing one, the primary instability leads to traveling waves with a high reflection coefficient atthe ends of the cell. The measured critical Rayleigh numbers and frequencies are in reasonableaccord with a stability analysis that includes corrections for the finite thickness of the cell andcross-diffusion effects. The weakly nonlinear waves that appear at onset do not stabilize, even veryclose to the transition, but continue to evolve, eventually becoming a packet of large amplitudeplumes. The packet travels back and forth along the cell in a nearly periodic manner. This behaviorand the absence of measurable hysteresis are consistent with the present weakly nonlinear analysiswhich predicts tricritical scaling (- e14 rather than the usual el~2) all along the instability boundary.However, the range of this scaling in e was found to be less than 0.005, which is inaccessible in thepresent experiments. © 1994 American Institute of Physics.

1. INTRODUCTION

Systems with two or more quantities diffusing at differ-ent rates exhibit a variety of phenomena of relevance to ma-terials processing, to mixing in the oceans, and to stellar andearth mantel convection.' The most extensively studied ex-ample is thermohaline convection, where the two diffusingspecies are heat and salt. The basic stability diagram forbuoyancy-driven convection in a double diffusive systemwith "free-free" boundary conditions2 is shown in Fig. 1:R, ow is the Rayleigh number proportional to the gradient inthe density of the slowly diffusing species (e.g., salt), andRraa, is the Rayleigh number proportional to the gradient inthe density of the more rapidly diffusing species (e.g., heat).Figure 1 was derived for the physically unrealizable freeboundary conditions at both horizontal surfaces: the surfaceswere taken to be rigid, with tangential slip but without tan-gential stress.3 In the present paper we analyze the convec-tive instability of a double-diffusive layer with physicallyrealizable boundary conditions at both the horizontal andvertical surfaces. The analysis is conducted for the Hele-Shaw (thin cell) geometry, which is used in our experiments.The analysis includes corrections for departures from idealityarising from the effects of cross-diffusion and finite cellthickness. Our analysis shows that the oscillatory instability(Fig. 1) leads to traveling waves.

Oscillatory convection in double diffusive systems hasbeen widely discussed, but only two experiments have exam-ined the onset of instability, and most of the key questionsare still open. The first laboratory study of the oscillatoryinstability was by Shirtcliffe, who established a stable con-centration gradient of sugar in a 10 cm high cell and then

')Also at Institute of Automation and Electrometry, Novosibirsk 630090,Russia.

b)E-mail: swinney~chaos.ph.utexas.edu

heated the lower surface.4 A few oscillations were observedbefore the flow became disordered. Recently Krishnamurtiand Zhu conducted the first experiment on sustained double-diffusive convection.5 The top and bottom cell surfaces weremade of a porous membrane in contact with salt solutions ofdifferent concentrations. Stepwise increases of heating at thebottom surface led to an irregular oscillation at about 10%above the critical Rayleigh number predicted by their analy-sis, but the observed frequency was 8-10 times lower thanexpected.5

Rather than using salt (or sugar) and heat as the twodiffusing species, we consider an isothermal aqueous solu-tion with two solutes of different diffusion coefficients.6 Inthis ternary mixture a concentration coo of a fast diffusingspecies is imposed at the top of the cell, and a concentrationc50 of a slowly diffusing species is imposed at the bottom, asillustrated in Fig. 2. The gradient in c, is stabilizing, that incf is destabilizing. Like Krishnamurti and Zhu, we use po-rous membranes at the top and bottom surfaces. In our cell,however, vertical mass flux is inhibited by filling the mem-branes with gel; a gel only 0.1 mm thick is sufficient toprevent the small pressure differences which are inevitablypresent from causing any significant mass flux, yet providesa well-defined boundary condition for the concentrations atthe top and bottom horizontal surfaces of the cell. We findthat the instability of the conducting state leads to travelingconvection cells, as predicted by our analysis; see Fig. 2.

One particular advantage of an isothermal convectioncell is that the boundary condition for the side walls-impermeability-is perfectly satisfied, making it possible toapproach the ideal Hele-Shaw geometry. This cannot bedone in thermal convection, where the corresponding condi-tion would be perfectly insulating side walls. The Hele-Shaw geometry restricts the pattern formation to a plane;

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Rft oscillatorfat instead A -.

stable

FIG. 1. Instability threshold for the conducting state of a three-dimensionaldouble-diffusive system with free-free boundary conditions on the top andbottom of a horizontal layer, as a function of the Rayleigh numbers of theslow and fast diffusing species.2

hence the analysis and observations are simpler than forthree-dimensional patterns.

Another advantage of the isothermal mixture is that oneof the important parameters in the problem, the Lewis num-ber r, which is the ratio of the diffusion coefficients,D2 Df, can be varied from about 0.1 to 1 by appropriatechoice of solutes. In contrast, the Lewis number is .10- 2 indouble-diffusive convection with heat as one of the diffusingquantities.7 Since values of r approaching unity can bereached in an isothermal mixture, it is possible to study ex-perimentally the neighborhood of the codimension-2 pointwhere the onset behavior changes from traveling waves tosteady rolls (see Fig. 1).

A problem closely related to double-diffusive convectionin an isothermal ternary system is thermal convection in abinary mixture, which has been examined in precision ex-periments that impose a vertical temperature gradient acrossimpermeable horizontal plates.8 '9 In a binary mixture thetemperature gradient is destabilizing, while the Soret effectleads to a stabilizing concentration gradient. Although thereare similarities between convection in a binary mixture withan imposed temperature gradient and convection in a ternaryisothermal mixture with imposed concentration gradients, theproblems differ in the governing equations, boundary condi-tions, and values of the principal coefficients; therefore, nodirect comparison of the two types of phenomena is possible.

We will show that even the primary instability in doublediffusive convection holds some surprises. Our linear andweakly nonlinear analyses of double-diffusive convection ina Hele-Shaw cell are presented in Sec. II. A tricritical con-

Cf =Cf 0, C', = 0

Cf 0, Cs = C80

FIG. 2. Snapshot of waves of concentration traveling to the left in a Hele-Shaw cell. The horizontal gradient of the refractive index field of the fluid isshown. Cell A', cfo=0.543 g/l of HCI, c~0=2.00 g/l of propylene glycol.

dition is found to be satisfied all along the oscillatory insta-bility boundary: the real part of the coefficient of the cubicterm in a Ginzburg-Landau model is zero everywhere alongthe boundary. The coefficient of the fifth order term in theGinzburg-Landau equation is small and positive; hence thebifurcation to traveling waves is supercritical, but unusuallyabrupt. This type of "vertical" bifurcation with vanishingcubic order coefficient has been predicted previously forphysically unrealizable boundary conditions.'0 Section HIdescribes our convection cell and the image processing tech-niques that enable us to observe relative concentration gra-dients as small as 10-7 times the applied concentration dif-ference. In Sec. IV we compare the observed values forinstability threshold, frequency, and growth rate with thosepredicted theoretically; the agreement is very good if correc-tions for cross-diffusion effects and finite cell thickness aretaken into account. The one-dimensional traveling waves thatform at the instability do not saturate but evolve into a"blinking" state of finite amplitude traveling plumes. Nohysteresis is observable, however, in accord with the weaklynonlinear analysis. The results of the experiments and analy-sis are summarized in Sec. V.

