instability of a viscous interface under horizontal ...ajuel/reprints/etalib/physfluids07.pdf ·...

Instability of a viscous interface under horizontal oscillationEmma Talib and Anne Juela

Manchester Centre for Nonlinear Dynamics and School of Mathematics, The University of Manchester,Oxford Road, Manchester M13 9PL, United Kingdom

Received 1 January 2007; accepted 22 June 2007; published online 18 September 2007

The linear stability of superposed layers of viscous, immiscible fluids of different densities subjectto horizontal oscillations, is analyzed with a spectral collocation method and Floquet theory. Wefocus on counterflowing layers, which arise when the horizontal volume-flux is conserved, resultingin a streamwise pressure gradient. This model has been shown to accurately predict the onset of thefrozen wave observed experimentally E. Talib, S. V. Jalikop, and A. Juel, J. Fluid Mech. 584, 452007. The numerical method enables us to gain new insights into the Kelvin–Helmholtz KHmode usually associated with the frozen wave, and the harmonic modes of the parametric-resonantinstability, by resolving the flow for an exhaustive range of vibrational to viscous forces ratios andviscosity contrasts. We show that the viscous model is essential to accurately predict the onset ofeach mode of instability. We characterize the evolution of the neutral curves from the multiplemodes of the parametric-resonant instability to the single frozen wave mode encountered in the limitof practical flows. We find that either the KH or the first resonant mode may persist when the fluidparameters are varied toward this limit. Interestingly, these two modes exhibit oppositedependencies on the viscosity contrast, which are understood by examining the eigenmodes near theinterface. © 2007 American Institute of Physics. DOI: 10.1063/1.2762255

I. INTRODUCTION

The periodic excitation of fluid interfaces can lead to awide variety of dynamics. Vertical oscillations, for instance,may either destabilize or restabilize fluid interfaces. Faradaywaves are commonly observed at the interface between sta-bly stratified fluids,1 while the Rayleigh–Taylor instability,which arises when a denser liquid is placed on top of alighter one, may be suppressed by vertical oscillations.2 Os-cillations applied parallel to the flow have also been found tosuppress instabilities, for example in a viscous film flowingdown an inclined plane.3 Similarly, Coward andPapageorgiou4 showed that the inclusion of backgroundmodulations parallel to a basic two-layer Couette flow couldstabilize long wavelength instabilities driven by viscositystratification, over a wide range of viscosity contrasts.

When superposed layers of immiscible fluids are oscil-lated horizontally, they are accelerated differentially becauseof their different densities. This results in a parallel, time-periodic shear flow, which also depends significantly on theviscosity contrast between the fluids.5 Interfacial instabilitiesin the form of standing waves have been observed in bothcylindrical geometries6,7 and rectangular vessels,5,8–10 wherethey are commonly referred to as frozen waves. In the rect-angular vessel, the presence of end walls imposes a zerohorizontal volume-flux, resulting in a streamwise pressuregradient which yields counterflowing fluid layers. Lyubimovand Cherepanov11 examined the stability of superposed lay-ers of inviscid fluids contained between horizontal oscillatingplates with a zero horizontal volume-flux. By decoupling fastoscillatory motion from the mean flow in the limit of large

frequencies and vanishing amplitudes of forcing, they de-rived, on average, a steady streamwise pressure gradient con-tribution to the mean flow equations. Their linear stabilityanalysis yielded a dispersion relation analogous to the steady,inviscid Kelvin–Helmholtz instability.12 For layer depthslarger than the capillary length, the most unstable mode ex-hibits a finite wavelength determined by the capillary length.Khenner et al.13 extended this inviscid analysis to arbitraryfrequencies and amplitudes of forcing. Similarly to Kelly14

who considered layers of infinite depth, they reduced theinviscid linear stability problem to a Mathieu equation. Thus,they found parametric resonant regions of instability associ-ated with the intensification of the capillary-gravity waves atthe interface.

In this paper, we extend the above linear stability modelto include the effect of the viscosities of the liquids in layersof finite depth. Our aim is to gain an overview of the modesof instability arising in this system, and their sensitivity tothe viscosity contrast. As demonstrated by the excellentquantitative agreement between experiments and computa-tions for the onset of the short wavelength frozen wave,5 thisviscous model is sufficient to quantitatively predict the onsetof instability observed in a vessel with end walls except inthe close vicinity of the walls. In this limit, however, theneutral curves exhibit only one mode of instability in con-trast with the multiple resonant parametric modes presentedby Khenner et al.13 for a wide range of parameters.

In contrast with the Faraday instability,1 the existence ofa basic shear flow necessitates the numerical solution of thelinearized perturbation equations governing layers of finitedepth for perturbations of arbitrary wavenumber. Analyticalsolutions have only be obtained in the specific limits of lay-ers of infinite depth7 and long wavelength perturbations.15aElectronic mail: [email protected]

PHYSICS OF FLUIDS 19, 092102 2007

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http://dx.doi.org/10.1063/1.2762255

http://dx.doi.org/10.1063/1.2762255

Kamachi and Honji15 investigated the stability to long wave-length perturbations of two fluid layers subject to a time-periodic horizontal pressure gradient in the limit of zero in-terfacial tension. The configuration they considered differsfrom our system in that the external oscillatory forcing wasimposed through the pressure gradient, so that their basicflow was neither driven by the density contrast between thelayers, nor required to satisfy a zero horizontal volume-fluxcondition. As a result, they found that layers of equal densi-ties could become unstable provided they had different kine-matic viscosities.

A numerical solution of the linear stability problem un-der consideration in this paper was previously obtained forarbitrary wavenumbers by Khenner et al.,13 using finite dif-ferences to approximate the spatial dependence of the flowand a Fourier expansion to resolve its time-dependence.They focused on layers of equal viscosities, but could notresolve low viscosity flows with nondimensional frequencies=d2

2 /2360, where is the angular frequency of forc-ing, d2 is the depth of the upper layer, and 2 is the kinematicviscosity of the upper layer. We employ a pseudospectraldiscretization scheme to approximate the spatial dependenceof the perturbed flow field. Rather than integrating the result-ing differential equation in time,3 we apply the fast iterativescheme introduced by Or16 to calculate the marginal stabilitythresholds. This method yields excellent resolution, and thus,enables us to reach the low viscosity limit with values of4104, as well as to investigate a wide range of viscos-ity contrasts 1N1=2 /13104, where 1 is the kine-matic viscosity of the lower layer.

