,.~.

! "f '(·Ie.'" frow hoppers.

··It ."ill-Hl' apptu'atu,;,

.. i>eriments. Puu,deT

I fur till' prrdiet ion of

rhe flow of granular

--I -CDJ. FI.,i" .\I(rlt. (lfInl). 1'01. :!:?i. flli. 81-101.,

I'rintpd ill (;"f'II/ Uril"i"

Square patterns and secondary instabilities indriven capillary waves

By S. T. MILNER...\'1'&'1' Bell Labonltories. ~llIrril"" Hill. X.J 0797..1.. CSA

81

IWlIlatic- modelling of~4. :!~'j-2(jG.

: durin!! steady-state

i:utl obstacles - rr.

"f h<1\"iour of flowing

1·"If'f.~ hy POlYl/91/1ial

'1)('1'''. ("Item. EtlY'1g

IllJlX'n: and bunkers.

1 eonieClI lllaSS flow

I'llll'dcl' l'"rll1lol. 42.

I to ~oil mechanics1},""rll(Hlic'i (ed. C.

-

(Hf'eein~d 10 .Junllnr...· 1990 and ill re\"j:.;ro form 9 .lul ...· 1990)

Amplitude equations (including nonlinear damping terms) arc derived whichdescribe the c\'olutioll of patterns in lal'gc-aspcct-ratio driven capillary wavecxperimcnts. For dri\"c strength just abO\·c threshold, a reduction of the number ofmarginal modes (frolll tra\'elling capillary waves to standing wa.ves) leads to simpleramplitude equations. which ha\'c a Lyapuno\' functional. This functional determinesthe \\"C.l\·C'nulIl!Jer and symmctr,\' (square) of the most stable uniform state. Theoriginal amplitude C'quations. howe\"el". ha\'c a secondary instability to truns\"erseamplitude modulation (T.-\~l). which is not present in the standing·wa\·c equations.Tlw T.-\:\l in;-.;tability announces the restoration of the full set of rnarginal modcs.

1. Introduction

The aim of this work is to gi\"e a sy:;tcmatic account -of the formation of regularpatterns and their secondary instabilities in parametrically driven capillary wavesystems of large u:;pect ratio. The parametric instability of a fluid-air interfacedri,·en b.,' "erlieal oscillations \\"as fi"st obselTed and i''''estigated by Faraday (1831)and is as:-::ociated with his name,

In cell, slIfliciently large and deep compared to the resonant \\"a"elength of thepattern. and with a sufficiently clean frce slirface. the many complex processes ofdamping at the \\"alls pliles 1967). meni,ells (Co, 1986: Hocking 1987). and sllrfaeela.,·er plile, 1967) of the eellmay be neglected ..-\ large enollgh system also allo\\"s theeffects of the fillite cell size on pattei'll selection (Douad~· & Fau"e 1988) to beneglected. Tht" present work explores mechanisms of pattern selection and ~econdal'Y

instability whith depend on nonlinear intcractions between modes in un infinitesystem.

\\"hill' many experimcnts on dri\'en HLll'face WU\'CS h,:1\'e been performed on smallcells. i.e. on parametric resonance of 101l"-order modes (Douady & Fau"e 1988;Ciliberto & c.:ollub 1985: Simonelli & c.:ollub 1989). some experiments have also beenperformed (Ezerskii. Korotin & Rabino"ieh 1985: Ezerskii et al. 1986: Levin &Trubniko,' 1986: Tufillaro. Ramshankar & Gollub 1989) in lI·hieh the cell is on theorder of 100 \\"a"clengths on a side. These lalter experiments display severalremarkable featul'cs. including (i) ~qual'e patterns. c\"cn in cylindrical containers; (ii)a secondaI',\' instability to a pattern with tran::i\'crsc modulations of the arnplitude ofstanding wa\·c~. but with long-range order retained: (iii) at still higher drivestrength. a tran:;ition to spatiotemporal chaos.

Estimates of hOIl" big a cell is required for neglect of boundary effects to be a goodapproximation ma,\' be gi\'cn in terms of the various lengths in the problem. i.e. thesystern sizc IJ. the \"iSCOlIS penetration depth I. and the wa\'elength A. These estimates

82 s. 1'. J/ilncr

al'C sati:-:nccl b~' (':'\perimellts :-:illlilal' to HlOSC of' Ezerskii ef 01. (1986) and Tufillal'oel al. (I[)89): the requirement' are deri,·ed. following Miles (1967) and Cox (1986)in ..\.ppendix C.

Additiollall~·. large-aspcct-ratio ('rlls pl'o\"ide all essentially continuous spectrumof modes in \\"hic:h the system may I'espond to the drive: complications resulting Croma discl'cte spectrum arc absent. (The more subtle effects of boundary conditions onWH\'cllumber selertion described in ('1'08:-: el ai. (1!J8~) may in principle be present inlal'gc-usPc(·t-l'atio capillary wan~ c:clls. but \n~ will not cOllsider this possibility here.)

In the absente of boundary eflect:'!.. nonlinear interactions in the bulk of the cellbetwC'ell gl'O\\" ing tH pi lIa ry \\'a \'C':; \\"j II dd"crm inC' the s)' mllletr)' and wa \'elength ofregu lar pn ttC'I'Il~. III uch as the Jlrcfel'C'llce fur rolls of a gi \'CI1 wid th in H: aylcigh-Benardcells i, dl'tcnnilll'c1 by interactions betlreen ilH'ipient rolls in the bulk of the cell. In§2. Ire employ the methods of ;';elrell &: \rhitehead (196B) to obtain amplitudeequation:-: for thc neutral 1110<1(';-; of thl' ".\·:-;tcm in the absence of damping. \\'hich aretran.'lling capillary \\'an~8 with frequcllcy w C'qllal to the subharmonie of the drivefrcqIH.'IlC,\· :2w, Ezel'~kii el aI, (IOSG) first dCl'i\'C'd equations of this form (but WiUllineal' damping terms onl,\'. ancl codTicicnts which cliffeI' from the results of theprcsent \\'ork) under the a"sumptioll of a square pattern (which is obselTedexperimC'ntall,Y), To undcl'~talld the origin of the square patteI'll, we derive c\'olutionequations for pattel'lls of general symll"\etry. Headl\rs who are not concerned with themethod of derinltion may skip to the discussion immediatel,\' preceding equations(1-1)-( 1U).

\\'e dcri\'l' lineal' and nonlinear damping terms for thc amplitudes of the travelling\\'a\'es in §3, I.t tUl'llS out to be l\sscntial to consider nonlinear damping as amechani:-:1Il stahilizing the amplitwJc of rcgular pa.ttcrns, in the ph,vsical situationthat the :-;,\'stel11 (rathcr than the experimenter) selects the wa\'elength of the pattern.:\ny non·dissipa ti \·c mcchan ism for shifting the frequency of a growing mode at finiteamplitude ma." be (otall.,· '''Jmpensut.ed b.'· the system choosing a mode sufficientlyo fl' rC'sollancC': this leads in thc absence of nonlinear damping terms to unboundedamplitudes at any ,·alue of dri,·e abOl·e threshold, if the system is free to adjust thewa\'C'number globally, (It i~ po:,>siblc that for ~omc set of boundary conditions, globaladj ustmont of the wa \'CIlUIll bC'1' is su ppressed or el im ina ted, The in tell t of this paperis to ill\'('stigate what happells if this is not so, .\1 lIch of the paper. i.e, the del'i\'ationof thc amplitude equations \,·ith damping for arbitrary angles bet\\'ccn wa\'es (§§2and :3), dues not depend on this assurnption,)

The sct of amplitude equations for the tra\'elling capillar~' W<.1,\'CS, including thenon-linear damping tel'lns. is neither Hamiltonian. 1101' del'i\'able from a LyapunovfUllctional. Howc\'cl', for dri\'(.~ :-:trength:-: sufficicntl,\' close to threshold (determinedby thc damping strength. which is assumed to be small), the damping splits thedegenel'acy betwcen the grO\dh rates oft\\'o standing capillary waves (both of whichare neutral modes at zero dri\'e in the absence of clamping), This results in twostanding \\"i.1ves of diffcrcnt gI'O\,·th rates. and a non-zerO drive threshold for thefaster-growing standing waye. Thus fo!' dl'i\'e strength close to threshold: the numberof neut.ral modes is reduced by haiL H ieeke (1990) in a recent paper obtained (for theone-dimensional case) new, simpler amplitude equations by assuming essentiallythat the amplitude of the deca,\'ing standing W<.l,\'e is determincd adiabatically by thegrowing standing WU\'C, \Ve constl'uc.:t these equations, for the present case ofarbitrary patterns in two dimcnsions. in §-!, These simpler amplitude equations are(with some restrictions) deri\'able from u I"''\'apuno\' functional. Thus: the symmetryand wavcnumbcr of the most stable pattern Illay be found from Ininimizing the

The constituti,'c cquation for the pressure due to surface tension is given by 1) =

E(l', +l',), where c, and c, Ilre the principal rlldii of eUI'\'IltUl'e, which lead. to

The eqll1ltion for the LIlplace pressure is mellnt to be e\'alll1lted at the frcc surfllee,of coursc. so that. is the most convenient place to enforce the Bernoulli equation.Howevcr. it is inconvenient to cvaluate rp on the surface (as is also required for

(I)

(4)

(2)

p = -EV'lVs/(1 + (VS)')l].