II. THEORY

A. Previous theoretical analyses

The discussion and linear stability analysis for thedouble-diffusive or the equivalent thermohaline problem isfairly extensive"1 1l 1 3 and serves as a textbook example' 4 ofan oscillatory instability for the case where the destabilizingsolute is more rapidly diffusing. Most of these analyses havebeen done for the case of a three-dimensional cell with un-physical free boundary conditions. An exception is thermo-haline convection in a porous medium.' 3 The results of thatinvestigation of the onset and frequency of the oscillatoryinstability are essentially the same as those obtained belowfor the Hele-Shaw geometry. A weakly nonlinear analysisfor the case of free boundary conditions near the oscillatoryinstability has been performed by Huppert and Moore,' 5 whoassumed a pattern in the form of a standing wave. Theseauthors predict either a forward or backward bifurcation, de-pending on values of parameters, and thus the existence of atricritical point. On the other hand, when traveling waves areconsidered,l0 the tricritical condition holds all along the lineof oscillatory instability. In what follows we find a similarresult for a model that approximates the boundary conditionsappropriate to the present experiment.

B. Hele-Shaw geometry1. Basic equations and boundary conditions

We start from the equations of hydrodynamics in theBoussinesq approximation for the thermohaline problem, asin Ref. 2 with the Soret term neglected. 16 The temperaturefield is identified with the fast species and the salt field withthe slow species. The term vV2 u in the Navier-Stokes equa-tion is replaced by -12vu/w

2 , as is appropriate for theHele-Shaw approximation' 7 when wld-41, thus reducingthe problem to two-dimensions (w and d are the cell thick-ness and height respectively).

3924 Phys. Fluids, Vol. 6, No. 12, December 1994

unstable

Codimension-22000

point-

stationaryinstability A . ."' . I ....... X :'.....'.. ..';'

Precitechensky et a/.


In our system the imposed concentration gradients foreach of the two species constitute independent control pa-rameters that can be sustained indefinitely. A concentrationco is imposed at the lower cell boundary (where Cf= 0) andCfo at the upper boundary (where c5 = 0). Both of the solutesare assumed to increase the density of the solvent; hence thegradient in c, forms a stable stratification in the field ofgravity and is stabilizing, while that in Cf is destabilizing.

The conducting state is

U= 0, c,= c50( 1 -ild), 6f = cfo(i/d), (1)

where U is the dimensional velocity, 6, and 6f are the dimen-sional concentrations, and d is the depth of the fluid; i is thevertical coordinate and is measured from the lower horizon-tal bounding plate.

We nondimensionalize time in terms of d2 /Df, length interms of d, concentration of the slowly diffusing species interms of c, 0 , concentration of fast diffusing species in termsof - Cfo, and the streamfunction in terms of Df . The equa-tions for the deviations of the dimensionless streamfunctionand concentrations from their values in the conducting state(1) are then

I 'AV !V, 1) VI+lg R. dc' R dcf\ardt dJ(x)- Am+ Ra -Rd) (x2)

(2a)

dt J( >Cf) + x= ACf (2b)

dc' d doatJ(qy"c5) + ax = TACs, (2c)

where x is the horizontal coordinate, A = d2/dx 2

+ d 2 Idz 2 , Ux= dqlaz, u5 =- d /'dx, operatorJ(a,b)=(da/dx) (dbldz)- (da/dz) (dbldx), o-=12vd2 1Dfw2 is a modified Schmidt number, w is the spacing be-tween the vertical bounding plates, and

a'iciogdw2

i 12vDf (3)

are the Rayleigh numbers for species i=f and s;aj=(l/p) dpldci for i=sjf; g is the acceleration due to

gravity, and v is the kinematic viscosity.The velocity boundary conditions are the same as for a

binary mixture in a porous medium,'8 i.e., the vertical com-ponent vanishes at the horizontal boundaries. Periodicity isassumed in the x direction.

\iS( l\ RCf( = Cf 1 e/yt-iqx

| -~i ttL

C+ f 1 ) t +iqx sin(Irrz). (4)

CS1

In the following 93{ } and J{ } will denote the real and imagi-nary parts.

Upon linearizing Eq. (2), we find the complex growthrate, y, is given by

(5)

where k2 = q 2+ IT2

. Our experiment corresponds to Schmidtnumber o- of order of 105. Therefore, we examine Eq. (5) inthe limit of o- going to infinity and the following results aregiven in this limit. The minimum (with respect to q) value ofRf which gives a zero value for 9i{y} and a nonzero value forJ{ y} = &), and thus corresponds to an oscillatory instability isgiven by

RfC =R + 47T2

( 1 + T). (6)

The linear analysis does not distinguish between travel-ing and standing waves. As seen from Eq. (6) and the defi-nitions of the Rayleigh numbers, the oscillatory instabilityoccurs when the overall stratification of the fluid is unstable.This is similar to the case of three-dimensional porous me-dia.

The frequency and wave vector at onset are given by

(O2 =W2 (1 -T)(R 5,-R5 c2), (7)

qcIT. (8)

The lines of oscillatory and stationary instabilities intersectat a codimension-2 point corresponding to an R5 value of

R,, 2 =47T27

2/(1 - 7). (9)

3. Weakly nonlinear theory

The calculations are essentially the same as the one forwhich standing waves were assumed 5 We have allowed forthe possibility of traveling waves as well as standing wavesin Eq. (4). We write

t' =pt,

Cf- = 77CnCfn

C5 n=1 Cmn

(l0a)

(1Ob)

(l0c)Rf=Rfc+ E 7nRfn X2. Linear theory n=1

To perform the linear stability analysis of the conductingstate we write the solution in the form of superposition ofright- and left-going waves

P=&)C±E ?7nPn,

Phys. Fluids, Vol. 6, No. 12, December 1994

(1Od)

Y/0-+ I = q 2� Rf RsJ T+_kP - T+-,r-k7

Predtechensky et a/. 3925


with the first order terms of the form in (4) with yt replacedby it'. The expansion parameter V in (10) is to be eliminatedin the end by solving (10c) for 7 in terms of Rf-RfC andsubstituting into (lob) and (1Od).