The model is formulated in Sec. II and details of thenumerical method are presented in Sec. III. When capillaryforces are significant compared to vibrational forces and thedensity and viscosity contrasts are moderate, our model pre-dicts a parametric instability with neutral curves comprisingsuccessive resonant tongues. These are discussed by com-parison with the linear stability predictions of the inviscidmodel13 in Sec. IV A. The lowest order mode, commonlyreferred to as the Kelvin–Helmholtz KH mode, undergoes atransition from long to finite wavelength, which is addressedin Sec. IV B, for arbitrary values of the viscosity contrast.We show in Sec. IV C that the viscous model is essential topredict the critical amplitude and most unstable wavenumberof each resonant mode accurately, particularly when the vis-cosity contrast is large. We find that the shape of the neutralcurves varies significantly with the relative influence of cap-illary, gravity and vibrational forces. The thresholds of theKH mode and the first resonant mode are often of the sameorder, compared with the larger thresholds of the higher or-der resonant instabilities. Thus, we focus on the evolution ofthese two modes with changes in the fluid parameters in Sec.IV D. In Sec. IV D 1, the decrease in capillary forces towardthe realistic values of the interfacial tension for which thefrozen wave is observed, indicates the persistence of the firstresonant mode and the disappearance of the KH mode usu-ally associated with the frozen wave. The variation of thedensity contrast in Sec. IV D 2 yields regions where eithermode is the most unstable and thus a value of R1=2 /1

where 1 and 2 are the densities of the lower and upper

layer, respectively for which the threshold of both instabili-ties are the same. Thus, the dynamics may be affected bynonlinear interactions between these linearly unstablemodes.17,18 Intriguingly, the two modes exhibit opposite de-pendencies on the viscosity contrast. Only the KH mode,which becomes increasingly unstable, persists for large val-ues of N1. The respective instability mechanisms are eluci-dated with the time-averaged perturbation stream functionsin Sec. IV D 3.

II. FORMULATION

A. Governing equations

A schematic diagram of the model is shown in Fig. 1.We consider two superposed layers of incompressible andimmiscible fluids bounded above and below by rigid plates.We choose a Cartesian coordinate system where x* is parallelto the undeformed fluid interface, which lies at z*=0, and z*

is parallel to the acceleration of gravity. The plates are oscil-lating sinusoidally, which in the frame of reference of themoving plates, corresponds to a sinusoidal external accelera-tion of the fluids in the x* direction. The superscript * denotesdimensional variables. The denser fluid is placed in the lowerlayer, so that the configuration is gravitationally stable. Eachfluid layer is characterized by a density , a kinematic vis-cosity, and a height, d, where the subscripts =1,2 de-notes the lower and upper layers, respectively. The interfacialtension between the fluids is denoted by . Using d2, a,−1, and 2a /d2, where 2=22 is the dynamic viscosityof the upper layer, as length, velocity, time, and pressurescales, respectively, the dimensionless equations governingthe two-layer flow, in the frame of reference of the oscillat-ing boundaries, are

v

t+ Av. v = − R p +

1

N

2v −G0

Ak

+ costi , 1

. v = 0, 2

where v= u ,w is the velocity in each layer.= d2

2 /2 and A=a /d2 are the dimensionless frequencyand amplitude of oscillation, respectively, R=2 / andN=2 /, where R2=N2=1, are the ratios of densities and

FIG. 1. Schematic diagram of the two-layer fluid system in the frame ofreference of the oscillating rigid boundaries. The unperturbed interfacedashed line coincides with the x* axis and the perturbed interface is shownwith a solid line. The superscript * denotes dimensional variables.

092102-2 E. Talib and A. Juel Phys. Fluids 19, 092102 2007


viscosities, respectively, and G0=g / d22=A2 Fr−1, whereFr= a2 /gd2, is a modified inverse Froude number.

Since the rigid boundaries are stationary in the oscilla-tory frame of reference, the no-slip conditions are given by

v1 = 0 at z = − d and v2 = 0 at z = 1, 3

where d=d1 /d2 is the ratio of layer depths. The primitiveform of the interfacial conditions, comprising the kinematiccondition, the continuity of velocity, and the balance of nor-mal and tangential stresses, at z=x , t, where measuresthe interfacial deformation, are expressed respectively as,

1

A

t+ v1 . = v1 . k , 4

v1 = v2, 5

n . 1 . n − n . 2 . n =

WeA · n , 6

t . 1 . n = t . 2 . n , 7

We= 2d232 /=We/A2, with We=2a2d2 /, is a

modified Weber number, denotes the stress tensor in eachfluid expressed as

= − pI +1

RN

v + vT ,

where I is the identity matrix. n is the outward, normal unitvector pointing from fluid 1 into fluid 2 and t is the tangent,unit vector on the interface given by n= − /x ,11+ /x2−1/2 and t= 1, /x1+ /x2−1/2, respec-tively.

In addition, following Refs. 11 and 13, we enforce a zeronet volume-flux in the x-direction,

−d

v1 · idz +

1

v2 · idz = 0, 8

in order to model the counterflowing layers generated in ves-sels with endwalls.

B. Base flow solution

The base flow, v , p, is periodic, parallel to the hori-zontal boundaries and the interface remains unperturbed. Ittakes the form,

uz,t = ReitAemz + iSR

− 1 + Be−mz , 9

FIG. 2. Base flow solution at t=0, cal-culated for R1=0.49, G0=1.9910−1,

We=2.0102, d=1, and A=0.4 anddifferent upper layer viscosities sothat N1 remains constant: a N1=1and =3.927104; b N1=102

and =3.927102; c N1=103

and =3.927101; d N1=104 and=3.927. Note the thinness of theboundary layers in the lower layer,where the nondimensional frequencyis 1=N1=3.927104.

092102-3 Instability of a viscous interface Phys. Fluids 19, 092102 2007


pz,x,t = R−G0

RAz + Sxeit + C , 10

where m=iN with i= −11/2, the integration constantsA, B, and S are determined by imposing conditions 3, 5,7, and 8 and C is an arbitrary constant. Their analyticalform is detailed in the Appendix. Examples of base flows att=0 are shown in Fig. 2 for four values of the viscosity ratioN1=1, 100, 1000, and 10000. The profiles evolve consider-ably in both layers despite the constant nondimensional fre-quency in the lower layer, 1=N1=3.927104.