~+V1>'Vs= i,V1>,

V'r/> = 0,

¢+!(V¢)'+p+!zeos2w/ = 0,

with the incomprcssibilit.'· condition

and the eon\'eeli\'e equlltion of motion for the height s(x. y) of the free surfllee

2. Amplitude equationsOur starting point for deri\'ing Ilmplitude equations for the FaradllY instability is

the set of idelll-flllid equIltions (Benjamin & Ursell /95-1) for the velocity potential¢(x, y, =),

Square lxlllprn.s ami secondary instabililies in drit:en f{llJillary wares 83

appropriate Lyapul1o\' functional. L"sing our results for the nonlinear couplings forplltterns of geneml ymmetry, we find that square plltterns indeed give the deepestminimum in the functional of the usual high-symmetry candidates (single wu,'cs,square and hcxagonal patterns).

In §5 we in"cstigate thc secondary instabilitics ofthc uniform square pattern, Thesimplified amplitude equation hl'lS both transverse (zigzag) and longitudina.l(Eckhaus) instllbilities (Eckhaus /965: Xewell & Whitehelld I!J69) of the usual sort;ho\\'el'er, the path of the most stllble state through the drive-wavenumber planenc"cl' intersects an Eckhaus boundary. Instead, it turns out that the full amplitudeequations, which contain the dynamics of the dccaying standing waves neglected inthe adillbatie approximlltion, display an instability of the uniform pattern totmns\'erse amplitude modulations (TAM) (Ezerskii ,I al. 1986) which is not presentin the sirnpler amplitude equlltions, Clearly, the smllil splitting between the twostanding wa,"cs induced by weak durnping docs not continuc to be important fol'inerellsingly strong dri\'e, The TA~1 instability explicitly depends on Ilnd indeed maybe said to announce the failure of the adiabatic approximation, and the return torelenlnce of the neglected modes, \\'e show thllt the pllth of the most stllble uniformstate collides with the TAM instability boundary.

Finall.'", in the concluding Scction wc make quantitative comparisons between thepresent theOl'y and the experiments of Tufilillro el al. (/989) which Ilppellr to sIltisfythe large-system requirements for "alidity of our theory. \Vc find semi-quantitativeagreemcnt in the value of thrcshold drive strcngth, as wcll as for the pattcrnamplitude, unstable wavenurnber, and drive strength at the TAM instllbility, Thepredictions for quantities at, the TAM instability, particularly the dril'e strength, arcrather scnsiti,'c to the ,'alues of the nonlinear couplings. Errors by a factor of thrcedue to neglect of higher-order terms in our expansions, or boundary effects, such asdamping within the ,'iscous penctration dcpth at the free surface due to surfacecontamination. would be sufficient to achic"e quantitath'c agreement.

fthc tl'an~J1ing

damping as a::oical sit uationortll(' pattern.

: mode at finiteIde suflicicntlyto unbounded, to Ildju,t theditiolls. globalI of this paperthe ch.'rin\tionell wa \"('S (§§ 2

including the'I a Lyapullo,"'I (dell'rrninedling :;plits theboth of which'CSlllt~ in two

'shold for the~1. the lIumberIlined (for the19 eS:icntiaJly,tiellll,\' by thecsent case ofequalions arche symmetryinirnizing the

llll\l!:-: Spl'l't I'll III

, re:-:llltilig from\" eondilions 011

IL· 1)(' pn..'H'nt in()s:-;ibilily here.)hulk of Ihe cell

\\"il \"l,It'llgth ofIylt·igh-Bt:nal'd'ofth,' eeli. Inrain ilmplitudeling. whi("h arcli(· of till' dri,-c·orm (but withI'e:,ult:-: of the

It is ObSCITCd

Pl'iH' ('\"olution

'emed with theling C'tjlliltions

i) :1/1(1 Tufilillroand Cox (/9S6)

S. '1'. .Ililllel"

.Y, = ?:Vi 9+K'il,Vi rp+ Vi ¢J. VS+!:VSC,' Vi rp. (8)s. = tLV' S(VS)' HLVs' V(VS)' +Sc, ¢-K'Vi ¢+t(Vrp)'+gc,(Vrp)'. (9)

and similarly for spacc and t,imc dcrinltivcs of ¢'We also cxpand thc prcssure to 0(1:"): thcn thc Bcrnoulli and sUl'face·convection

equations may be written

(5)

(7)

(6)¢-rv's+fScos2wf+.V.(rp.S) = 0,

. crp .S- cZ +·\(9· s) = o.

with thc nonlinear tcrms expandcd to third OI·der in the fields as

cornputing the motion of the slirface): instead. we expand 9 on the surface in termsof ¢J at-i":.= 0:

",. y. _. . crp(x,Y,Z)1 lY'C'rp,..(x. y. ,,(x. y)) - 9(·t. y. 0) + S(x. y) c= ,-0 +", c='"''

11'0 follow tho procedurc of "o\\'cll & II'hitehead (1969) to derive amplitudeequations for the <.1bo'"c h.nIJ'odynamics. The marginally growing or neutral solutionsurc the solutions of the linearized equations. In the present case. these arc the set oftl'a\'clling capitlar,\' waves. with any WUVcycctor. Because we arc driving the systempal'l1metricall!' (Landau & Lir.shitz 1976) at frequency 2w. we expect the rcsponse tobe at w. Hellce we take us the lowest-order solutions a sum over capillary waves atfl'cqucnc.," w with some set {kJ} of WU\"C\'cctOI'S:

So = Lap,-. '1')exp (ikJ.x-iwf)+e.e ..J

(10)

(II)rpo = -/w exp (kz)Lap;.'1')exp(ik j .X-iw/)+c.e.J

Horo k = IkJI and k_J = -kJ. tho "ariables x and X are two-dimensional. and thenuiables X ancl '1' arc slowly nU,\·ing.

.-\n expansion bookkeeping parameter € is introduced. with X = ex, l' = ft, V-+V+fVx ' c,->c,+fC'l" Hero V= Vx+Zc,. and Vx and Vx arc gradients in the plane."ote that the z-dependence of the fields is not slow. since the flow decays into thefluid on a Icngthscalc /.:-1: hence there is no 'ex ',

The fields a re ex panded as S= fSo+f'S, + f3S, + .... the dri ve as f = ffo + f'f, + ....The lineal' operator of (6) and (7) is expanded as L = Lo+fL, +f'L,+ ... , and thenonlinear purts as e.g...\i, = e:!'\11) +€aN~':.!) + ....

Equations (6) and (7) rna~' then be expanded ill powers of € to give

(12)

whoro If' '" (s. ¢J) and -" '" (.Y,. -"~). The first equation gi"es the neutral solutions. Thesecond equation has a soh'ability condition. which determines the group velocity inthe equation of motion for the {a}: when this condition is enforced, the secondequation gi\-es the second-harmonic response of lfI. The soh'ability condition for thethird equation gives further terms lineal' in gradients of aj' which shift the frequency

Square ]Ja/lerll" and "erondary ins/abilifies in tlri"ell capillary Icat'e8 85

( 15)

(14)

(13)

~ = C(o( k,M (M)'w Ck ( )+ 2k

or. with 'I = k,~k and ~k' = '1'+'1';, the relation

:3," 3w 0 3w °~'" = -,'1 +,..,,'1; +8'0'1-+ ""2,.. -1,,'" - II'·

The resulting equiltion of motion for aJ is gi\'en as (this result has the same forlllas the expression reported b." Ezcrskii e/ ai, (1986), but we find different ,'alues forthe coefficients 'l~p for thc case they consider. namely square patterns,):

, ilJ * :30) - 3iw 0" 3iw k- °o= aJ+ 40) a_J+ 2k (kJ, V)a i - 4k' v -aJ + 81c' ( i' V)-ai

- i111-' 10.1' aJ- i1ji,' a. a.• a~i (16)

with tho expressions for 1j~ relegated to Appendix A, The beha,'iour of thesenonlinear coefficients us a function of the angle between the jth and kth waves isdisplayed in figure I,

of the capillary wa\'{~:.; along the dispcr:':iion relation: in addiLion. the secondharmonics mix \\'ith the ncutral modes to gi\'c a resonant three-wavc response. whichgi\'es cubic terms in the equation of motion for Lhe {OJ}'

The incompressibility eondition V'!.¢ = 0 must be enforced order by order in € as\\'ell. The expansion generates the equations

'i/'1'" = o.}'i/'1', + 2V, V.I" 1'" = o.