We find from Eq. (2) at second order in 7

020, (11a)

+ gi~ R+(lib)1 R ~

Cs = 91{ +Sin(2 lrz)( (i/'I) 2

1 2 icc4\ ± -R-g4 + qh i/cL± 4c+) (ic

Rf = 0, (lid)

p=0, (lie)

where an overbar means complex conjugate.At third order we find, by imposing a solvability

condition's in which we assume that 04 is not zero, that

{ d k 2 2\2 2.

i +k}2 (ico~k2 )(ico+ rk2) i q2 (Rf1~-R,) pz q(i&)+ rk2)Rf2

42( iw~+ 2 -R i(O+ k2\ 1 0112± 1j

~~k \ R s w~ + T2 k 4 fc w +~k4 } 8

+q4I21 R,(ioj,+k) Rfc (i c~+ Tk 2) 112_(ico,+ TO 2)(2ico + 47T2 Tr) -(iwc+k2)(2iw,+4 IT2) 2

where q is to be evaluated at qc and co is to be evaluated atscc. Note that Eq. (12) holds for arbitrary o-.

If we set I VA 2 =I l 2

= 1/4 for standing waves, thenEq. (12) agrees with Eq. (4.5) in Ref. 15 if we allow fordifferences in notation and the Hele-Shaw rather than planarcell.

For traveling waves we set 0=0 in Eq. (12), since wealready assumed aid is nonvanishing. Then we make use ofEq. (5) and find for arbitrary a- and R,

(13)Rf 2 = 0

andk2q2

P2= k~cqcI OII

with kC= 2 + q2.It follows from this analysis that double diffusion in the

Hele-Shaw geometry provides a physically realizable ex-ample of an interesting behavior that has been previouslypredicted for unphysical boundary conditions in other con-vecting systems: 10' 20 the tricritical condition holds not just ata point but everywhere on the boundary of oscillatory insta-bility. That is, if we consider left traveling waves (e, = 0)and solve Eqs. (10) and (12) for aid as a function of

Rf -RfC(R,)6= Rf,, (15)

then nowhere along the line of oscillatory instability does theamplitude beyond the instability increase as e .

We evaluated terms through the sixth order in (10) forthe case of traveling waves and o-=-. For all the values of T

and R, which we examined, Rf2 was computed to be lessthan 10-30, consistent with the analytic calculations; Rf4 was

(14)

(12)

found to be positive; hence the bifurcation is supercritical,and very near the bifurcation AL must increase as e"'4. How-ever, for r=0.62 andR3 =400, we foundp 4 /p 6 =5.06. Thevalue of 72 , 2.53, at which the sixth order term becomesone-half the fourth order term and the expansion (1Od)breaks down, yields e=0.005. Similar calculations atR,=40 and R,=4000 gave E=10-6 and e=10-4, respec-tively. These small values of e explain why the eJ14-regionmay not be accessible in our experiment, which at thepresent time has a resolution of 0.003 in e.

4. Complex Ginzburg-Landau equation

Problems of stability and pattern selection in a systemwith an oscillatory instability with nonzero wave vector areconveniently studied using the complex Ginzburg-Landauequations.2 1 '22 These equations for the amplitudes of right-and left-traveling waves are

T7(AR+sA R) = e(1 + ic0 )AR+ 2(1 + ic,)A R - (KIAR12

+MIALI2+NIARI4+...)AR, (16a)

rO(AL-sA L) = e(l +icO)AL+ (2(1 +ic)A LX-(KIAL 2

+MIAR12 +NIAL 4 + ... )AL. (16b)

The coefficients of the linear terms in these equations aregiven in Sec. II B 5. The coefficients of the nonlinear termsmay be obtained by assuming the normalization of AR andAL such that they are the amplitudes of streamfunction as inEq. (4) and then comparing solutions of the spatially homo-geneous Ginzburg-Landau equations to solutions of Eqs.(10), (lid), (lie), and (12) for the amplitudes as a functionof e. In particular this implies 9t{K} = 0 everywhere (for all

3926 Phys. Fluids, Vol. 6, No. 12, December 1994 Predtechensky et al.


o-) along the line of instability. We find 9%{M}>0 (at leastfor infinite a-) along the line of instability, and 91{N}>0.

The nonlinear analysis can determine whether the oscil-latory instability at onset leads to traveling or standingwaves. The analysis follows from Ref. 21 with an obviousmodification for the case when the fifth order term is domi-nant. The result is that spatially homogeneous travelingwaves are stable with respect to spatially homogeneous per-turbations while spatially homogeneous standing waves arenot.

We have computed and considered only the fifth ordercoefficient, which provides a self-coupling in the above.Deane, Knobloch, and Toomre23 have considered the threecouplings between left and right traveling waves at fifth or-der. By imposing different relations among the couplings,they obtained the generic bifurcation diagrams shown on theleft side of their Fig. 2. Our analysis given above indicatesthat the region in e where this behavior occurs is probablytoo close to onset to be accessible in the present experiment.

5. Absolute and convective Instabilities

Instabilities involving propagating modes can be distin-guished according to whether they are convective (distur-bances grow in time only in a moving reference frame) orabsolute (disturbances grow in time even at a fixed position).This distinction is important for infinite systems and for fi-nite systems with periodic boundary conditions, but is not asclear for small aspect ratio systems like the one studied in thepresent experiments. Nevertheless, it is of interest to use theresults from the complex Ginzburg-Landau equation [Eq.(16)] of the previous subsection to determine how far theonset for absolute instability is above the onset for convec-tive instability.

The parameters needed for the analysis include the groupvelocity

___a = CX (17)8q

and the curvature of the onset value of Rf with q

1 a 2 Rf(q) _4(1 +T) (8{°=Rfc RR ~~~~~~~~~(18)

where RfC(q) gives 9%{y}=0.The slope of the instability exponent near the threshold

Rfc, defined as Tj =RfCi{8yIdRfl, is

T IRfcI4* (19)

Other parameters are:

1 d 2wc(q)

, 1 2r 1, 2q (20)

-1cc 1= -2r)ir2I @ (21)

Toc=-RfC2{-J-} 2To, (22)

In Ref. 24 the above quantities are combined to give thethreshold of absolute instability in an infinite system as:

0.3

0.2

0.1

0.00 1000 2000

FIG. 3. Threshold of absolute instability.

s2r24 S( c0 (23)

A graph of ea vs R, for T=0.6 2 is shown in Fig. 3. Forour system ea grows rapidly from zero at the codimension-2point to a constant value of about 0.2. However, as we showin Sec. IV B ,the amplitude of the convective motion in ourexperiments increases so rapidly with e that even at e=0.005the Ginzburg-Landau expansion is not applicable. More-over, because of the high reflectivity at the end walls, theonset of absolute instability, Rfa = (1 + Ea)Rfc does not play

an essential role in the pattern formation.22

C. Nonideal Hele-Shaw geometry

In this section we examine the effects of nonideal Hele-Shaw geometry, i.e., the effect of finite wid, on the linearstability analysis. In order to examine these effects we studythe model in the Brinkman approximation,2 5 obtained by re-placing the Laplacian in the viscosity term of the Navier-Stokes equation by

V2 =~ ( + 92 _ 12) (24)

(instead of V2 = - 12/W2 as in the Hele-Shaw limit):

1 d (w/d)2 \ I dc5 Rdc

-8 at 12 +R 5-a- - (25a)89Cf d11'

- + - = Ac8t ax Cf

8c3 ± '9 = 'rACe .