C. Perturbation equations

According to Squire’s theorem, which was extended totwo layers by Hesla et al.,19 it is sufficient to consider onlytwo-dimensional disturbances to the base flow. This choice isalso justified by the experimental observation of a two-dimensional frozen wave. In deriving the stability equations,we express the governing Eqs. 1 and 2 in terms of thestream function, , defined as

u =

z, w = −

x,

and seek a normal-mode solution20 of the infinitesimally per-turbed base flow of the form

,p, = , p,0 + z,t,Pz,t,hteikx + c.c.,

11

where k=2d2 /* is the dimensionless wavenumber of thedisturbance, * is the dimensional wavelength, and c .c. de-notes the complex conjugate. By substituting 11 into theNavier-Stokes equations 1, as well as the boundary andinterfacial conditions 3–8, subtracting out the base state,eliminating the pressure term, and neglecting higher orderterms, we obtain the following Orr-Sommerfeld equationsfor the two-layer flow:

t+ ikAu − k2 − ikAu

+1

N

2k2 − k4 − = 0, 12

where the prime denotes partial differentiation with re-spect to z. The no-slip boundary conditions become

1 = 0, 1 = 0 at z = − d and 2 = 0, 2 = 0 at z = 1.

13

The interfacial conditions are posed at z=x , t, the un-known position of the disturbed interface. However, since weare interested in the linear stability of the undisturbed inter-face, we can expand the flow variables and theirz-derivatives as a Taylor series in about z=0, and retainonly the leading order terms in . This approximation is validprovided the Taylor series remains well ordered. The linear-ized interfacial conditions, applied at z=0, become

1

A

h

t+ iku1h + ik1 = 0, kinematic condition, 14

1 − 2 + hu1 − u2 = 0, continuity of velocity, 15

1 − 2 = 0, continuity of velocity, 16

R1

1

t−

ikA

R1u11 +

ikA

R1u11 +

3k2

N1R11

−1

N1R11 +

ikG0

AR1h −

2

t− ikAu22

+ ikAu22 + 3k22 − 2 +ikG0

Ah +

ik3

We Ah

= 0, normal stress balance, 17

1 + hu1 + k21 − N1R12 + hu2 + k22

= 0, tangential stress balance. 18

III. NUMERICAL SOLUTION

The partial differential system 12, with the associatedboundary and interfacial conditions 13–18, is solved nu-merically for arbitrary wavenumbers using a spectral collo-cation scheme, and an iterative method based on Floquettheory16 to resolve the time-dependence.

A. Spatial discretization

The solution of 12 is approximated by expandingz , t as a Lagrange polynomial series, truncated at theLth term,

z,t z,t = j=0

L

jtCjz , 19

where jt are the unknown time-dependent amplitudes tobe determined, and Cjz are the Lagrange basis functionsdefined by

Cjz = i=0,ij

L z − zi

zj − zi, j = 0,1, . . . ,L, 20

satisfying

Cjzi = ij . 21

In the spectral collocation method, the function z , t isrequired to satisfy Eq. 12 exactly at L+1 collocationpoints zi. The collocation points are chosen to be theChebyshev-Gauss-Lobatto points defined on −1,1, in orderto achieve high resolution near the boundaries and the inter-face. For this purpose, the lower and upper layers aremapped onto Chebyshev space, −1,1, with the transfor-mations

=2z + d

dand = 1 − 2z , 22

respectively, so that the interface is placed at =1. It followsthat the system of Eqs. 12–18 in remains the same ex-cept the nth derivative in z must be replaced by



n

zn = qn n

n ,

where q1=2/d for fluid 1 and q2=−2 for fluid 2. Thus, 19becomes

i,t = j=0

L

jtCji, i = 0,1, . . . ,L, 23

where

i = cosi

L

are the Chebyshev-Gauss-Lobatto collocation points, whichare the extrema of the Lth order Chebyshev polynomialgiven by

TL = cosL cos−1 .

The Lagrange basis polynomials based on the Chebyshev-Gauss-Lobatto collocation points are defined by

Cj =− 1 j2 − 1TL

cjL2 − j

, j = 0,1, . . . ,L,

where c0= cL=2 and c1= ¯ = cL−1=1.

Differentiation of with respect to is expressed interms of the derivatives of the basis functions evaluated atthe collocation points,

n

ni,t = j=0

L

jtDijn , 24

where

Dijn = q

n n

n Cji, n 1, i, j = 0,1, . . . ,L.

i , j are the row and column indices of the nth order derivativematrix Dij

n , respectively. The first order derivative matrix,Dij, for the Chebyshev-Gauss-Lobatto collocation points canbe written in explicit form as

Dij =ci− 1i+j

c ji − jif i j ,

−2L

2 + 1

6if i = j = L,

2L2 + 1

6if i = j = 0,

− j

21 − j2

if 1 i = j L − 1.

25

Higher order derivatives are multiple powers of Dij, that is,

qn n

n Cji = qnDij

n = qnDijn, 26

where Dijn is the nth power of the matrix element Dij.Following Lanczos’s method,21 the governing equations

12 are evaluated at =i,

q2Dij

2 − k2Iij j

t

= 1

L

q4Dij

4 − 2

L

k2 + ikAuiq2Dij

2

+ 1

L

k4 + ik3Aui + ikAui Iijj , 27

where i=2,3 , . . . ,L−2, j=0,1 , . . . ,L and I is the identitymatrix, and the boundary conditions 13–18 are evaluatedat the boundary collocation points =1 and =−1,

ILjj = 0, 28

qDLjj = 0, 29

1

A

h

t= − ikI0j,− iku101j

h , 30

q1D0j,− q2D0j, u10 − u20 1j

2j

h = 0, 31

I0j,− I0j1j

2j = 0, 32

R1q1D0j,− q2D0j,01j/t

2j/t

h/t

= 1

N1R1q1

3D0j3 − 3k2q1D0j −

i

R1kAu10q1D0j

+ ikAu10 I0j,− q23D0j

3 + 3k2q2D0j + ikAu20q2D0j

− ikAu20 I0j,ik

A 1 −1

R1G0 − k2

We1j

2j

h ,

33

1

N1R1q1

2D0j2 + k2I0j,− q2

2D0j2 + k2I0j,

1

N1R1u10 − u20

1j

2j

h = 0. 34

Thus, we obtain an L1+L2+3 L1+L2+3 system ofthe form

Bx

t= M0 + Mtx , 35

where x is the unknown vector

xt = 10,11, . . . ,1L1,20,21, . . . ,2L2

,hTt ,

Mt=Ms sint+Mc cost is a 2-periodic matrix, and Band M0 are constant coefficient matrices.