'i/'¢, + 2V, V.I' 1', + V',- 1'0 = o.These ma,\' be :soh'cd order by ordcr, b.\· finding particular solutions ¢Jl Pl which satisfythe ith incompressibility equation. and then writing the ith non-resonant correctionas ¢, = ¢~h' +¢:P', whore 'i/'1',h, = 0, The solvability condition will be alfected by 1"P',but the non*reSOTHlnt corrcction rp(hl cxtracted at z = 0 frolll the l:th equation ofmotion ma," tri"iall." be oxtended to gi"e 'i/'¢l'" = 0 (and hence no elfeet on the ithincolllpre:':isibility equation) b,\' multiplying b,\' cxp(qz) with the appropriate q.

Some details of the expansion arc gi\'cn in Appendix ..\, The follo\\'ing aspects ofthe procedurc arc \\'OIth noting:

(i) The form of tht:> nonlinear terms in thc equution of motion for a j up to cubicorder must bc i'l1~.)la}..I2aj + iTlii,) al.' a _I.' a~J' \\'ith 'liP and 'Ijr? real as we ha \'C not yetincluded damping. Thc rcason is that thc capillary wa\'cs that these amplitudesmultiply must ha\'c frt:>qllcn(;ic~ and \\,a\'C\'ectors which sum to thc frequcncy andwa \'O\'ector of the wa \-e c:oITt:>sponding to (lj' i.e. the tCl'm~ must be resonant. Becauseof the requirement that the frequencies of the wa\'cs slim to w, there can be 110 termsof el'en order in the {a i }, This absence of. for exa ml'lo, quadratic terms suggests thathexagonal pattel'lls lI'ill not be fa"oured.

(Our expansion to third order in € neglects tcrms studied by Zakharov, L'\,o\' &~tarobinets (1971) of the form!a: a~.ai' lI'hieh gi,'os an amplitude dependence to thedl'i\'c coupling and a mechanism for limiting patteI'll amplitudes different from thoseof the present lI·ork. Theso torms aro 0(£') and aro negligible here,)

(ii) The form orthe lineal' gradient terms in tho equation for ui is prescribed by thedisporsion relation, That is, alt'./) = exl' (i~k,x- i~w(~k) /) should be a solution ofthe equations of motion. The diRpcl'sion relation fJ)'!. = I:/1·3 implics an cxpansion of theform

--

£'!, + "". and the

(12)

)ns, Theloeity in. second, for the'quene)'

( II )

=£/, V

'he plane., into the

(10)

(7)

(6)

(5)

I. and the

amplitudeal solutions'0 tho set ofthe systemresponse to.\' wa \'CS at

(8)

'¢)', (9)

e-con\'cction

{acc in tcrms

,~ yU'\\ 10II-- ,

\ 5 ill,,,

S. T . .Ililner

,,,III;

T(2, I:'-'-'-'_._._.-1' .

IIII

.'-::-~-:!:-:'-~~-:------'-'-'-'c::':-'~~:-',- 5-0.5 0 0.5 1.0

cos (0)

86

4

3 THI

2 '-Tlil

1.:;:----

""0-

-I

-2-1.0

FIGL:Rt: 1. The nonlinear rrc(llUm('~'-shift e<>efficient:; 7W and damping .c~mcients yW of thetravcllillg-Wa\-e amplitude equations (22) are displayed as a function of kJ·kk' the cosine of theangle between thejth and kth tnl\-elling wtln;s. Solid cun'e. '/1(11: dnshed curve. 1'111: dot-dashedcun'e. 'f~:): dot-dot-dashed cun·e. yC21.

3, DampingThe prel'ious section considered parametrically driven capillary waves in the

absence of "isaotls dissipation. The resulting amplitude equations arc Hamiltonian,and in fact arc cquh-alcnt to an expansion of the total mechanical energy of thesystem (kinetic plus sudilCe tension) in powers of (aj ),

',"c may add damping in the following way. First note that in the experiments ofTufillaro el ai, (1989) the ,-iseous penetration 0" skin depth is much smaller than thewa \"elength of the eapillar,\" wa,-es: hence the flow in the fluid is ideal over almost allof the "olume, An argument due to Landau (Landau & Lifshitz 1959) concludes thatthe contribution to ,-iseous dissipation from within the skin depth is small comparedto that in the bulk. So, to compute dissipation in the bulk, we must substitute thecomputed expansion for the flow, if> = 90 + ¢, + ¢" into the expression for the viscousdissipation in an incompressible Auid. which is

(17)E=-2'ljdl'( a'¢ )'.ax/axi

The resulting expansion for b' takes the form

E = -D'o'lal- DW lalla,.I'-D)Z' a'j a'_1 (l.Z a~" (18)

with D'O) = 8,/Ik", and explicit expressions for DW, i = 1,2 arc relegated to AppendixB.

One ma~' then assume a form for the damping terms in the amplitude equation.

<il = -(y,o'+yWla,nal-y)Z)a"a_,a~I+"" (19)

);ow we take the explicit deri,-ati,-e of the Hamiltonian implied by the amplitudeequation, the rclc\'ant terms of which arc

H = h'o, lall' + h)1' la)' la,I' + h)Z' aj (I._jaZ a~, (20)

with h'o, = 2Ik', h)~ = -Ik' TW for ; = 1,2. (The quadratie part of the Hamiltonian11 gives the trivial part of the equation of motion for aj' namely uj = - iwal , whichhas been absorbed into the time-dependent of al ,)

(\r c 11<1\"e neglected for si mpi ici t~· the II"a I·en um bel' dependence of the dam pingcoefficients, which b\' an cxtension of thc abovc rnethod would lead to an expansionin gradients of the li·near damping term as i·'( 1+ 3(k. V)'/ki),,) at this order. Thesecorrections are ~mall because thc relati\'e WU\'cnumber shift is small in the region of'interest.)

'1'0 summ'Hi7.c: \\·c ha\'e explicitly computed the coefficicnts of this amplitudeequation. including both nonlinear frequcncy shifts ('Jji~) and nonlinear damping(yj;'). b~· the sy'tematie expansion of Xell"ell & \\·hitehead (1969), for an wrbilrary setof' pattern wa\'e\"eclors {kilo The expressions for the nonlinear coefficients may befound in .'Il'pendiees A and B. The dependence on k).f" of these coefficients is shownin figure t. Jn the next seC'tions. we shall make usc of these general results to examinethe loeill and global stabilit,'o- of regular patterns of eapillar.\' wa\-es.

4. Standing waves

Thc amplitude equations with damping terms ilrc not H<:tmiltonian: rather, theyarc del'il'able from a Hamilt.onian plus Ihe dissipation functional (18). Mostimportant. they are not of ·relaxational· form: thus in general we cannot makesimplc arguments about the mOtit stable state of the dl'i\'en system as the cxtremumor some functional.

'J'he presence of damping splits the degencraey at threshold between the growthrates of the t\\·o neutral eigcnmodes at each kj (two degencrate standing waves): for1 = y. t.he groll"th rate of the f,"·ourable standing lI"al·e just becomes positive. II"hilethe other is - 2y.

Thus sufficiently close to threshold (noli" at1= y). the number ofrelel·ant neutralmodes is half the number of tra\'elling wa\'cs, This suggests dcri\-ing equations forthe amplitudes of standing lI"al·es. i.e. applying the Xewell-Whitehead formalismanew to thc POEs for the {aJ}. In particular. we may write new ncutral modes as

Square 1JallPrl/s and secondary il/slabililies in (fri,·CI' capillary w(wes 87

Equating the two expressions for the dissipation results in I' == 1'(0) = 2vk2, and the

following results for the yW:yj[' = Dji'/12h(o')-J)(o'hj['/lh'Ol)'. (21)

The beh3\"iour of this nonlinear coefficient as a function of the angle between thejthand kth \\"(;\\'C8 is shown in figure I. Some comments about these functions arc inorder. First. though indiddual coefficients ,(I) take on negati\'c \'alues for someangle:--. this docs not lead to negati\"e \"alues for the nonlinear contribution to thecnerg.\' dissipation. Second. the divcrgence of 'JJ(l) and y(1) at cosO ..... !I'esults from ashortcoming of' the computational method. For an angle such that k l +k2 has lengthtwice k o' a quadratic nonlincarit,'o' produces wa\'es at the dri\'c frequency which arcresonant. i.C'. tiatisfy the dispersion relation. Because equa,tions of motion 1'01' thesewa\'es were not retained in the calculation, spurious divergences appeal' in thecoefficients. This leads to di\-crgent damping of pairs of wa vc~ separated b.v an angleof 0 = COS-I 121-1) ~ 1-t.9°.

The eomplct"e equation of motion for the tra \'clling W3\"e amplitudes then takes theform

(22)

ilJ :3w - 3iw 0 3iw - 0o= (i) + i·'a) + -tW a~) + 2k (k)· V) a)- -!k' V-a) +8k' (k)· V)- aJ

+ (yj).' - i1')1') la,fa) + Iyj~' - i 'lj~') a, ,,_, "~j.