(25b)

(25c)

A similar method has been used in Ref. 26 for binary mix-tures. Equations (25) are solved subject to rigid boundaryconditions on the velocity at the bottom and top plates:

V/= = c =c= 0at i=0 and id (26)

assuming the system is infinite in x. We have used themethod of Refs. 27 and 28 to solve the equations.

In order to test the numerics for wid = oo and the useful-ness of the model (25) for studying the case of finite wid, wehave studied the case of ordinary Rayleigh-B6nard convec-tion, i.e., Eqs. (25) for R,=0. This critical value for the


, * , I , I . I I I , I

Predtechensky et al. 3927


10 R 0 /4i

Hele-Shaw

limit , X

10.1 1.0

wid

FIG. 4. Instability threshold for Rayleigh-Binard convection as a functionof wid, the ratio of the cell height to thickness: a comparison between theBrinkman approximation (solid line) and the direct Galerkin analysis29 (dot-ted line). The dashed line is the function F in Eq. (30).

stationary instability as a function of wid is given by thesolid line in Fig. 4. The dotted line is the result of the Galer-kin analysis for this problem. 29 The close correspondence ofthe two lines indicates that the model defined by Eqs. (25) isa reasonable one for exploring the effects of finite w/d.

The model (25) was then used to calculate the values ofRf and the frequency at onset for an oscillatory instability,for values of wid corresponding to experiment and o-=-.The results of these calculations are compared to the corre-sponding quantities in the strict Hele-Shaw limit as given byEqs. (6) and (7) and experiment in Sec. IV. For thermal con-vection in binary mixtures2 8' 30 there is a jump in the fre-quency and wave number at the codimension-2 point. Foro-=, we find no jump in our model, just as in the binarymixture problem with zero Lewis number28

D. Cross-diffusion effects in cell with finite wid

In order to investigate the effects of cross-diffusion, wehave extended Eqs. (25) to include cross-diffusion terms. Weintroduce the rescaled cross-diffusion constantsrij=(Dtj/Dff) (a/a 1) , (i,j=f,s), where Dij aredimensional-diffusion constants and rfs describes the flow ofthe fast species due to the gradient of the slow species (andvice versa for Tsf); re -r is the conventional Lewis number.There is an Onsager relation 31 between off-diagonal kineticcoefficients, but we have no knowledge of the relevant ther-modynamic derivative matrix; hence the off-diagonal ele-ments are introduced as independent parameters. We assumeDij to be independent of concentrations.

The stationary instability threshold is given by

fc 'r- T'f 7

-7sf

(27)

where Ro is the critical Rayleigh number for a one-component Hele-Shaw system with finite wid. This formulacan be obtained utilizing the fact32 that the full linear double-diffusive problem for steady instability can be converted tothe single-component case under proper transformation ofthe concentration fields.

SALT SOLUTION GEL-FILLED MEMBRANES(thickness -0.1 mm)

FIG. 5. Diagram of the convection cells. The long side walls are made ofpolished Amersil T-12 quartz and the ends are stainless steel. The 0.1 mmthick nitrocellulose membranes are impregnated with polyacrylamide geland form the top and the bottom of the cells.

To find the location of the codimension-2 point and theonset of oscillatory convection, we consider an asymptoticexpansion of the solution at the codimension-2 point in pow-ers of frequency p. Since the odd terms of the expansionmust vanish due to symmetry, one can obtain the location ofthe codimension-2 point (for details see Ref. 32):

Rfc2 =R( Tfs) - Tfs(r- rsf)RfC2=Ro~ 1-r-rf,+ r-sf

7(7- Tf)-r 2 f (1-Tf5 ) (28)

Rsc2=Ro I1- 7r- Tfs+ rsf

The onset of oscillatory convection and the correspond-ing frequency are given approximately by

R05c=R,+R 0 (1 + r),

W 2=(I - r-Tfs + sf) (R, -R 5 c 2)F(w/d).

(29)

(30)

The critical Rayleigh number Ro and the function F(w/d)are shown in Fig. 4.

Diffusion measurements on ternary fluids33 with compo-nents similar to those in the present experiment suggest thatthe off-diagonal elements can be of order 10% of the diago-nal ones. In this range and for R,<2400 the above expres-sions are a good (to within 0.01% or better) approximation tothe results of our numerical analysis. Equation (30) was thenused to find a fit to the experimentally determined frequen-cies. The results will be given in Sec. IV.

Ill. EXPERIMENT

A. Apparatus

The convection cell is shown in Fig. 5. The innovationhere is the use of thin gel-impregnated membranes for thetop and bottom surfaces of the cell. The gel membrane isprepared by saturating a 0.45-1.0 Itm pore membrane (ni-trocellulose or Anopore) with acrylamide monomer, and thenpolymerizing it to a gel in place. The recipe for the polyacry-lamide gel is given in Ref. 34. The gel prevents mass flow in

3928 Phys. Fluids, Vol. 6, No. 12, December 1994 Predtechensky et a/.


TABLE I. A set of cells used for experiments.

Cell d(mm) Lid w/d Chemicals e

A 3 20 0.254 NaCI/propylene glycol 0.63A' 3 20 0.254 HCVpropylene glycol 0.31B 6 10 0.25 HCIlpropylene glycol 0.31C 1 20 0.127 NaCI/glycerol 0.62D 3 20 0.069 NaCI/glycerol 0.62

'Computed from data in Table H.

the membrane. The outer surface of each membrane is incontact with a continuously refreshed reservoir so that well-defined concentrations are imposed. To check the diffusivequality of gel-filled membranes, we performed a measure-ment of the volume flux of sodium chloride across the cell ina linear stable-stratified regime and found no significant(<5%) deviation from ordinary diffusion, which indicatesgood permeability of the membranes. The thickness of themembranes with a nitrocellulose substrate was 100 ptam, and60 /.um for Anopore substrates. The upper reservoir containsthe fast diffusing species, typically sodium choride (concen-trations in the range 0.2-6.0 g/l), and the lower reservoircontains the slowly diffusing species, propylene glycol (con-centrations in the range 0.3-45 g/l). Propylene glycol waschosen because of its small specific gravity relative to purewater in conjunction with a fairly good index of refractionincrement relative to the variation of the solute concentration(see Table II).