B. Temporal solution

The Floquet problem 35 is solved using a fast iterativescheme based upon the Newton-Raphson method that wasdeveloped by Or16 for a viscous single layer under oscillatingshear. According to Floquet theory,22 the solution of 35 isof the form

xt = Ztexpt , 36

where Zt is a 2-periodic vector function and is theFloquet exponent. Zt is expanded as a complex Fourierseries, truncated at order K, so that

xt = n=−K

K

xneint+t, 37

where =r+ ii and xn are constant vector coefficients, andsubstituted into the differential Eq. 35. We focused on har-monic solutions with i=0 because subharmonic solutionsare not generally found in this system due to constraintsimposed by the conjugate-translation symmetry.23 By substi-tuting solution 37 into Eq. 35 and employing the iterativeprocedure of Or,16 we obtained the following polynomial ei-genvalue equation of order K+1 on x0,

M0 − B + M*R−1 + MR1x0 = 0, 38

where M= 12 Mc+ iMs and M* denotes the complex conju-

gate of M. Within the formulation of 38 we have used thefollowing recurrence relations: for 1nK−1,

xn = Rnxn−1, Rn = − M0 − Bin + + MRn+1−1M*,

x−n = R−nx−n−1, R−n = − M0 − B− in +

+ M*R−n+1−1M ,

and for n=K,

xK = RKxK−1, RK = − M0 − BiK + −1M*,

x−K = R−Kx−K−1, R−K = − M0 − B− iK + −1M*.

Equation 38 admits a nontrivial solution if and only if thedeterminant vanishes.

To locate the critical values of A, Ac, on the marginal

stability curve, we fixed ,R1 ,N1 ,We,G0 ,d ,k ,i=0 andincremented the value of A from A0=0.001 until we identi-fied the value of A for which the growth rate, r, satisfyingEq. 38, first crossed zero. r was obtained by iteration of asuitable initial guess, guess=2. This value was found to besuitable for the entire parameter range investigated, since thesystem was only solved up to the point of marginal stability,i.e., for values of A, A0AAc. By iterating on guess, weevaluated successive values of the determinant and deter-mined an estimate of r accurate to two decimal places,approx, at which the determinant first crossed zero. The ap-plication of a Newton-Raphson iteration to approx yielded r

for each value of A. Once the first zero-crossing of r waslocated, the associated eigenvector x0 was evaluated usingsingular value decomposition x0 is the singular vector cor-responding to the smallest singular value.

In order to reconstruct the stream function in each layer,, the solution vector, xt, was formed using Eq. 37 andthe relevant component part substituted into Eq. 23. Theremaining Fourier vector components, −KxnK, were de-termined using the recurrence relations give above.

C. Convergence tests and validation of the code

All numerical computations were carried out inMATLAB. Each linear stability result was checked for con-vergence by consecutively increasing the number of Fouriermodes and the number of Chebyshev modes in the lower andupper layers by 2. The calculations were subsequently per-formed with the minimum number of Fourier and Chebyshevmodes required to reach linear stability results accurate to thethird decimal point. The number of Chebyshev polynomialsrequired for a converged solution increased with the nondi-mensional frequencies in each layer, 2= and 1=N1,because the boundary layer thicknesses are reduced. Thespacing of the Gauss–Lobatto–Chebyshev collocation pointsnear the boundaries is OM

−2, and it was found that atleast one collocation point had to lie within the boundarylayer to ensure the convergence of the solution. The resultspresented in this paper were calculated for 74N1148,30N290, and K=14. The code was validated by success-fully reproducing several known results for both oscillatorysingle-layer and two-layer flows. The stability results ob-tained using the Chebyshev collocation code were found tobe in excellent agreement with those of Ref. 13 in the limitof large viscosities 250. We were also able to repro-duce Or’s16 results by setting the upper layer flow parametersto that of air 2=15.1110−6 m2 s−1 and 2=1.29 kg m−3,and omitting condition 8.

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

k

Ac

k0

k1 k2

Ω=360Ω=560Ω=3000Ω=10000inviscid theory

FIG. 3. Marginal stability curves showing the critical amplitude, Ac, againstthe wavenumber, k for 3.6102104. The other parameter values are

R1=510−1, N1=1, G0=1.6010−1, We=6.25 and d=1. The inviscid pre-diction of Ref. 13 is plotted for comparison with a gray line. Note that theseflow parameters are the same as in Ref. 13 Fig. 7.



IV. RESULTS

A. Parametric instability

We determined the critical value of the amplitude of os-cillation, Ac, at which the parallel shear flow becomes un-stable to standing waves of wavenumber k. A series of neu-tral stability curves is shown in Fig. 3 for fluids of equal butdecreasing viscosities, i.e., N1=1 and increasing . WhereasKhenner et al.13 were unable to resolve calculations for3.6102, the pseudospectral method employed here al-lows us to reach values of up to =4104, which is thelargest value presented in this paper. Each curve separatesthe Ac ,k-plane into a region of stable solutions below thecurve and unstable growing harmonic solutions above the

curve. As increases, the marginal stability curves con-verge toward the stability boundaries of the inviscid model,13

which are shown with gray lines in Fig. 3. The inviscid sta-bility boundaries were obtained by Floquet analysis of theMathieu equation13 and comprise a succession of tonguesof harmonic response, i.e., of the same period as that ofthe external oscillatory drive. The long wavelength modek0=0.001, which was the minimum value of k used in thecalculations in Fig. 3 is primarily due to the average effectof the oscillatory drive and is associated with a Kelvin–Helmholtz instability mechanism.13 The tongues centered onthe modes k1 and k2 delimit the first two regions of resonantresponse of the flow. As Ac tends to zero, the resonant re-sponse approaches a single Fourier mode, while at finite Ac

results from the superposition of several Fourier modes; seeFig. 5a. The convergence of the viscous stability resultstoward the inviscid result as increases suggests a similarparametric instability in the viscous model. Although thelong-wavelength KH instability is virtually independent ofviscosity, the resonant response at finite wavelength, cen-tered on the modes k1 and k2, is strongly stabilized by in-creasing the viscosity of both layers, and the most unstablewavenumbers k1 and k2 are displaced toward higher valuesof k. This dependence on viscosity is similar to that uncov-ered by Kumar and Tuckerman1 in their stability analysis ofthe Faraday problem with equal viscosity layers.