(20)

( I!))

(18)

ude equation.

d to Appendix

( Ii)

, Hamiltonian- iwoJ' which

the amplitude

'xpcl'iments oflUller than the>H'r almost all('oncludes thatnail compared,ubstitute thefor the \'iscous

·i('llts yj~~ of thetht.' tOi'ine of theill: dot-dashed

wa \'e~ in the. Hamiltonian.enerC1 \' of theo.

88 8. 'I'. J/ilner

(24)

(27)

proportional to the neutral cigcllstatcs of the linearized llj equations (as in Hieckc(990).

tn this \\'<.1,\" we arc led to examinc t he neutral stability elln"c of the llJ equationswith damping. The Iinearizl'd equation::; t:oupling (lj and a!J. which we assume to varyin space as cxp (iq·x) (i.e. we examine the growth of modes sl.ightly djtfercn~ inw<.l\"cnumbcr from k). an.'

o= [- iw +Y- iu . - if . ] [ "J ] . (23)if -Iw+y+ 1U a~J

Here the· detuning· u is gi,·en by u = - Ilw(q) '>; - (awjaq) Ilk. That is, the detuningis the difference in frequency between the subhal'll1onic of the drive, and the naturalfl'cquency of a \\,<1\'C with Wi:l'"C\"cctOI' kJ+q.

With the dri,·e strength f at the thrcshold valuef* = (y'+u')l, which defines theneutral stabilit.\" CUIT(-' in the (f. o-)-planc. the lincarized equations have eigenvaluesiw = 0 and iw = 2y. The ('orrc~ponding eigcll\"cctors arc (1, iQ) and (i.-iQ*)rcspecti,-el.,-. Hcre the phase Q is gi"cn b.,'

_ (y- iU)1Q= - ..Y+IO"

Hence the distinction between the two standing wa,'cs is thcir temporal phase withrespcct to thc dri,·c. since

S= (ClJcxplikJ·x)+CI_Jcxp(-ikJ.x))cxp{;-iwl). or Sec eo,,(kJ·x)expl-iwl+iO,).

whcrc cxp (2iO,) = iQ for thc ncutral modc and exp (2iO,) = - iQ* for the decayingmode.

The ncutral Illodcs may thus be written

((lJ.CI~J) = (I. iQ).·lp'. 'l')exp(iq·x). (25)

The derinltion of the stal1ding-\\'an~ amplitude equat.ions is straightforward tothird order. as Hlcrc is no second-hurmonic- gcneration. no complications due to a frccboundar.,'. and no subsidiar." equations to soh-c. Lt will turn out that \\'c arcintcrc;(cd in u of O(y'). For u"" y we obtain to third order

. u ( :litu - :3w· 0 ;lw 0) I (3W. )'O=AJ-;; -2k(kJ.V)+8k,(kJ.V)--4k'V- A J- 2y 2k1kJ.V) Ai

-(-Y€+ u')AJ+(fik+'!.'l'i,.)IAkl'A J. (26)2y Y

where TjA; == 'l11.1+'/1~)A.+'lji.I+'lj:?k' and r similarl.'· in terms of the yO). and thedimen,ionlcs" dri,·c ,trength € '" (j - y)jy. (This € i" unrelated to the bookkeepingparameter of §:!. '\'hich will not appear further in this paper.)

Here. as thl'Oligholit the di:-5f'llSsioll of the A J equation. indices in implied sums runonly oYcr 'po:-:;ili\'c' \·<.lIl1c:o; of the indC'x k: i.e. each wavcvcctor appears only once.

Again we may rnake a fC'\\' (;OrnnH'fltR about thc structurc of the AJ equat.ions,(i) The linear tcrllls are of thc f01'1Il

[ - y€+...!... (llw(V) - U)']rl J::ly

s,"stcmatic expan:--ion to higher order reproduces successi\-e terms in the expansionof l!1w in gradients.

Square patte.rns and sefolldary iusfabilif,ies in dril.:en capillat·y waves 89

(ii) The cubic term abo,'e can change sign for sufficientl:' negati"e u. suggesting theneed for a fifth-order tcrm. By carrying the expansion to higher orcler, the fifth~ordcl'

term can be obtained (1'01' u ~ y) as

(31 )

(29)

(28)

1'I'YE:u* = ----.

2 r

F(.4.u*(A)) = -yw.·J'+t"rA'. (30)

where r;:; L j r JA" and II j:-; lhe number of modes in the pattern, As the summands !JJ.:depend on the angle between rolls. r takes different "alues for each regular patteI'll,The fUIlC'tional (:10) ha~ the deepest minimum fol' a square pattern, among the choicesof single W~l\-C:-; (rolls), :-iquarc patterns. or hexagonal patterns. For rolls (n = 1),

r = 6I'k': 1'01' ,;quare,; (II = 2). r = I.nl'k': and 1'01' hexagons (II = 3). r = 32.291'k'.(Here II is the kinematic' ,·iscosity.)

The optimulll amplitude A for a gi\"en dl'i\"c is thell (neglecting O(A 6 ) terms)

~l("")+'l") )'" +('1"'+"'" ) ('J''' -'" )]i' 1'1' i'A4y.Ijl.: -jA;.Ijl jlt .I_jA; -.II.:l .ljl .'"1 A: 11.1 j'

The amplitude equation at cubic order is derinlble from a Lyapuno\' functional;the 0(.-15) tCrlll is not in general derinlble from a functional. Sufficiently neal' tothreshold. \\"here the 0(.4') term may be negleeted, \\"e may determine the most stablestate. To O(A .1). the functional is (omitting gradient terms)

~ol1lineal' damping. Ileglected by Ezcrskii ef ai, (1086). thus enters in an essentialway to determine the :-;table amplitude. \\'ith the detuning parameter u to bedetermined Ilot by the exppl'imenter but by the s,\'stem through its choice ofwavenumber. wc requirc r> 0 to ha,'e a finite response to dri,'e strength just abovethreshold.

Farther from threshold. \\"here the OrA') term in F matters. \\"c may still havc afunctional in a limited sense, under the assumption that the pattern is regular andof a known number of modes. \\'ith this :-5trong restriction we may determine theoptimal state. i.e. thc angle between standing Win'CS, the best ,'alue of u, ancl theoptimum amplitude, This is useful for analysing the stability of a square pattern withrespect to a shearing distortion. as well as the bchu,-iollr of u*(yE:) for lurger dri"eamplitudes.

We no\\" suppose a regular pattern (all i.·I)1 equal). ender the assumption that amechanisrn exists for the s,\'stem to adjust wa"elength (hence detuning) continuously,we may minimize F with respect to u to (ind the best deLuning for it given amplitude,as u* = -t'l,.-tp:! with '/';:; L". 'ill." (For square patterns. we hu,'e from Appendix Athat T = II.79wk'.)

Of course. it is pm-sible that the system arri"e5 at a pattern in whieh differentstanding wa,'es ha\-e different amplitudes, 01' c,-en different detunings; thesepossibilities may be in"cstigatcd on the basis or the standing-wave amplitudeequatiuns (26). (28) (generalized tn the ca,;e of dilferent detunings u) 1'01' the dilfcrentwa"c5). e,-en in lhe case when no functional exists. This possibility will not bepursued here.

Proceedings under the assumption of a regular pattern, and disregarding for themoment the ~Illall ncgati\'c term of 0(._1 6

) which is generated, the optimizedfunctional is

-

I phase \\"ith

(24)

tfonn.ll'd to~11I(, to a freehat we arc

(23)

(26)

HI. and theookkeeping

(27)

d :-lIms I'lIll0111:' once.IUaliull:-5.

expansion

,(-i,,,I+iO,).

he decaying

(25)

h defines the, cigen\'alucsnel (I. - iQ*)

the dctuningI the natural

(as in Riecke

, {IJ equations,:otl!ll(' Lo nu',\'

dillerent in

90 s. 7'. .IIi/"er

-2 -I 0 2UfJJ/y"

FIGl:R~; Z. Shown are the various stability boundaries in the (e. cr)-plane together wit.h the path oft.he most :stable uniform square pattern. ~olid parabola. neutral stability boundary (A = 0): solidline. locus of saddle node bifurcations (8:'\): long dashes. trans\·crsc·amplitude instabilityboundun" (T.-\~I): dotted line. Eckhaus instabilitv bOlllldan" (Eckhaus): short dashes, the moststable u~liform state. U*(f) (optimal). TIlt' mO:'it stable tll;iform stale collides with the T:\~1boundal'.".

,j Eckhaus.,...-,:Ii(iiJ .....A~O

)

\\<E- TAM

\,,,, .\ \ Opllm;lI

\ "Y\ ",

...... \ ""

J

o SN-'"

L"nc!e(" these assumptions. the functional becomes

P(A, u) = -Y€I1.~' +~ (u+~7'A ')"1.~'+~nT.4' +_J_n'l~ A'. (32)21' - - 2-!y

The optimum dctuning is u* = -rr.-l 2 as beforc; the optimum A is the solution of(;FlcA' = O. 01'

- T + (T' +1<7")1-I*'= ' (33)- 7"/41'.