This combination allowed us to reach the region of thecodimension-2 point with well-controlled levels of soluteconcentration and without loss of sensitivity. For the cellswith smaller w (0.2 mm) a glycerol solution was used. Thecell parameters are given in Table I. In calculations the vis-cosity of solutes was assumed equal to 0.01 cm2 s-1 exceptfor propylene glycol at 45 g/l, where v=0.0116 cm2 s5 t. Thespecific gravity increments a and the diffusivities of specieswere taken from Ref. 35 and are shown in Table 11.

B. Observation techniques

The convection patterns are visualized from the side us-ing a double pass schlieren scheme shown in Fig. 6. Verticalorientation of the knife edge was chosen to detect a horizon-tal gradient of the refractive index of the flow. The side wallsof the cell were made from highly homogeneous AmersilT-12 quartz optical windows; one gold-plated surface formsa mirror. All equipment was mounted on a stable Invar opti-cal bench. The entire setup (with both 4-liter solution reser-voirs) was enclosed in a thermally insulated Styrofoam box

TABLE II. Properties of the fluids at 25 'C (Ref. 35) used for experiments.

Solute D (cm2 s--') a (g'1) dn/dc (g-1l)

NaCI 1.51x10-5 7.0X10-4 1.8x10-4HC1 3.07X10-5 5.0X 10-4 2.3 X10-4Glycerol 0.93X10- 5 2.3X10-4 1.2X10-4Propylene glycol 0.95X10-5 a 0.6X10- 4 1.0X10-4

'Estimated value based on molecular weight.

MIRROR SURFACE

/ LENS/ -......... BEAM SPLITTER ZOOM LENS---------- '7 --.-- o ob

X em_ __ alp wS S ~~~CAMERAKNIFE EDGE

EXPERIMENTAL CELL

PINHOLE

W LIGHT SOURCE

FIG. 6. Double-pass schlieren system. The focal length of the achromaticf/10 lens is 750 mm.

with a thermoelectric temperature controller. Pumps and alight source coupled by a fiber optic bundle were mountedoutside the box to avoid undesirable heating. The tempera-ture of the setup was maintained at 22.8±0.1 'C. The con-centrations of solutes were controlled using a volumetrictechnique. In addition, the concentration of conductive spe-cies was monitored by a digital conductivity meter.36

Images composed of 512X25(X8 bit) pixels were ac-quired with a Cohu CCD video camera connected to a mi-crocomputer. An MVP-AT image acquisition and processingboard3 7 was used to perform most of the image processing inreal time.

When very weak solutions were used in order to inves-tigate the convection in the vicinity of the codimension-2point, the observed signal was often significantly smallerthan the sum of the optical inhomogeneities of our schlierensystem. A modified background subtraction technique wasemployed to improve the quality of pictures: to detect themoving patterns, each output image was formed from a cur-rent sample image minus an image taken a few samples ago(typically with delay of about 1/2 of period of oscillation). Acharacteristic image is shown in Fig. 2. This "sliding" sub-traction is a high-pass filter that eliminates a long-term driftin the opto-mechanics, thus providing better signal/noise ra-tio. Of course, this technique with fixed time delay intro-duces some amplitude distortion proportional to the fre-quency of local oscillations in the signal and cannot be usedfor stationary convection.

Two basic techniques were employed to determine thestability boundary. The first technique was a direct observa-tion by continuous monitoring of space-time diagrams. Eachrow in such a diagram was formed sequentially from spatialamplitude of image intensity determined as a line of pixels(512Xl), typically in the central region of the side image.The typical shape of such a function is shown in Fig. 7. Eachimage was vertically averaged to enhance this amplitudefunction. This leads to additional improvement of the signal-to-noise ratio. To achieve a better contrast and keep a fulldynamic range of pictures, the amplitude functions were nor-malized to the value 255, and the norm was stored in the firstfour pixels of each row in a floating point format. A decisionon whether or not any pattern has been formed was made byvisual recognition of inclined strips (or other patterns)against an initially featureless noisy image. For better recog-nition, an additional enhancement of the space-time dia-grams was achieved by "clipping" the resulting image near

Phys. Fluids, Vol. 6, No. 12, December 1994 Predtechensky et a/. 3929


(a) 0 "

(b) 20

~s 10

'u 0

-10

-200

(a)

1

2-

Time (Ih)

3-x (min)

4FIG. 7. (a) Clipped image of the convection pattern in Fig. 2. (b) Spatialintensity of transient traveling wave at the midline of Fig. 2. Cell A',R,=55.3, Rf=l2 3 , E=0.004.

zero threshold. The resulting sensitivity of the method to thegradient of concentration is estimated as 4X10-4 glV' cm-1

for cell A. Measurements were usually made for several daysat each control parameter setting.

Accurate determination of the bifurcation point Rf, wasachieved using a relaxation technique similar to that in Refs.38 and 39. Instead of fitting data to a wave packet and ex-tracting the amplitude of its envelope, we characterize theconvection by its "total energy" E, defined as the energy inthe spatial Fourier modes in a range 10(X/2) to 30(X/2). Thespatial harmonics were obtained with resolution to X/2 viathe standard Fourier transformation of the amplitude functionexpanded to double the length by addition of zeros.

An abrupt change in control parameter (salt concentra-tion) was used to create a test perturbation; it appearedmostly near the edges of the cell. An example of space-timegrowth of such a perturbation and its decay after switchingback to a subcritical value of Rf is shown in Fig. 8(a). Figure8(b) shows the corresponding time evolution of the "totalenergy" of this motion; regions with exponential growth anddecay following increases and decreases, respectively, in Rfare clearly seen. The two slopes give two independent valuesfor growth rates: one for a supercritical value of Rf, andanother for a subcritical Rf value. Measurements at severalRf yield the dependence of growth rate on solute concentra-tion, as shown in Fig. 9. A linear interpolation was used toestimate the critical value of Rf .