In Fig. 4, the neutral curve plotted in Fig. 3 for=360 is extended to wavenumbers up to k=10. It exhibitsa succession of local minima at k0 ,k1 . . .k6, including a long-wavelength KH instability and six resonant tongues. Thetime dependence of the solution at the first four minima isshown in Fig. 5a. In the presence of viscosity, Ac is finite ateach local minimum, and the temporal dependence of theneutral solution is a superposition of several Fourier modes.The temporal response for the instability modes k0 and k1 is

FIG. 4. Neutral curve showing the first seven instability tonguesk0 ,k1 , . . . k6 for =3.6102, R1=510−1, N1=1, G0=1.6010−1,

We=6.25, and d=1.

FIG. 5. a Interface height versus time: time-series of the eigenfunction at the modes k0–k3 in Fig. 4. b Time-averaged perturbation stream functions at themodes k0–k3 in Fig. 4.



approximately sinusoidal, whereas Fourier modes of up toorder three and five contribute significantly to the time-dependence of the response of the instability modes k2 andk3, respectively. The form of the time-averaged perturbationstream function, whose physically significant real part isplotted in Fig. 5b, changes continuously with k. The streamfunction at k0 exhibits on average a single peak across thetwo layers, centered near the interface, which indicates onaverage a single vortex acting to deform the interface. Thepeaks in the perturbation stream functions of the modes k1

and k2 indicate the presence of counter-rotating vortices oneither side of the interface, so that the interfacial deformationis driven on average by the net difference in vorticity on bothsides see Sec. IV D 3 for further details on the interpretationof the stream function plots. The instability correspondingto k3 has a more complex structure, whereby co-rotating vor-tices in each layer are separated by a thin, counter-rotatingvortex near the interface in the upper layer. Although theperturbation is distributed across the entire channel in thecase of the long-wavelength mode k0, the peaks are in-creasingly concentrated near the interface for higher valuesof k, so that the instability is clearly interfacial, similarly tothe “soft” wave uncovered by Kamachi and Honji.15

In Fig. 4, the mode k0 corresponds to the absolute mini-mum of the neutral stability curve and thus, a long-wavelength instability is excited at onset. While the thresh-olds of the modes knn2 are always higher than that of k1,either k0 or k1 may be the most unstable modes depending onthe values of parameters. We examine this competition infurther detail in Sec. IV D.

B. Long versus finite wave instability

Depending on the choice of parameters, the mode k0

may be found at k=0.001 the minimum value of k for whichcalculations were performed, i.e. long wavelength or atk0.001 finite wavelength. An example is given in Fig. 6

where k0=0.001 for We=9.18, but k00.001 for We=15.3.

A selection criterion was established by Khenner et al.13 forfinite-depth layers of inviscid fluids. In our notation, ifH=d2 / / 1−2g is the layer height scaled by the cap-illary length H is also the capillary wavenumber in our for-mulation, finite wavelength KH instabilities are only foundif H3 and

We We* =64H2 − 321 + R12

9H21 − R13G03 .

Our calculations suggest that this criterion remains valid forthe viscous analysis when N1=1. When N11, however, wefind that this criterion is no longer strictly valid. For ex-

ample, for We=10, R1=4.9010−1, N1=102, G0=1.9910−1, =1.57102, and d=1.0 see Fig. 9 below,H=1.433 but k0=0.30.001. This is not surprising sincethe viscosity contrast affects the most unstable wavenumberof the frozen wave significantly see Ref. 5, Fig. 7. Theeffect of the viscosity contrast on the neutral curves is dis-cussed further in Secs. IV C and IV D 3.

The long wavelength instability is virtually independent

of individual changes in We, , and N1, but dependsstrongly on the value of the modified inverse Froude number,G0, indicating that gravity acts to stabilize long wavelengthperturbations. Thus, an increase in the frequency of forcing

, which corresponds to increasing and We, while de-creasing G0, destabilizes the limit of small k more stronglythan the resonant modes at higher k, whose most unstablewavenumbers also increase significantly with frequency seeFig. 7. Thus, in the limit of large frequencies, the KH insta-bility mode k0 is observed at the onset, thus validating theresults of the time-averaged model of Lyubimov andCherepanov.11 A similar decrease of the threshold of the longwave instability with increasing frequency was also found byKamachi and Honji.15

0 1 2 3 4 5 60

0.5

1

1.5

2

k

Ac

Long wave mode k0

Finite wave mode k0

Resonant modes k1

We=9.18, viscousWe=15.3, viscousWe=9.18, inviscidWe=15.3, inviscid

FIG. 6. Neutral curves for We=9.18 and 15.3. The other parameter valuesare R1=4.9010−1, N1=102, =2.11102, G0=2.9110−1, and d=1. Thecorresponding inviscid results Ref. 13 are plotted for comparison with graylines.

FIG. 7. Neutral curves for increasing frequencies of forcing: — =1.1

103, G0=1.1610−2, We=2.29102, –·– =1.4103, G0=6.510−3,

We=4.08102, ¯ =1.6103, G0=4.910−3, We=5.40102. Theother parameter values are N1=100, R1=4.9010−1, and d=1.



C. Fluids of unequal viscosities

In experimental flows5,10,9 the viscosities of the fluids aregenerally different, and the viscosity ratio can be large. Thus,we focus hereafter on the general case where N11. Talibet al.5 have shown that the instability threshold is surpris-ingly reduced over a wide range of viscosity contrasts, whenN1 is raised by increasing the viscosity of either layer. Overthe range of N1 investigated in Ref. 5, 1N16104, thenonmonotonic variation of the most unstable critical ampli-tude and wavenumber with N1 is reflected in the averageperturbation stream function. The neutral curves shown inFig. 8 are for increasing viscosity contrasts achieved by in-creasing the viscosity of the upper layer similarly to Ref. 5,so that 50N1103 and 3.1710216. The value ofthe wavenumber k1 increases with viscosity contrast. Thiseffect is much stronger than when both viscosities were in-creased simultaneously in Fig. 3, i.e., reduced withN1=1. Thus, the inviscid theory is unsuitable to model para-metric instabilities in viscous layers, since it can neither pre-dict the instability threshold, nor the value of the most un-stable wavenumber of the parametric modes.