\rc may then compute the depth of the minimum in ".... as a function of anglebCt\'"ccn standing waves in a cantcd squarc pattol'n, using the results of AppendicesA and B for l' and T as functions of the angle between the wa,·es. We find that thesquare pattern is indeed stable to changes in the angle away fl'olll ~rr. The trajectoryof the optimal dctuning u* as a function of dri\·c strength is displayed in figure 2.

The functional of (32). n,lid for regular patterns. has a saddle-node bifurcationwhen the fourth-order term has a negati\"c cOCtliCiCllt, 01' u'/'/y+T < O. (Sec figure2.) The saddle node occurs when the dri,·e decreases to the point at which P loses thepair of extrema which arc associated with the metastable state at finite amplitude A.This OCCurs when

(34)

The finite·amplitude uniform solution becomes metastable on a transition line at aslightly higher n,lue of dri,·e. Between the neutral stabiht." boundary and thetransition line. the A = 0 uniform solution is mctastable.

Perhaps unfortunately. it is not eddent that the physical system may beconstrained to respond to the dri\"c with patterns of a given wavelength,corresponding to some desired dctuning u. Left to its own devices, the system is freeto choose u = u*(y€). which ,,,·oids the region of metastability. (The detuningparameter u. ,,·hich was undetcrmined and phenomenological in the work of Ezerskiiet at. (1986). is in fact gi,·en by minimizing the functional of (32).)

(3.5)

, the path of1= U): solid

ill;<;tahility':'. till" mosth the 'LUI

:olution of

n of an~le

.ppendic·esd that thet rajet'l"or.\·\ fi¥u re :!.,iful'eatiunSee figu re

L"' lose:, theplitude.~ .

1 line at aand the

I may bel \-elt.'ngth.-('Ill i:< free

detllning)f Ezcrskii

--

-

Square jXltfer1l.S and secondary in.slabilitie.s ,:n driven capillary waves 91

5. Secondary instabilitiesFollowing standard methods. we may find the boundaries of stability of the

uniform pa.ttern of the standing-\\'a\'e amplitude equations by searching fortransverse (zigzag) and longitudinal (Eckhaus) instabilities (Eckhaus 1965). \·Veknow the relation between detuning and amplitude. from the funetional "alid forsquare patterns; and. from the equation for stationarity of the pattern, we have theexpression for drive in terms of detuning and amplitude. Thus we know the path ofthe most stable square unifonn state in the (e. u)-plane. One ean then eheek to seewhether the path collides with a stability boundary.

Lt turns out that the path collides not with an Eekhaus boundary, but rather witha boundary for a transverse amplitude modulation (TA:II) instability. deseribed byEzerskii el (Lt. (t986). The T.-\)L instability is nol found in the standing-waveequation, but rather in the full tra"elling-wO\'e amplitude equations, for thefollowing reason. The standing-\\'a\'c cquations were derived essentially by assumingthat the decaying, non-neutral combination of travelling wa\'es was a degree offrcedom whose amplitude was sla\'cd to the ncutral mode. i.e. tha.t its dynarnics wcreadiabatic. The full tra\'clling-wave equa.tions for the a} are (since a} and a~} arceoupled) seeond order in time. while the standing-wave equations omit half of themodes and arc first order in time. The TAM instability cxplicitly involves thcdynamics of the decaying standing wave, and hence the breakdown of the adiabaticapproximation which gave rise to the standing-wave equations. This secondaryinstability thus answers the question of how the system, at some drive f abovethreshold, reintroduces the degrees of freedom which were irrelevant at smaller 1-

\Ve linearize about the stationary solution of thc standing-wave cquation (withfifth-order terms included). i.e. the minimum of the functional of (32), which satisfies

u' ( (7) 7"0= -ye+?"+ r +- .-l;;+?"A~.-y y -y

'I"e perturb thejth standing wa,'e by writing AI=.-l.+(B'+iB"). where B' and B"arc small space- and tirne-dependent perturbations on the uniform wa\'C amplitudeA., and .-l,. = A. for k *j. With thejth roll taken along:i:. this leads to

o=/j,+...f..[-JlC'+Y'+"uC'lB,_3u iJ B"+'J[(r. +~T)A'+1"A']B' 12y ·I;r 16 y :! Y 2y I -}} Y JJ rT Y rT •

(36)

0 -1'>- t l .', .', 3 "1 /"" 3u, B"- '.) +-;- -i(.r+T6(y+:;uvy '.) --l.x ."y - 2y

with no implied sum on j. and wherc T! is given by

'I"~ = 7: T+'~ ('r.') +7'" )(7: - T. ) (:37)- II "-' II -II II II'I

L?or square patterns. the "alues of these eoefficients are computed from AppendicesA and B to be T = 9.I9wk'. 7il = 1.:33wk'. and rll = 4.82vk'.

As usual. the B' and B" a"e deeoupled in the ease of modulations tmnsverse to thewa"e"eetor k l of the roll we are perturbing (iJ, -> 0), and eoupled in the longitudinalcase. The transverse case gi"es a phase instability for rr> O. The longitudinal casegi\'cs an Eckhous instabilit\· when

(38)

92 S. 7'. Jlli/ner

As a matter of J1umerica,! convenience, the Eckhaus boundary may be plotted inthe (e. tT)-plane parametrically as a function of A, by solving (38) for tT(A), andcom pu ti ng I(A. tT(04 )) from the stationarity cond ition for a fin ite-am plitude pa ttel'll,

l' = (y+ro4')'+(tT+TA')'. (39)(See figUl'e 2.)

Some comments on the unusual shape of the Eckhaus boundary are in order. Vcrynear threshold. where the last term of (36) may be approxima.ted by 2riJ A;,B' and(35) similarly approximated, the Eekhaus boundary and neutral stability curves aregiven respectively by

(T'2 Tn (j2

-""2r.+[;01, -""oj.2y Ii 2y

Thus the usual factor of tis I'Cplaeed in this limit by 0/(2r+01) when dealing withpatterns with more than one set of waves.

The asymmetry of the Eckhaus boundary occurs in the vicinity of the emergenceof the line of saddle nodes (sec §4). This occurs at tT/w = -(y/w)(rw/y7') which forsquare pattel'lls is -0.16y/w.

Obse.'\"e finally that the optimal-state trajectory does not cross the Eekhausbounelal',\': this is intuitive. since the Eckhaus insta.bility is a mechanism for thesystem to locally adjust the wavenumber when global adjustment is impossible.

The instability analysis of the full uj equations is much more complicated, becausethe linearized equations result in a 4 x + system in the general case. However. in thecase of disturbances of one standing wave in a pattern transverse to its wavevectork j . the linearized equations decouple into two 2 x 2 systems. One system describes thecoupling of spatial phase and relat'i\'e amplitude degrees of freedom for left-andI'ight·mo\"ing parts of the standing \\'a\-e, and gives in the long-wavelength limit an. undulation' or transverse phase modulation mode, already encountered above.

The second 2 x 2 system describes the dynamical coupling of temporal phase ando\"el'all amplitude degrees of freedom of the left- and right-moving waves, and givesrise to the transverse amplitude modulation (TAiI'l) instability, which is not presentin the ..t j equations. It can be shown explicitly that the series of corrections to thestanding-\\'a\'e neutral mode (equation (25)) generated in the Newell-Whitehead­type deri\"ation of the standing-wu\-e equations is the same as an adiabaticapproximation on the dynamics of the temporal phase in the travelling-wave"quations (16). (17).

\\'e linearize about the stationar." solutions of (22). perturbing thejth tra\-elling\\'a\'e by writing G±J = exp (iO,)(A +b zJ ). and G±I = exp (iO,)A for k *j. Thedeeoupling of the 4 x 4 equations for the real and imaginary parts of b±J is achievedby a simple change of nll'iablc. Defining b == bj+b_j , we obtain

0= &' + ;;, Vi b" +2(tT+ T, A')b" +2T,b',

o = &" - ;;, Vi b' + 2(y+ r, A')b" - 2T,A'b',

(40)

(41)

where we have defined

1~ == L: 1'W + '1'jJl + 'l'~'!k J; == TW + T}]> + 11JJl + '1'~2Jj',.and similarly for r; in terms of ylIJ, For square patterns, the values of thesecoefficients computed from Appendices A and B are 1~ = 8"79wk2,l~ = 8.5wk2

, ~ =2.9+vk'. and T, = Gvk'.