When propagating waves reach the end of the cell, re-flection and interaction between the counter-propagatingwaves complicates the picture. The time evolution will con-sist of pieces of exponential growth broken by regions ofconstant energy or even with some decay, but local slopes ofthe exponential regions remain nearly the same as for short-lived waves. Therefore, the influence of reflections on thethreshold of instability is negligible in our system, in contrastto the situation in thermal convection in binary mixtures.39

IV. EXPERIMENTAL RESULTS

A. Comparison with linear theory

The measured onset Rayleigh numbers for oscillatoryinstability are shown in Fig. 10 for several combinations of

(b)12

10

8

bS 6

4

2

0I0

U a~ 20

1 2 3Time (h)

FIG. 8. Growth and decay of convection pattern following changes in con-trol parameter. (a) Space-time diagram of growing and decaying pattern; (b)the corresponding behavior of the logarithm of "energy." The dashed linesshow exponential growth (1) and decay (2) of the energy; the arrow marksthe moment when the concentration was changed to a subcritical value. CellA, R,= 965 (15 g/l of propylene glycol).

sample cells and fluid mixtures. The curve in Fig. 10 is fromlinear theory with correction for finite cell thickness [Eq.(29)], for o-= and T'0.63 (see Table I). For the range of7=0.4 -0.8 and w/d=O.l - 0.25 appropriate for our ex-periment, the stability curves on a log-log scale are nearlythe same, so we can show data for all mixtures on the same

4.

0t

2

I

0

-1L

1.55o 1.55 1.60Concentration of NaCG (g/l)

1.65

FIG. 9. Growth rate near onset as a function of salt concentration. The datapoints (1) and (2) correspond to those shown in Fig. 8. The salt concentra-tion at the onset of instability (1.560±0.005 g/l) corresponds toRf = 10 4 2 .


0 x/d 20

Onset -

(2) .-

O�

Predtechensky et al.


1o'

Rfas

R103

. codimension-2point

1 o2 ~Rs 10

FIG. 10. Stability diagram for double diffusive convection as a function ofthe Rayleigh numbers Rf and R. for species that diffuse fast and slow. Thecurves were obtained from a stability analysis: solid line, convective insta-bility to traveling waves; dotted line, instability to stationary convectionrolls. Four data sets are shown: NaC/propylene glycol in cell A (0); HClI/propylene glycol in cells A' (V) and B ([2); NaCI/glycerol in cell D (0) (seeTable I).

graph (Fig. 10). The measurements yield onset Rayleighnumbers typically a few percent higher than those predictedfrom linear stability theory, as Fig. 10 illustrates.

Several experimental points in Fig. 10 are fairly close tothe codimension-2 point, where the frequency of oscillationsshould vanish in accord with Eq. (7). The frequency datashown in Fig. 11 confirm this square-root dependence semi-quantitatively at small R,, but deviate considerably at largeR, (large concentrations). This deviation may be due tocross-diffusion effects, which can be significant at higherconcentrations of the diffusing species.4 0 Therefore, we fitthe data in Fig. 11 with the rij as free parameters. Figure 11shows the best fit of Eqs. (30) and (29) to the frequencies atonset.

2.0

1.5

~1.0

L,0.5

0.0

FIG. 11. Comparison of the measured frequency of the traveling waves atthe onset of instability with the prediction of a linear stability analysis with-out cross-diffusion effects (dashed line) and with cross-diffusion (solid line).Without cross-diffusion the fit was provided with T-0.8; with cross-diffusion 7-0.58, fs=°0.35, and rsf=0.07 provided the fit. The opencircles show data for NaCl and propylene glycol in cell A, the filled squaresare for NaCI and glycerol in cell D.

500

250

0100 1000

FIG. 12. Differential threshold for oscillatory instability as a function ofR_. Data sets are the same as in Fig. 10. Lines are drawn in accord with Eq.(30): solid line is for cell A with wld = 1/4, dashed line is for cell D withw/d= 1/16. The asterisks mark the location of the codimension-2 point.

The values deduced for the cross-diffusion coefficientsare reasonable: DfS=0.11DNaCG, Dsf=0. 2 3 DNaC]. Wefound no direct measurements of D, for propylene glycol,but surprisingly the data from cell A (w/d= 1/4) with NaCl/propylene glycol with the estimated value of diffusion coef-ficient fit the solid curve for NaCl and glycerol (Fig. 11).

Considering the departures from the analysis at higherconcentrations of stabilizing species, we note that Eq. (6) canbe rewritten as RfC-Ra=4iT2(1+T). The data in this rep-resentation are shown in Fig. 12 on a linear-log scale. Thedeviation increases with the increasing concentration of theslow species. Data for cells with aspect ratios 10 and 20show no significant difference in Rf, within our resolution.For cell D, which is close to ideal Hele-Shaw geometry,wid = / 16, however, the open circles in Fig. 12 are muchcloser to theoretical predictions.

We also considered the influence of thickness of themembranes. Assuming that the diffusion constants in themembranes are at least half as high as in the mixture itself,we found the corrections in Rf, to be smaller than 1%. How-ever, the effect of the membranes can reduce the frequenciesby up to 10%.

Measurements of the growth rate at onset agree fairlywell with the predictions of the linear analysis [Eq. (19)], asFig. 13 illustrates.

B. Development of weakly nonlinear waves

When the critical value of Rf is exceeded, nearly regularstripes develop in the initially featureless space-time dia-grams described in Sec. III B. The side views (see Figs. 2and 7) show weak traveling waves of concentration at smallE, where the growth rate of the instability is small comparedto the inverse time for wave propagation across the cell.These waves have a characteristic crescent shape; see Fig.14(a). The same shape is apparent in a contour plot of thelinear eigenfunction of our problem, derived for a nonidealHele-Shaw cell; see Fig. 14(b). A curvature of the eigen-function has also been found for a similar problem. 27


0 LI0

* El

~- t7 ? _ ~ - - - - - - -

Predtechensky et a/. 3931


(a)

400

20200

0 I - . . . , . . . . .0 1000 2000Rs

FIG. 13. The slope of instability exponent near threshold vs the Rayleighnumber R, of the slow species. Solid line is obtained from Eq. (19). Thedata are the same as in Figs. 8 and 9.

From run to run, these waves may propagate to the leftor to the right, and always form a clear standing wave struc-ture near the edges of the cell, as the spatiotemporal plot inFig. 15(a) illustrates. This indicates a reflection coefficientnear unity, in contrast with the situation in thermally drivenconvection in binary mixtures.3 9' 41

For nearly the same e, the whole pattern sometimes ap-pears in a form that looks very close to a standing wave, asshown in Fig. 15(b). However, a detailed inspection of thewindow marked by a rectangle in the upper part of this figurereveals the special structure shown in Fig. 16(a) in enlargedformat. A computer analysis of these data4 2 showed that thisstructure consists of counter-propagating waves of about thesame amplitude but with different wave numbers, q1 =3.0and q2 = 2.5.