It is interesting to note that we have not been able to findany set of parameters corresponding to realistic fluid proper-ties, where the neutral curve comprises more than a singleminimum. The parameters chosen for all the calculationsshown above correspond to physically unrealizable fluidproperties. In order to achieve the moderate values of the

modified Weber number We750 necessary for multipleresonances to emerge while retaining experimentally practi-cal frequencies of forcing, the interfacial tension has to be atleast one to two orders of magnitude larger than that mea-sured in common pairs of liquids. In Sec. IV D, we investi-gate the changes in the neutral curves when the physical

parameters R1, We, and N1 are varied, and determine limitswhere the multiple minima vanish.

D. Parameter study of the finite-wave instabilities

The large number of nondimensional groups 7 govern-ing the dynamics prohibits an exhaustive investigation oflinear solution classes. In order to establish the physicalsignificance of the k0 and k1 modes, it is convenient to re-strict our attention to flows based on the physical propertiesof a perfluorinated liquid Flutec PP9 in the lower layer1=1.973103 kg m−3, 1=2.010−6 m2 s−1 and siliconeoil in the upper layer 2=9.670102 kg m−3, 2=2.010−4 m2 s−1. The interfacial tension between the layers istaken to be =4.010−3 N m−1, from the measurements ofinterfacial tensions for similar pairs of fluids.5,10 Layerheights of d1=d2=0.05 m and an angular velocity of=10 rad s−1 yield the following nondimensional param-

eters: R1=4.9010−1, We=2.98104, N1=102, G0=1.9910−1, =3.927102 and d=1.0. For these values of pa-rameters, the neutral curve Ac ,k exhibits a single minimumat a nonzero value of k. In order for local minima corre-sponding to the k0 and k1 modes to emerge, the interfacial

tension needs to be significantly increased so that We7.5102 see Sec. IV C. We choose =5.9610−1 N m−1

which is approximately two orders of magnitude larger thanthe interfacial tension measured between Flutec PP9 and sili-

cone oil, i.e., We=2.0102, and investigate the evolutionof the k0 and k1 instabilities with changes in the densitycontrast R1 and viscosity contrast N1 and .

1. Dependence on interfacial tension

For =5.96510−1 N m−1 We=200 and =1.988

10−1 N m−1 We=600, which correspond to unphysicallylarge values of the interfacial tension, both the modes k0 andk1 are found in the neutral curves shown in Fig. 9a. Anexamination of the eigenmodes reveals that for =1.193

10−1 N m−1 We=1000 and =5.96510−2 N m−1

We=2000, only the k1 mode remains. Note that all thecritical amplitudes converge to the same value as k tends tozero. This is because long wavelength perturbations of theinterface are virtually independent of interfacial tension,which like viscosity, predominantly acts on the growth ofshort wavelength disturbances. Note also the presence of thek2 mode for =5.96510−1 N m−1. The minimum of theneutral curve indicating this parametric-resonance mode van-ishes for decreasing values of .

As seen in Fig. 9c, the wavenumber of the KH modek0 remains approximately constant with respect to the cap-illary wavenumber denoted by H=d2g1−2 /, while therelative wavenumber k1 /H increases monotonically with .Although the dimensional critical amplitudes of the modes k0

and k1, denoted by a0 and a1, respectively, and shown in Fig.9b, both decrease as is reduced, a0 decays rapidly andconverges towards a1, until it vanishes for 2.010−1 N m−1. Thus, the k1 mode is the most unstable overthe entire range of investigated. Note that it is the k1 modewhich persists when is reduced toward experimental valuesfirst point in Figs. 9b and 9c, rather than the k0 modegenerally associated with the KH instability.11,13

FIG. 8. Neutral curves for We=9.18 and fluid layers of unequal viscosities.The viscosity contrast is increased by increasing the viscosity of the upperlayer. Thus, N1 increases and decreases. The other parameter values areR1=4.9010−1, G0=2.9110−1, and d=1. The inviscid result Ref. 13 isplotted for comparison with a gray line.



2. Density contrast

The density contrast is varied by changing the density ofthe lower layer, 1, which yields a change in the nondimen-sional parameter R1=2 /1 only. A series of four neutralcurves Ac ,k are shown in Fig. 10a. The shape of theneutral curves evolves considerably when R1 is decreasedfrom 0.58 to 0.1. For R1=0.58 and 0.4, there are two localminima corresponding to the modes of instability k0 and k1,but for R1=0.2 and 0.1 only the local minimum associatedwith k1 remains. Detailed plots of the critical amplitudes andwavenumbers of the modes k0 and k1 are shown in Figs.10b and 10c. A0 and A1 both increase monotonicallywhen the density contrast is reduced, i.e., R1 increases. Themode k1 is generally the onset mode except for fluids ofnearly equal densities R1=0.9, where the mode k0 is mostunstable. The crossover of the Ac ,R1 curves occurs be-tween R1=0.8 and 0.9, so nonlinear mode interaction mayprevail in this region.17,18 Note that for R1=1, there is noflow nor instability.

While the wavenumber of the mode k0 remains close tothe capillary wavenumber H, the wavenumber of k1 de-creases strongly to values below that of H when R1 is de-creased, resulting in the disappearance of the mode k0 forR10.3. Although the threshold of the mode k1 decreasesmonotonically over the entire range of R1 investigated, the

neutral curve shown in Fig. 10a for R1=0.1 is locatedmostly above that for R1=0.2, and thus indicates a tendencytowards restabilization as R1 decreases strongly. Althoughthe density contrast enables the relative acceleration of thefluid layers, the stable density stratification also induces sta-bilizing buoyancy forces. Thus, decreasing R1 lowers the in-stability threshold except for small values of R1, i.e., verydifferent densities, where the flow is increasingly stabilized.A qualitatively similar dependence of the instability thresh-old on R1 is predicted by Ref. 11 in the case of the time-averaged counterflow of inviscid fluids in the limit of highfrequencies and vanishing amplitudes of forcing.

3. Viscosity contrast

The most intriguing behavior is found when varying theviscosity contrast. As in Sec. IV C, we choose to increase theviscosity contrast by increasing the viscosity of the upperlayer beyond that of the lower layer, which is kept constant.This leads to a combined increase in N1 and decrease in .This means that the nondimensional frequency analogous toa Reynolds number divided by A in the upper layer,2=, decreases while the nondimensional frequency in thelower layer, 1=N1 remains constant.