(+3)

,Iutted in':..1). andiJutlcrn,

IIp!"_ 'Orry~; B' andII rye:;: arc

ling with

nergencerhich fnr

EokhausI for the:o',.... iblc.bcca lise

'I'. in the\'c\'edorribes theleft-andlimit anbo'·e.ta:;:e andnd gi\-e:-:pr ,ent

's to theilehead­diabati(·ng-wa \"(:'

'.I\'cll iIl!!j. The

.ehic\-ed

(+0)

(+ I)

,f these, .., ~ =

--

-

Square palJerns and secondary instabilities in dricen capilla.ry wa.ves 93

The condition for (+0). (·1L) to gi"e positi,-e growth rutes for disturbances is

+J;A '(y + r, A') + G:,q'- (7; + 7;H' -0")' - (0"+ ('I; - T,) .4')' < O. (42)

If 0"+('1',+'1;)..1' > O. then the left-hand side of the inequality is smallest forinstabilities at wa\'cnurnbcr q gi\-cn by

301. '" T. ...-,-'I' = (,,+ .).,'+0".+h', -Otherwise. the least stable q is q = 0: this leads to an instability for 0" < - (7;+'1;)..1' when 0" < -7;..1'-'1',' J;(y+r;)A'. This is ne'·er satisfied for the optimalstate,

Just as for the Eekhaus boundaries. (38). the limits of (+2) may be convenientlyplotted by ,oh·ing for 0".

0"" = -('I',-7;)A'±[+J;A'(y+r;A'lll. (4+)

and plotting yc:(u) paramctricall.\' as a function of A, (Sec figure 2,)\\'e may find analytically the location of the crossing ofthc optimal state. (33) ,,·ith

0"* = -~TA'. and the left-hand T.-\~l boundary. for y ~ (d: this occurs at

-2'1'yr..

(45)

:\ote that at this point. all terms in the standing-,,·a,·c functional!" of (32) are ofthe same order. Terms nut ('aleulated in F. of higher order in A/y, would all bc ofcomparable magnitude, Hence thc nllmerical \"allies of the coefficients of (45) maynot be [·eliablc.

Proeeeding with the calculated ,"allies for the nll'iolls cocfficients, the secondaryin"tability occurs at a dri"e "lrength abo"e thrcshold (j'-y')/y' = 2. = 5.81(y/01)'.with a detuning 0"1'0 = ~.2.'j(yl(u)'. an amplitude kA = 0.018YI01, and a modulational,,·a'·eIHimber q/k = ~.-lly/01.

For pr('senting the locations of the \"arious stability boundaries in the (c:, u)-plane,lI"e note thaI for y ~ o1. lI"e ma.'" scale kA = A'(y/01). r = ryk'. 0"/01 = O"'(y/01)', and• = e(y/01)'. and all dependence on y and 01 scales out (with eOITeetions of order(y/01)'). The resulting neutral stabilit~·. saddle node. Eekhaus. and 'L-Dl boundariesal'(' shown together with the optimal state trajcctol'~" in figure 2,

6. Conclusion0.1. Summary of Iheory

,rc hu\"e dC'ri\"cd two different amplitude equations to describe the evolution ofgeneral patterns in parametrically driYell capillary wa\"o s~'stcms, The first amplitudeequation is fol' tru\'clling ('upillal''y \\'a\'es, which arc the neutral modes in the absenceof dissipation, with a ,·anishing threshold dri,·c strength. These equations aresystematieall~"obtained in an expansion in the width of the band of excited modes,following ~cwell &, Whitehead (1909). Ezerskii el al. (1986) previously presented

9-J. S. 1'. .IIilner

these equations fol' the particular case of square patterns: we agree as to the form ofthese amplitude equations but disagree as to the values of the coefficients for squarepatterns.

To these Hamiltonian equations we add both linear and nonlinear damping terms,which arc calculated from the I'iseolls dissipation of energy in the blilk of the Auid,expanded in powers of the travelling W;l"C amplitudes, using the flow field results ofthe amplitude equation expansion. (Onl~' linear damping terms are p,'esent in thework of Ezerskii el 01. (1986); it turns out that the nonlinear terms play an essentialrole in selecting the square pattern. :see below.) The resulting equations arc notdcrhoablc from a functional. and thus may ha,'c dynamics which is not merelyI'claxational: one may not infer from these equations in any simple way the moststable uniform pattern.

The effect of I'iseous damping is to split the degeneracy of the growth rates of thetwo standing w(l\'cs (which arc both neutral modes as zcro drive in the absence ofdamping). resulting in two standing capillar," waves with different growth rates,Bieeke (1990) obsen'ed in a recent paper that for d"ive strengths sufficiently near thethreshold for the faster-growing mode, thc dynamics of the slow-growing mode maybe adiabatically eliminated. \\'e employ this idea to obtain a second amplitudeequation, for standing W3\'CS, with half the number ofdcgrees of freedom of the firstamplitude equation. This second cquation is deri\"ablc from a functional (for drivestrength ncar enough to threshold): thus, the most stable uniform pattel'll and itswu\'cnumbcl' may be found.

The quartic term in thc corresponding functional is a mcasure of the strength ofnon-linCH I' damping, which 'nters is an cssential way in determining the most stablepattern, '\'e find that squarc pattcl'ns (which are obscl'\'ed experimentally) areprefelTed among rolls, squarcs. and hexagonal patterns, Thc basic reason for this isthat square pattcrns ha\"e the weakcl:)t nonlinearity in the damping, which allows thcfunctional to attain the dec pest minimum (and the pattcrn arnplitucle to grow thelargest).

\\'e examine cach set of amplitude equations for secondary instabilities; thestanding-\I'l1\'e equations ha\'e the usual longitudinal Eekhaus and transvel'se phaseinstabilities (Eekhaus 1965: );e\\'ell & \\'hitehead (969). However, patterns of thefull tnl\'elling-wu\"e equations hu\'c a trans\"erse amplitude modulation (T.-\..i'l)in'tability. described b,l' Ezerskii el 01. (1986). \\'e find that the uniform patternwhich minimizes the functional from \\'hich the standing wave equations are derivedcollides lI'ith the TAM instability boundary. and compute the dl'ive stl'ength(j- Y)/j' = £. instability lI'a\'enumber q. and pattern amplitude A for which thisoccurs. Fa" weak damping (ylw ~ I), the quantities (k,.1)'. (qlk)'. and e all seale as(Ylw)' - ,"(pIL)I,"i.

6.2. C'ompar;/ion It:illl experimel/t

Two rctcnt expcriments 011 parametrically drh"ell capillary \\"a\"e8 (Douady &. F'atl\'c1988: Tufillaro el al. 1989) aI'(' candidates fol' eompal'ison to the pl'esent theol'y ofsystems of large aspect ratio, in which \'iscolls damping at the walls and the freeslirface oCthe fluid are neglected, as are cfreets due to a discrete spectrum of excitablemodes, \\'c re\'icw in AppendiX C thc \'al'iOlIS boundary contributions to dampingwhich arc neglected in the present calculatiOlI. to determine which of thc cxperimentsmay sensibly be compared with the prosent theOJ','"

\\'e conclude that the system of Tufillal'O el al. (1989) is indeed large enough thatdissipation at the walls of the containel' may be neglected. We find that sUl'faeecontamination cOllld possibl,\' contribute appreciably to the damping, but no history

he form offor square

in~ tCI'I11S.

the flilid.I'cslllt~ of'nt in thl'I essenl ial:) arc notIt merel,'"the most

IcS of theb~('nCl' ofth rates.. ncar thelode IlUl!'

mplitude'I he firstfor dl'h"C'I and its

cngth ofst stableIII.,·) are)I' thi~ is100n, the~I'OW the

It'S: the..:c phases of the(T.-\~I)

patterndCl'in'dtrcnf!thieh this:,calc a:s

Fa II '"l'

eor!" ofhe free:c:itablclIll pi ng-imcnts

!h that5urf'acchistory

-

Square pallems and sefOndary inslabi/ilies in driren Clll,ill"ry waves 95

dependence has been reported (.J. P. Gollub t!J89, private communication) fol'meaSllrements on n-butyl alcohol used in these experiments. If nonlinear surfactantdamping were important. its dependence on the angle between waves would have tobe ill\"cst igatccl to prescn-e the conclusions of §4 as to the global stnbility of squarepatterns.

The cxperiments of Douad)" & Fauve (1988) may not satisfy the assumptions ofthe present theory that the damping is confined to the bulk of the fluid. sing theexperimental parameters. we find that the ratio of dissipation at the walls of thecontainer to the bulk contribution is about 0.25. which may be neglected; and thereis no contribution to dissipation from a mO"ing contact line, since the meniscus ispinned in these experiments. Howe,"cl'. Lhe surface of thc water lIsed in theseexperiment:; is intentionally contaminaed with sUJ'face-acth"e molecules to stabilizethe surface tension. This Illay lead to large surface contributions to the damping. Ase,·idenee of this. thc threshold for the initial instability is reported to be about200 em/s'. \\·hile the theoretical prediction assuming only bulk damping is about60 cm/s'.