Using the linear analysis given in Sec. II, we can com-pare the wave numbers of the counter-propagating waveswith the predicted width of the stability interval. Near thecritical point e=(q-q,)2t, so the possible wave vectorrange of growing solutions will be Aq=2fe/i,. Makinguse of Eq. (18) and the parameters given in Fig. 15(b), onemay obtain Aq=0.9. Hence the stable range of waves dueto the Eckhaus instability, 52Aq, is 0.6, which is close toexperimental value of 0.5. This means that the observedcounter-propagating waves lie near the opposite ends of thestability interval, which suggests that some nonlinear inter-action leads to a wave number repulsion. With our resolutionin e (about 0.2%) we did not observe a supercritical asymp-totic state with symmetrical standing wave structure.

(a)

(b) [ElII

.I

i

0 x/d 20 0 x/d 20

FIG. 15. Evolution in space and time of quasilinear wave patterns nearonset: (a) traveling waves (R,=55.3, Rf =123, e=0.004); (b) quasistand-ing waves (R,= 138, Rf= 223, =0.007). The data are from cell A'; 1-hourdivisions are shown on the time axis.

(b)

P IAIW in'm , -

r- -a. Aw-.

a T~iw } I .

I

0 x/d 20 0 x/d 16

FIG. 14. Comparison of (a) the observed shape of left-propagating growingwave with (b) a contour of the linear eigenfunction of the most unstablemode for a cell with finite wld=114.

FIG. 16. Sequences of images showing quasistanding wave patterns. (a)Observed in the cell A'; the set of images corresponds to the solid rectangleshown in Fig. 15(b); the separation in time is 50 s. (b) Amplitude plots of thestream function obtained in a numerical calculation. 43


(b)600

(a)____ s ___

___=_8_I . . . - 'A__

_

__i = _ _ . m Y

: | EI = .. . _

=

:.; .: :: 7': 1:.

Predtechensky et al.


A numerical simulation of two-dimensional thermo-solutal convection 4 3 yielded patterns very similar to to thosewe observe: compare (a) and (b) in Fig. 16.

C. Traveling plumes and intermittency

We emphasize that the patterns in both Figs. 15(a) and15(b) are unstable: the waves grow very slowly in time, andeventually a modulational instability transforms the motioninto a blinking state (similar to that found in Ref. 9) consist-ing of localized wave packets that propagate back and forthin the cell, as Fig. 17 illustrates. These blinking states arealways slightly asymmetric and irregular in time, even after aweek of observations. Similar spatiotemporal behavior wasfound by Cross4 4 in a simulation of coupled Ginzburg-Landau equations. Variations in the local frequency and spa-tial wave number are quite noticeable in Fig. 17.

The amplitude of the waves in the packets is 30-100times larger than the waves at the quasilinear stage shown inFig. 15. The wave packets in the asymptotic state containtraveling plumes, as shown in Fig. 18. The explicit two-dimensionality of the plumes would be missed if the patternswere observed from the top, as in most experiments on trav-eling waves near onset: vertical averaging would yield simi-lar waveforms for Fig. 7 and Fig. 18. Another importantfeature of the convection is a noticeable change in the veloc-ity of wave propagation when the quasilinear waves turn tothe nonlinear traveling plumes.

D. Absence of hysteresis

The Hopf bifurcation in a dynamical system must pos-sess a characteristic amplitude dependence on s if the bifur-cation is supercritical, or the system can jump to an unpre-dictable state if the bifurcation is subcritical. As we havedescribed, above threshold our system goes asymptotically toan irregular high-amplitude blinking state. Therefore, welooked carefully for hysteresis, as would be appropriate for asubcritical bifurcation. However, this hysteretic behavior wasnot observed within the resolution of our experiment, 0.3%.We made repeated measurements along the stability bound-ary with increasing and decreasing Rf to try to determinewhether or not there is hysteresis in the primary instability.The hysteresis if any is small, less than 0.6%.4546 Immedi-ately above the threshold for instability, the quasilinear statecan persist for several horizontal diffusion times, but even-tually there is always a transition to the blinking state. Thesame is true when moving back below the threshold: thetransients are long if the state is close to the threshold, so therange of hysteresis estimation is mostly limited by observa-tion time.

To characterize the typical behavior of the system nearthis critical point, we plot in Fig. 19 the dependence of the"total energy" on time when changing the salt concentrationfrom e= -0.003 to e= +0.003 and back. The motion belowonset consists of sporadic short wave packets generatedprobably by external noise (e.g., nonuniform pumping).These wave packets propagate in one direction and rapidlydecay, leaving a characteristic bump in the plot. When theparameter Rf was set above the threshold, the waves started

16

18

20

22_-

0 x/d 2

FIG. 17. Asymptotic state of convection that started as shown in Fig. 15(b).

to grow exponentially in steps (see Fig. 19) with growth rateincreasing from step to step. Eventually the "energy" stopsgrowing at the last step and stays fairly stable during morethan two days of observation. When the concentration of thesalt was decreased to a slightly subcritical value, a long de-cay process was observed. It took a long time and occurredin a rather irregular fashion, but finally the system came backto the initial state (see Fig. 19).


0

2

4

6

8

10

Time (h)

12

14

Predtechensky et al. 3933


(a)

(b)IT v E E 131

FIG. 18. Schlieren images of a saturated region in a blinking wave packet:(a) gray scale image; (b) the same image with enhanced contrast.

E. Subcritical noise amplification

In runs where the concentration of slowly diffusing spe-cies was high, noise patterns could be observed below theonset of instability; see Fig. 20. These very weak patternsappear smoothly when approaching the critical point frombelow. The space-time diagram in Fig. 20 possesses a typicaltexture reflecting the propagation of waves with a character-istic velocity and wave vector. This texture exists in a rangeof e about 2% below onset and evolves into patterns de-scribed above when the critical value of Rf is exceeded. Theestimated peak-to-peak amplitude of concentration fluctua-tions is about 4X10-5 g/l for the data shown in Fig. 20,corresponding to fluctuations of about A Cl/CIo- 7.

V. CONCLUDING REMARKS

Previous laboratory studies of double diffusive systemshave almost always been conducted under transientconditions-for example, first a stable concentration gradientwas established and then the system was heated from below.Such experiments have yielded some insight into layeringand fingering phenomena but have not provided quantitativedata even for the primary instability. We have demonstratedthat a convection cell with gel-filled membranes at the hori-zontal surfaces provides well-defined boundary conditionsthat can be sustained indefinitely. We have conducted experi-ments and analyses for the case with a destabilizing densitygradient of the more rapidly diffusing species and a stabiliz-ing density gradient of the slowly diffusing species. The pri-

14 . a . ..... .. .

12

10 # ~~~~~(b) 1 4791 0

614.8

4

2

0

-20 24 48 72 96 120 144 168 192 216

Time (h)

FIG. 19. Log plot of "energy" vs time. Cell D, R,=633 (40 g/l of glyc-erol). At arrow (a), e was increased from -0.003 to +0.003 (concentrationof NaCI was increased from 14.8 to 14.9 g/l); at arrow (b), e was decreasedfrom 0.003 to -0.003 (concentration of NaC1 was decreased from 14.90 to14.79 g/I).