In Fig. 11a, neutral curves calculated for increasingviscosity contrasts indicate that the k1 mode is increasingly

FIG. 9. a Neutral curves Ac ,k for

We=2102, 6102, 1103, and 2103. b Dependence of the dimen-sional critical amplitudes of the modesk0 and k1 on interfacial tension. cDependence on interfacial tension ofk0 and k1 relative to the capillarywavenumber. The other parameter val-ues are R1=4.9010−1, N1=102,G0=1.9910−1, =1.57102, andd=1.0. The first point in both band c corresponds to the experimen-tal value of the interfacial tensionfor Flutec PP9 and silicone oil dis-cussed in Sec. IV D, =4

10−1 N m−1 We=2.98104.



stabilized, but also that it vanishes beyond N1=750. Thethreshold of the k0 mode, however, decreases as the viscositycontrast is raised. A more detailed picture of these dynamicsis presented in Figs. 11b and 11c, where the critical am-plitudes and wavenumbers of the modes k0 and k1 are plottedagainst N1. In fact, A0 exhibits a nonmonotonic dependenceon N1 with four regions of distinct dynamics similar to thoseuncovered by Talib et al.5 The flow becomes successivelymore stable, unstable and stable again as N1 is increased, andthe four regions of distinct dynamics are reflected in thevariation of the critical wavenumber of k0 with N1. The k0

mode is only weakly sensitive to variations in the viscositycontrast for values of N1102, while for 102N1104 itsthreshold decreases sharply. By contrast, the k1 mode isstrongly damped when increasing N1 within the interval1N15102. It is then destabilized within 5102N1

7.5102 and vanishes beyond N1=750. Thus, the mode ofinstability at onset is k1 up to N13.25102, and k0 beyondthis value see Fig. 11b. Note that the wavenumber of k1

varies significantly more than that of the mode k0, whichremains close to the capillary wavenumber H.

The nonmonotonic dependence of the onset of instabili-ties cannot be correlated to simple features of the base flowprofiles shown in Fig. 2. In these, the thickness of the inter-facial boundary layer remains approximately constant in the

lower layer since 1=N1 is constant, while it increasesmonotonically in the upper layer with the reduction in 2

=. In order to gain insight into the opposite variations ofthe instability thresholds with increasing viscosity contrast, itis useful to examine the unstable eigenmodes. The time-averaged perturbation stream functions of the marginallyunstable modes k0 and k1 are shown in Fig. 12 for increasingvalues of N1. We choose to represent them for −0.04z0.04 since the modes of instability are interfacial. Thepeaks in the perturbation stream functions indicate the pres-ence of vortices, which are counter-rotating if the peaks areof opposite sign. This is illustrated in Fig. 13, where contourplots of the data shown in Fig. 12a are shown over onewavelength of the instability. Dark light shades correspondto negative positive values of the stream function respec-tively, while the medium gray horizontal and vertical linesindicate the locations where the stream function is equal tozero. In the case of the mode k1 Fig. 12a and Fig. 13,there are counter-rotating vortices present on either side ofthe interface, which is located at z=0. In the lower layerz0, the amplitude of the stream function peak hardlychanges with N1. In the upper layer z0, however, theinitially small stream function peak increases by a factor ofapproximately 2.5 for 10N1400 to reach values of simi-

FIG. 10. a Neutral curves Ac ,k forR1=0.58, 0.4, 0.2, and 0.1. b Depen-dence of the critical amplitude of themodes k0 and k1 on the density con-trast. c Dependence of k0 and k1

on the density contrast. The other pa-

rameter values are We=2.0102,N1=102, G0=1.9910−1, =1.57102, and d=1.0.



lar magnitude to those in the lower layer. As a result, the netdeformation of the interface is sharply reduced, since thedeformation resulting from the presence of the lower vortexis increasingly cancelled by that of the upper one, which iscounter-rotating. The sharp increase in the critical wavenum-ber with N1 appears to be governed by the thickness of thelower layer vortex, which becomes increasingly localizednear the interface as N1 increases from N1=10 to N1=400see Fig. 13. The concentration of vorticity near the inter-face in the lower layer is accompanied by the emergence of a

secondary vortex which is visible for N1=400 in Fig. 13.Thus, as the viscosity ratio is increased to N1=400, thestream function of the k1 mode shown in Fig. 12a smoothlychanges to resemble that of the k0 mode shown in Fig. 12b.

Indeed, the perturbation stream function of the k0 modeFig. 12b comprises diffuse co-rotating vortices in bothlayers, separated by a concentrated counter-rotating vortexright below the interface. We do not show the streamlinerepresentation in this case, as it is not well suited to visualizechanges in the very thin interfacial vortex present near the

FIG. 11. a Neutral curves Ac ,k forN1=2.5102 ,=1.57102, 7.5102 ,5.24101, 1.0103 ,3.93101, and 2.5103 ,1.57101.These successive parameter values areobtained by increasing the viscosity ofthe upper layer, and thus raising theviscosity contrast. b Dependence ofthe critical amplitudes of the modes k0

and k1 on the viscosity contrast. cDependence of k0 and k1 on the vis-cosity contrast. The other parameter

values are R1=4.9010−1, We=2102, G0=1.9910−1, and d=1.0.

FIG. 12. Time-averaged perturbationstream functions in the vicinity of theinterface plotted for 0.4z0.4: ak1 mode with N1=10, 100, 340, and400; b k0 mode with N1=100, 400,7500, and 5000.



interface in the lower layer. The considerable growth of thestream function peak corresponding to this lower layer vor-tex with N1 for 102N17.5102 leads to a significantincrease of the difference in magnitude of the disturbanceson either side of the interface. Thus, the net deformation ofthe interface is sharply increased. For N1=5103, the flowhas entered a different regime whereby the lower layer per-turbation remains unchanged but the upper layer perturbationis reduced. This yields a continued increase of the differencein magnitude of the disturbances on either side of the inter-face. Thus, the threshold of the k0 instability continues toexhibit a sharp reduction with increasing N1. The depen-dence of the k0 mode on the viscosity contrast discussed hereis analogous to that of the frozen wave instability investi-gated by Talib et al.5 and we refer to their paper for a com-prehensive discussion.