More important ma," be the eflects of a discrete spectrum in the experiments ofDouady & FaUl·e (1988). They obselTe in systems of smaller aspect ratio (£ - lOA)a great \'ilriety of pos~ibl.,· metastable states. First. they obsen'ed 'symmetric'superposit ions ofeigenstates sin(n1lx/,_) sin (II"'!I/") and sin (","x/£) sin (n1ly/L) withn and In ranging from 1 to about 15. These ma~' be regarded as four rolls of equalamplitude with W,1\'('\'('ctOI'S k = n.i±my and mx±nY. Second, they observed'dissymmetric' state:.:. ofa single cjgcnstatc sin (11rrxlJ....) sin (mrryi/"'), i.e. two rolls ofequal amplitudc and WU\'C\'ectol's k+ = mX±lly for angles between k+ and k_ greaterthan tn. -

Douad.,· & Fau,·c (1988) report that the modes with the smallest ,·alues of "'. i.e.tho:;e most nearly :-;quare (with thc lcast o\'cl'all modulation). hav"e the largestexistence uomain in frequency: this may be consistent with the global stability of the:;qual'c pattern in a large system. The~' also rcports that the threshold drive is lowestfol' the ::;mallc:;t rutio min for 111,2+ 11 2 constant: this is indicative of finite·systclllcHects. ~illce thc thrc:-:ihold dri\"c in a large system docs not ill\'oh'c nonlinear effects.and depends only on frequency (and not. for example. on pattcl'n symmetry).

The pl'c:;ent theory predicts that such nnitc·11I states arc not globally stable. in theabsenet> uf boundar"· etlects (including discrete spectrum etleds). In particular, the'dissymllletric' states cOl'respond to canted square patterns which the present workshows are not globall"· stable. The metastabilit.,· of these states could be in,·estigatedby repeating the anal,\'sis of §5 for these non·squHre patterns.

lI'ith high.,·iseosity fiuids (e.g. glycerol). Douad.\' & Fam·e (1988) observe anesscntially hexagonal pattern. For this case, the \'iscous penetration depth / is suchthat k/ <: I : most of the \'iscous damping is within the skin depth. and the presentcalculation of the frequency shift parameters 'lj~~ and nonlinear damping coefficientsyW· which a:-:sulIle Jd ~ I. arc not \·alid.

OUI' predictions for the primal'~' and secondary instabilities are in qualitati"eagreement with the experiments of Tufillal'o et a/. (1989); thc quiescent state givesway to a square pattern. which for slightly higher dri\'c becomes unstable totnlnS'"Crse amplitude modulation (TAM) of the standing \Hl\"es in the pattern.Howc\"er. thel'c are quantitati"e discrepancies. \\'ith experimental parameters w =3201l. ,,= 0.03. 11 = 0.8. and E = 2.J. (e.g.s. units), they measure a thresholdacceleration amplitude f* = 5..1y. This is lower than the theoretical ,·alue of f* =~wy/k = 8(f.

96 s. '1'. .I1i/",,.

This discrepancy cannot be due to damping processes not included in the theory.which \\'ould lead to an experimental threshold higher than the predicted one. Thediscrepancy may be due to non-uniform acceleration of the container: thedisplacement at threshold is only 9 Jllll. A direct measurement of y from the decaytime of free oscillations i~ difficult because of the finite response time of theelectromagnetic shaker used lo produce the wu\'es (J. P. Gollub 1989. pri\'atccommunication). (J n the cxpcrimcnts of Ezcrskii e/ at. (1986), thc observcd thrcsholdis rcported as f* = 4.2 g. higher than the appropriate theoretical value of 2.8 g.Ezerskii e/ al. (I !)86) claim good agreement without explicitly making thecomparison.}

The trans'·erse amplitude modulation ('L\~I) instability, first observed by Ezerskiie/ al. (1985. 1986). is indeed exhibited in the experiments of Tufillaro el at. (1989). [ntheir experiments. the TA~l instability occurs experimentally at € = 0.10, comparedto the theoretical prediction of € = 0.0 I. This could be explained by nonlineardamping coefficients which were a factor of three higher than the values predictedfrom bulk damping only. The corresponding values of pattern amplitude Smax = 8.1,instabilit.,· Wu\'cllumbcr q. and frequency detuning u (a measure of the wa\"cnumbcrshift of the pattern from its threshold ,·alue) are in better agreement withexperiment. The theoretical ndlles for the parameters of Tufillal'o el al. arc: {max =

95/1. k/g = 6.6. ane! u/w = 9x 10-'. Experimentally (J. P. Gollub. prinlte com­munication). k/q ...... 8. u/w is ullmcasurably small, and {max is consistent with

- 1001'.The mechanism fol' the determination of the globally optimal wavenumber of the

square pattern and the e,·cntual 'L,M instability are .-obust to changes in the valuesof the nonlinear coefficients which may result from the importance of highcr·ordcrterms in the standing-wu"e amplitude equation or new damping processes. Thescaling of the secondary instability with y/w for y/w <'! I is also robust and may betested by measuring the ~ccondal'Y threshold as a function offl'cqucnc.v (or \'iscosity,or slirfaec tension).

1 am indebted to Boris Shl'aiman and PielTe Hohenberg 1'01' a patient introductionto the subject of nonlinear dynamics. and the suggestion of this problem. I thankJCIT~' Gollub and R. Hamshankar for many useful discussions of their experiments:and acknowledge helpful cOIl\"ersatiuns with Paul K.olodncr: Jim Stokes, and EricHerbolzheimer. This research was supported in part by the :\ational ScienceFoundation under Grant :\0. PH182-1 ;853. snpplemented by funds from theXational Aeronautics and Space Administmtion.

Appendix A. ExpanSion of (ij

Here we prcsent some interlllediate results of the amplitude equation expansion:including the expressions for 'l1:': the terms in the equation of motion for aj generatedat each order: the non-resonant corrections tF'f to the SUIll of resonant modes tF'o: andthe particular solutions ¢lP'.

The expressions foJ' the '/I(i) are

'l"" kO [.' 3 I (5-Dc-16c'-4c')-(HJ +c))1(6-14C-4C')]= w ,. c- c--+ ~

2 4-(2(I+c))1 '(1\ I)

(1\ 2)

(E 3)

(E 2)

IE I)

(A i)

(A 6)

(A 5)

(A4)

(A 3)

'P, = -7 Cwl/ k) [ ~/: Ikl · 'iI) "I + i~, aj ]exp [ilkl · x -wi)] + c.c.

HIc L; (l)aj",exP[ilkl-k,).X]I" 0

+ tk f, (_i~b/k) "I (/" cxp [i(lk} + k,)' x - 2wl.)] + C.C.,

d = dl , '" WII +0))\13-0)-2(1 +,c)] [2(W +0))1- 1]-1,}b = b

l, '" [1-.50-20'][2(~11+C))'_I)-I.

c = el , '" kl·k,.Two terms generated to second order in the expansion of (i j not shown in (16) are

Integra tions by parts leads to

11: = - ~'Ifd' x[o; q,' + IV' + 0;)' Is + K')rjJ'l

where

Appendix B. Damping expansionEquation (16) for thc dissipation of energy c1uc to viscous clamping in the bulk may

be written using incompressibility as

which represent c1ctuning of the driving terll1 and second-order response to the drive,I'espeeti\·ely. These terms arc both small ill the experimenta.l cases of interest,becausc the dri,-e strength at thrcshold is O(y/w), \\'hich is small.

q,\P' = - ~~ exp (lcz) L; (k. 'iI) "lexP [i(kl · x -wi)] + c.e.,I

q,\'" = r;~' Ikl · 'iI)'''I+ ~~:~[V'''d(kj' 'iI)'''I+ y~: (k l · 'iI)"j]}x exp l i(kl · x - wi)] cxp kz

11-50-20') . - - .+lwz , [(k + k ). 'ilj("1 a. ), (W +o))\[(HI +0))'-2)] }' ,

x exp [i((k l +k,)' x- 2wl.)] exp Ik} +k,l z+ c.e ..

where in the above c = kjok j ,

The non-resonant corrections to lJIo = (~o, 1'0) are

the limits of integration depend on the wavy free surface of the liquid: and so maybe expanded

Squa're pallen18 and secondm·y instabilities i'n (hiven capilla'ry waves 97

The particular solutions ¢lUl) llsed in enforcing incompressibility for spatiallyvarying patterns Iwhcn thc z-dcpcndcnce cxp (lcz) docs not satisfy V'q, = 0), are:

nberofthethe ndues

ighel'-ol'c1el'·esses. Thenel may beI'riscosity,

expansion ..generated('s !Po: and

l] (A I)

(A 2)

ll'Oductioll11. 1 thankpel'imellts,;. and Erical Stience

from the

11'1': ~max =inlte COI11­~tellt with

the theol'y.·d one. The:<lIllCI': the1 the decay;me of the~g, pl'iYatcd thresholdIt' of 2.8 g.laking thc

by Ezel'skii. (1989). In" compared

nonlinears predicted~m:tx = 8.-1,

<.l H'llllll1 bel'lllcnt \\'ith

98 S. T. Milner

where in the above, V' aets only on (; and a, acts only on </>, which is evaluated atz = O.