0

(h)

2

0 x/d 20

FIG. 20. Structured spatiotemporal noise slightly below the onset of insta-bility in cell A, R,=2190, Rfa 2618, e= -0.015.

mary instability leads to traveling waves with an onset Ray-leigh number and frequency in reasonable accord with theanalysis, which includes corrections for cross-diffusion ef-fects and for finite rather than infinitesimal thickness of theHele-Shaw cell. Standing wave patterns near the ends of thecell indicate that the reflection coefficient is near unity, incontrast to what is observed in thermal convection in binarymixtures.4 tX39

We made careful measurements designed to determinethe convection amplitudes near the onset of instability. De-spite repeated attempts, however, with increasingly finerresolution in control parameter and increasing sensitivity forpattern detection, we never observed a small amplitude as-ymptotic state, nor did we observe any hysteresis. At suffi-ciently small e, the theory presented predicts that, in an infi-nite system, the bifurcation should be forward and theamplitude should increase as eY'4. Calculation reveals, how-ever, that at e-0.005 , a value not reliably resolved in thepresent experiment, higher order terms become important.This means that the description of the phenomena in terms ofamplitude dependence versus e is not applicable to our ex-periment.

The Lewis number T in our experiments ranged from0.31 to 0.63, appreciably larger than in the binary mixturesthat have been studied, where 10-2. As a consequence ofour relatively large value of r, we have been able to examinethe behavior near the codimension-2 point, and we find thatthe observed frequency dependence is in reasonable accordwith the prediction [Eq. (7)].

Our novel convection cell design can be used in futurestudies of higher instabilities and turbulence in double diffu-sive convection for wide parameter ranges. This type of cellcould also be used to study two-dimensional Rayleigh-Benard density driven convection. Although three-dimensional Rayleigh-B1nard convection has been exten-sively studied for many years, two-dimensional convectionremains unexplored because no technique has previously ex-isted for accurately approximating the insulating side wallboundary condition.

3934 Phys. Fluids, Vol. 6, No. 12, December 1994 Predtechensky et a/.


ACKNOWLEDGMENTS

The cell design with gel-filled membranes was devel-oped by Zoltan Noszticzius. We thank Paul Kolodner for ananalysis of the data in Fig. 15(b), and Edgar Knobloch,Michael Cross, and Pierre Hohenberg for helpful discus-sions. This work was supported by the U.S. Department ofEnergy Office of Basic Energy Sciences and the Office ofNaval Research.

J. S. Turner, "Multicomponent convection," Annu. Rev. Fluid Mech. 17,11 (1985).

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7P. Kolodner, H. Williams, and C. Moe, "Optical measurements of theSoret coefficient of ethanol/water solutions," J. Chem. Phys. 88, 6512(1988).

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'J. Fineberg, E. Moses, and V. Steinberg, "Spatially and temporally modu-lated traveling-wave pattern in convecting binary mixtures," Phys. Rev.Lett. 61, 838 (1988).°0 C. S. Bretherton and E. A. Spiegel, "Intermittency through modulationalinstability," Phys. Lett. A 96, 152 (1983).

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"See Sec. IX.A of M. C. Cross and P. C. Hohenberg, "Pattern formationoutside of equilibrium," Rev. Mod. Phys. 65, 851 (1993).

17 H. Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cam-bridge, 1932), p. 582; L. L. Green and T. D. Foster, "Secondary convec-tion in a Hele-Shaw cell," J. Fluid. Mech. 71, 675 (1975).

18H. R. Brand, P. C. Hohenberg, and V. Steinberg, "C'odimension-two bifur-cations for convection in binary fluid mixtures," Phys. Rev. A 30, 2548(1984).

191. Stakgold, Green's Functions and Boundary Value Problems (Wiley, NewYork, 1979).

20E. Knobloch, "Oscillatory convection in binary mixtures," Phys. Rev. A34, 1538 (1986).

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22 M. C. Cross and E. Y. Kuo, "One-dimensional spatial structure near aHopf bifurcation at finite wavenumber," Physica D 59, 90 (1992).

23A. E. Deane, E. Knobloch, and J. Toomre, "Traveling waves and chaos inthermosolutal convection," Phys. Rev. A 36, 2862 (1987).

24R. Tagg, W. S. Edwards, and H. L. Swinney, "Convective versus absoluteinstability in flow between counterrotating cylinders," Phys. Rev. A 42,831 (1990).

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2M. C. Cross and K. Kim, "Linear instability and the codimension-2 regionin binary fluid convection between rigid impermeable boundaries," Phys.Rev. A 37, 3909 (1988).

29 H. Frick and R. M. Clever, "Einfluss der Seitenwande auf das Einsetzender Konvektion in einer horizontalen Fliissigkeitsschicht," Z. Angew.Math. Phys. 31, 502 (1980).

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32A. G. Rossberg, M. A. thesis, The University of Texas at Austin (1994),see Appendix A.

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36Cole-Parmer Instrument Co., Chicago, IL, Model 1481-90.3 7 MATROX Electronic Systems Inc., Quebec, Canada.38P. Kolodner, A. Passner, C. M. Surko, and R. W. Walden, "Onset of

oscillatory instability in a binary fluid mixture," Phys. Rev. Lett. 56, 2621(1986).

39E. Kaplan and V. Steinberg, "Measurements of reflection of travelingwaves near the onset of binary fluid convection," Phys. Rev. E 48, R661(1993).

40 G. Terrones, "Cross-diffusion effects on the stability criteria in a triplydiffusive system," Phys. Fluids A S, 2172 (1993).

41 C. M. Surko, and P. Kolodner, "Oscillatory traveling-wave convection ina finite container," Phys. Rev. Lett. 58, 2055 (1987).

42P. Kolodner (private communication). Similar splitting in k also was ob-served in thermal convection in binary mixtures, see Refs. 41 and 46.

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44M. C. Cross, "Structure of nonlinear traveling-wave states in finite geom-etries," Phys. Rev. A 38, 3593 (1988).

45A finite amplitude jump with no hysteresis was also observed in thermallydriven convection in binary mixtures: P. Kolodner, C. M. Surko, and H.Williams, "Dynamics of traveling waves near the onset of convection inbinary fluid mixtures," Physica D 37, 319 (1989) and Ref. 46.

46V. Steinberg, J. Fineberg, E. Moses, and 1. Rehberg, "Pattern selection andtransition to turbulence in propagating waves," Physica D 37, 359 (1989).

Phys. Fluids, Vol. 6, No. 12, December 1994 Predtechensky et al. 3935


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