V. CONCLUSION

Horizontally oscillating viscous interfaces can be lin-early unstable to a Kelvin–Helmholtz mode and successiveparametric-resonance modes. The viscous model is essentialto predict the onset of each mode of instability accurately,particularly in the limit of large viscosity contrasts. Since thelarge number of nondimensional groups prohibits an exhaus-

tive investigation of the linear solution classes, we have fo-cused on characterizing the evolution of the neutral curvesfrom exhibiting multiple resonances to only a single mini-

mum, as found in the limit of practical flows. When We isincreased toward experimental values, the first resonantmode is found to persist, rather than the Kelvin–Helmholtzmode usually associated with the frozen wave observed ex-perimentally. Depending on the value of the density contrastR1, either the Kelvin–Helmholtz or the first resonant modemay be the most unstable, and similar critical parameters inthe limit of small density contrasts suggest the possibility ofnonlinear mode interaction. Interestingly, the two modes ex-hibit opposite dependencies on the viscosity contrast with asharp stabilization of the first resonant mode, while thethreshold of the Kelvin–Helmholtz mode exhibits a sharpreduction, analogously to the frozen wave instability dis-cussed in Ref. 5.

ACKNOWLEDGMENTS

The authors wish to thank Dr W. Y. Jiang and Dr S. P.Lin for their help with the numerical method. This work wasfunded by an EPSRC project studentship E. T. and anEPSRC “Advanced Research Fellowship” A. J..

FIG. 13. Normalized streamline con-tour plots over one wavelength2 /kc, calculated using the datashown in Fig. 12a. Dark light grayshades denote negative positive val-ues, respectively, while the horizontaland vertical medium gray lines indi-cate where the stream function is equalto zero. Thus, these plots indicate thepresence of counter-rotating vorticeson either side of the interface, which islocated at z=0. Their most strikingfeature is the narrowing of the lowerlayer vortex near the interface as N1

increases, which correlates with thesharp rise in the wavenumber seen inFig. 11c.



APPENDIX: INTEGRATION CONSTANTSOF THE BASE FLOW SOLUTION

The integration constants A1, B1, A2, B2, and S of thebase flow solution, determined by substitution of the baseflow solution into the boundary conditions, take the follow-ing form:

A1 = − iSR1

− 1em1d − B1e2m1d,

A2 = − i S

− 1e−m2 − B2e−2m2,

B1 =

B21 − e−2m2 + ie−m2 − em1d +iSR1

em1d − 1 −

iS

e−m2 − 1

1 − e2m1d,

B2 = S2i

m1R1em1d −

i

m1R11 + e2m1d −

i

m1e−m2 − 11 + e2m1d −

i

NRm2e−m21 − e2m1d + iNRm2e−m21 − e2m1d

+ im1e−m21 + e2m1d − 2em1d NRm21 + e−2m21 − e2m1d − m11 + e2m1d1 − e−2m2−1 =SE + F

D,

where E, F, and D correspond to the terms in the three square brackets,

S = 2im1m2 − im2em1d − 1 − im11 − e−m2 + im2He−m2 − em1d −GF

D i

m1m2R1 + 1 −

i

m11 − e−m2

−i

m2R1em1d − 1 +

i

m2HR11 − em1d + e−m2 − 1 +

GE

D−1

,

where

G = m11 − e−m22 −m21 − em1d21 − e−2m2

1 − e2m1d,

and

H =1 − em1d2

1 − e2m1d.

1K. Kumar and L. S. Tuckerman, “Parametric instability of the interfacebetween two fluids,” J. Fluid Mech. 279, 49 1994.

2G. H. Wolf, “Dynamic stabilization of the interchange instability of aliquid-gas interface,” Phys. Rev. Lett. 24, 444 1970.

3S. P. Lin, J. N. Chen, and D. R. Woods, “Suppression of instability in aliquid film flow,” Phys. Fluids 8, 3247 1996.

4A. V. Coward and D. T. Papageorgiou, “Stability of oscillatory two-phaseCouette flow,” IMA J. Appl. Math. 53, 75 1994.

5E. Talib, S. V. Jalikop, and A. Juel, “The influence of viscosity on thefrozen wave instability: theory and experiment,” J. Fluid Mech. 584, 452007.

6C. K. Shyh and B. R. Munson, “Interfacial instability of an oscillatingshear layer,” J. Fluids Eng. 108, 89 1986.

7H. Yoshikawa, “Instabilités des interfaces sous oscillations,” Ph.D. thesis,Université Paris VI 2006.

8G. H. Wolf, “The dynamic stabilization of the Rayleigh–Taylor instabilityand the corresponding dynamic equilibrium,” Z. Phys. 227, 291 1969.

9R. Wunenburger, P. Evesque, C. Chabot, Y. Garrabos, S. Fauve, and D.Beysens, “Frozen wave instability by high frequency horizontal vibrationson a CO2 liquid-gas interface near the critical point,” Phys. Rev. E 59,

5440 1999.10A. A. Ivanova, V. G. Kozlov, and P. Evesque, “Interface dynamics of

immiscible fluids under horizontal vibration,” Fluid Dyn. 36, 362 2001.11D. V. Lyubimov and A. A. Cherepanov, “Development of a steady relief at

the interface of fluids in a vibrational field,” Fluid Dyn. 21, 849 1986.12S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability Oxford

University Press, Oxford, 1961.13M. V. Khenner, D. V. Lyubimov, T. S. Belozerova, and B. Roux, “Stability

of plane-parallel vibrational flow in a two-layer system,” Eur. J. Mech.B/Fluids 18, 1085 1999.

14R. E. Kelly, “The stability of an unsteady Kelvin–Helmholtz flow,” J.Fluid Mech. 22, 547 1965.

15M. Kamachi and H. Honji, “The instability of viscous two-layer oscilla-tory flows,” J. Oceanogr. Soc. Jpn. 38, 346 1982.

16A. C. Or, “Finite wavelength instability in a horizontal liquid layer on anoscillating plane,” J. Fluid Mech. 335, 213 1997.

17A. A. Golovin, A. A. Nepomnyashchy, and L. M. Pismen, “Interactionbetween short-scale Marangoni convection and long-scale deformationalinstability,” Phys. Fluids 6, 34 1994.

18S. J. VanHook, M. F. Schatz, W. D. McCormick, J. B. Swift, and H. L.Swinney, “Long-Wavelength Instability in Surface-Tension-DrivenBénard Convection,” Phys. Rev. Lett. 75, 4397 1995.

19T. I. Hesla, F. R. Pranckh, and L. Preziosi, “Squire’s theorem for twostratified fluids,” Phys. Fluids 29, 2808 1986.

20P. G. Drazin and W. H. Reid, Hydrodynamic Stability Cambridge Univer-sity Press, Cambridge, 1981.

21C. Lanczos, Applied Analysis Prentice-Hall, Englewood Cliffs, 1956.22A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations Wiley, New York,

1979.23T. P. Schulze, “A note on subharmonic instabilities,” Phys. Fluids 11,

3573 1999.



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