To expand (83) to the order required in (18)," we must also have the non·resonantcubic correction lfI2 , which turns out to be

(B 6)

(B 5)

(B4)

where

'P, = L;4Wk'(. I .){B,ajlc'tI'+B,afa_fa!j}eXP[i(kj.x-wt)]+c.c.,jf ,w/k

13, = 1-2C+2C'+(!(!(I+C))ll5-8C-5C:+(I+C)'(1-8)]}

-;\(1 +c)( 1-8) - (1 +c)')(2(!(1 +c))'-lt',

13, = ~+c'+W -8_Jl ).

In the above, C '" kj .kf as before, and 8 = 8jf, the Kronecker delta.A tcdious and straightforward expansion of (B 2) then yields

DO' = -4l'w'k' [ -13, + 2b'(!(1 +c))l+ 2b( I +c)(1 + (!( I +C))l)'}

-¥l( I-c)' +;\(1 +c)'(1-8) + 8(1 +c)],

D'" = [ - 13, +!(I- c)'( I - L j ,.) - 2( 1- c')],

with 13" 13" b, (I, c, and 8 as in (B 5) and (A 6) respcetively.Equation (21) may then bc evaluated numerically for any cases of intercst.

Relc"ant numerical results for the various damping parameters are given in the maintext.

The above procedure may be extended to give damping eoeffieients which dependon the spatial derivatives of the a. j , terms neglected in the expansion (18) of theenergy dissipation. At linear order we obtain finally

(B 7)(ij = -2l'k'( 1+ 1:'" (kj .V),)a j+ ....

Rotational in variance arguments require the linear damping of a mode to }lependonly on its wavenumber, and not on the direction of its wavcvector. Hence (kj ' V) inthe above must be rcplaced in general by i6.kj(V), the expansion of which is

i6.kj(V) = (k j . V) - i(V'/(2k) - (k j .V)'/(2k)) +... . (B 8)

Thus we ha.\·e an expansion for y(V) which begins

(B 9)

(C 1)

Appendix C. Damping at boundaries[n addition to the bulk eontribution to energy dissipation computed in Appendix

B. there are three additional sources of dissipation in a deep cell of finite size L, withsome amount ofaclsorbed surface contaminants present on the free surface. These are(i) damping y", at the walls of the container; (ii) damping 1', at a moving contact linc(meniscus) along the wall of the container; and (iii) damping 1', due to a surface layeroffinite compressibility. We considereaeh of these in turn, following Miles (1967) andCox (1986).

At the sidewalls of the eontainer, the fluid velocity changes over a viscouspenetration depth 1= (21'/w)1 from the ideal-fluid bulk value v = V</> to v = O. Thusvelocity gradients are of order I-I; then (17) leads to

Ii; = (!l'W)IJdS' v',

Squa.re lXJfterns and ,'"Secondary instabilities in driven capillary waves 99

where the integral f dS' is o,·er the wall surfaccs. The integral o,-er depth f dz leadsto a factor (2k)-'. and we ma.,- replace ,. along the \1-,,11 at z = 0 by k9; this leads to

Ii:" = (~I'W); Ie'fdl/ \Q' (C 2)

The tolal energy of small o~cillations is twice the kinetic energy. which leads to

(Hcre 0" is thc static contact anglc.) The result (C 6) may be anticipated bydimensional analysis. as it must be proportional to III/-, which then forces alogarithmic dependence on the slip scale cutotl'. This gi\'cs risc to a contribution tothe lineal' damping coefficient y of the form

y, = -lwln,-'j(O.)KI'L-'. (C 7)

With the parameters of 'I.'ufillaro el. al. (1989). 00 - ~rr, and 8 = 10 A, the ratioy,IY = 0.17. This only gi'·es an upper bound on the dissipation at the contact line,because we hu\·c not taken into account contuct·angle hysteresis. Experimentalsystems t.vpicall~· ha\'e functions O(v) with a discontinuity 0h at v = 0 between a fewand a fcw tens of dcgrees. If the amplitude of surface wa'·es is small, so that thecontact angle for a fixed meniscus is within 0h of 00. the meniscus will not move, andno dissipation will occur at the contact line.

Finally. we considcr the effect on dissipation of a laycr of contamination (e.g.surf'acc-llcti\'e adsorbed molecules) at the free surface. Even if the two-dimensional,·isco,ity of this two-dimension,,1 liquid is neglected. the finit", compressibility of the

The rutio of Yw to the bulk contribution y to the lineal' damping coefficient is theny,jy = (Uk')-'. For thc e'periments of Tufillaro e( al. (1989), where D = 8 cm,w/2n = 1608- 1, I' = 0.03 em:.! S-I. and u ...... 2.1, erg cm-:!. we ha\-e I ........ 8 X 10-3 em,k = 32 em-I. and y,jy - 10-'.

Dis.~ipationat a mo\-ing contact line is a morc subtle quantity, as it depends on themicl'oseale at which the no·:slip boundary condition is \'iolated: a hydrodynamicdescription of a mO\'ing contact line without slip has a infinite dissipation rate(Bretherton 19(1). The slip ,c"le may bc set by some combination of molecularlengths and surfa(;c roughness. and may vary from material to material. Cox (1986)has analy-sed thc ,·elocity dependence of the contact angle to first order in e"pillar."number Ca. For the case of a dscous Auid displacing air he obtains

t>.O = j(O) Ca In .-'. (C 5)

\I·here 60 is the ,·cloeity-depcndent change in the macroscopic contact angle, j(O) =

2 sin (0)[0- sin (0) cos (O)r'. al1d thc capillary number is Ca = 'Ivl". The cutoffparameter, is, = sIR. where s is the slip Icngth and R is a macroscopic length atwhi(;h the surface is no longer flat. For the present case we make a conservativeestimate., = 10 A. and H = I.

To determine the dissipation at the mO\'ing contact line, we may compute thework done b.\· surface tension as the contact angle changes and the contact linemo\·c~. This gin~~ a dissipation pcI' unit length of order

/"

(C6)

(C 3)

(C4)

ti'll~ - ,.,,[,·os(00+60)-cos(00-6O)] -j(Oo) 'I 111 ,-1

J:: = !,pkfdS 9'·

Then the damping coefficient Yw '= £/(2E) is gi,-en by

yw=,~IID.

--(C I)

(138)

(135)

(B 7)

(B 6)

(139)

(B~I

;cous

Ihus

'nelixwith

:c arc

t lincla,\'cl'

I "1ll1

'pend.'1) in

epend)f the

~Cl'cst.. main

,ted "t

"onant

layer changes the flow in such a way as to increase dissipation within I of the frcesurface. (The extreme case is an incompressible surface la~·cr. which gi\-es additionaldissipation just as the side walls do.)

The situation has been analysed in detail b.,' ~Iilcs (1967): we brieRy summari ....c hisresults. l'he eomprcssional stresses in the surface layer must be ba.lanced byboundary-layer stresses resulting from some amOLlnt of slipping between the surfacelayer ancl the bulk over the skin depth I. That is. the stresses

must balance. here v is the fluid \'clocity a distance I from the frce surface; X is thebulk modulus of the surface la.'"er; and Cn is the relati"e "elocity of the fluid and thesurface layer.

Defining ~ '" (k'IPw'r1k'X to be a characteristic ratio between "iscous andcompressive stresses, Miles finds C = U(~ - 1 + i). Summing thc "iscous dissipationwithin the skin depth and the work done against the cornprcssi\"c stI'Cf'SCS. thecontribution 'Ys to the damping coefficient is found t,o be

I. lntroduc

The st udyproblem for\'('Ioeity ehal:-;urface appr(Gerwin JnUtsituations. ijyoung stellalactive gahl('l

TIl(' SllPl'I'~

analysis by ,\

Numerin

By JEJDepartlllt'lll

t ('urn'lll I

\\"e IU1\'c pelto confil'm ;structure:.-..!'eported inMajda (IDS,of' ~ll1all-UIll

indicate th"upon the )I

descri bed Ill'parabolic III

:-:i mlila tion~

nonlinear bl'WtH"CS is 11111

obsclTcd I'C:-:

incident \H\\

The natu!that disctls~l

:ipeeds neal' 1

disagreemen:,imulation:.:analytic l'('SI

dominate thThc role"

that the sta'

.1. FllIid .lhrll.

Prilllrft ill (;,w

(C ll)

(C S)

S. ']',1Uiiner

fy, = ~wJd (~_I)'+ I'

100

~ote that Ys is maximized for E= 2. which corresponds for the parametcl':i ofTufillaro el al. (I llSll) to X - 1.5 erg CI11-'. (This modulus would also be e"ident as areduction of the slIIface tension of a clean intcrface pli!es 196/): hence. a earefulmeasurement of the dispersion relation can place a bound on how large a X ma.\" bepresent.} The corresponding largest Ys/y ratio is about 2: hence a contaminatedsurface can lead to an obsen'ed y three times the bulk contribution. For y, to bc lessthan ty. we require X < 0.3 erg em- t .

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,.