Dynamics and rupture of planar electrified liquid sheets

PHYSICS OF FLUIDS VOLUME 13, NUMBER 12 DECEMBER 2001

Dynamics and rupture of planar electrified liquid sheetsB. S. Tilley, P. G. Petropoulos, and D. T. Papageorgioua)

Department of Mathematical Sciences and Center for Applied Mathematics and Statistics,New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102

~Received 19 March 2001; accepted 20 August 2001!

We investigate the stability of a thin two-dimensional liquid film when a uniform electric field isapplied in a direction parallel to the initially flat bounding fluid interfaces. We consider the distinctphysical effects of surface tension and electrically induced forces for an inviscid, incompressiblenonconducting fluid. The film is assumed to be thin enough and the surface forces large enough thatgravity can be ignored to leading order. Our aim is to analyze the nonlinear stability of the flow. Weachieve this by deriving a set of nonlinear evolution equations for the local film thickness and localhorizontal velocity. The equations are valid for waves which are long compared to the average filmthickness and for symmetrical interfacial perturbations. The electric field effects enter nonlocallyand the resulting system contains a combination of terms which are reminiscent of the Kortweg–de-Vries and the Benjamin–Ono equations. Periodic traveling waves are calculated and theirbehavior studied as the electric field increases. Classes of multimodal solutions of arbitrarily smallperiod are constructed numerically and it is shown that these are unstable to long wave modulationalinstabilities. The instabilities are found to lead to film rupture. We present extensive simulations thatshow that the presence of the electric field causes a nonlinear stabilization of the flow in that itdelays singularity~rupture! formation. © 2001 American Institute of Physics.@DOI: 10.1063/1.1416193#

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I. INTRODUCTION

Liquid films are encountered in many physical situatiowith applications in cooling systems, coating processes,biological applications. Free liquid films arise in both natuand technological applications and a fundamental undstanding of their nonlinear response is of interest. Tpresent work is concerned with the dynamics of free liqfilms in the presence of electric fields which act in the plaof the film. Normal electric fields have been used to manilate the stability of liquid films flowing down inclinedplanes, in an effort to control heat transfer in devices ussuch cooling agents~see, for example, Gonzalez anCastellanos1!.

The effect of an electric field on the linear stability ofsharp interface separating two nonconducting dielectricids of infinite extent has been studied by Melcher aSchwarz.2 A constant electric field is applied in the planethe undisturbed interface and the linear stability of viscoand inviscid fluids is carried out. The electric field producdispersion and hence linear stabilization of short wamodes. Similar stability findings are established in the stuof El-Sayed3 who considers the linear stability problemthe presence of an air stream outside the sheet.~This is incontrast to the case of normal electric fields interacting wperfectly conducting liquid films as shown in Ref. 1.! Aspointed out by Melcher and Schwarz, the electric field

a!Author to whom correspondence should be addressed. Also at Departof Mathematics, Imperial College of Science, Technology and MedicLondon, SW7 2AZ, United Kingdom.

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duces a polarization force which tends to displace the inface from its initially flat position.

Our study is concerned with the nonlinear stability oftwo-dimensional inviscid liquid sheet lying between two pependicular electrodes which have a constant voltage potedifference across them. The field acts in the plane of thefilm which initially has a constant thickness, is at rest agravity is absent.~The latter condition assumes that the Bonumber,rgd2/s, is small: Herer is the fluid density,d theundisturbed film thickness,g the acceleration due to gravityand s the surface tension coefficient.! This work includeselectric field effects into problems that have received conserable attention in recent years. Erneux and Davis4 have con-sidered the nonlinear stability of a viscous liquid film undegoing symmetrical perturbations when surface tensionwell as disjoining pressures~Van der Waals forces! are com-peting effects. Using long wave asymptotic expansions tobtain a system of coupled nonlinear evolution equationsthe symmetric film shape and the induced local horizonvelocity. They also use a weakly nonlinear analysis to sgest film rupture, even though the latter event is beyondrange of validity of weakly nonlinear theory. This problemstudied further in a series of articles by Ida and Miksis,5–7

who develop equations for the leading order evolution of tfilms in the presence of surface tension and Van der Waforces by using matched asymptotic expansions whereratio of the film thickness to its lateral extent is a smparameter; the general three-dimensional equations areplied to specific geometries in Ref. 6 and the case of a plafilm ~the problem studied in Ref. 4! is solved numerically.Rupture is found and the numerical solutions indicate t

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7 © 2001 American Institute of Physics

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3548 Phys. Fluids, Vol. 13, No. 12, December 2001 Tilley, Petropoulos, and Papageorgiou

the dynamics is dictated by similarity solutions of the secokind.

If viscosity and Van der Waals forces are excluded frothe equations derived in Ref. 4, the nonlinear stability wsurface tension present becomes a conservative systemno linearly unstable modes. In fact the dispersion relationidentical to that of the well-known beam equation. It mayexpected, therefore, that the system will respond differedepending on the initial perturbation amplitudes; in additiothe flow can be unstable to modulational instabilities. Tnonlinear inviscid sheet equations were first derived andtially analyzed by Matsuuchi;8,9 in the first article a weaklynonlinear ~disturbances being small compared to the shthickness! stability analysis is carried out and a nonlineSchrodinger equation is derived for the amplitude modution, indicating modulational instability. The weakly nonlinear assumption is dropped in Ref. 9 and the nonlinear lwave equations are derived and used in a numerical expment that indeed shows that the sheet can break up by inducing a weak long wave perturbation to a class of nonlintraveling wave solutions for the problem. The rupture phnomenon was observed but the singularity structure wasaddressed.

A more extensive recent study is that of Mehring aSirignano.10 In this work a more general system of equatiois derived that can be used for sinuous motions of the salso. This is done by defining a local sheet centerlineexpanding flow variables in a Taylor series in the vertidirection in order to derive simpler one-dimensional modeTraveling wave solutions are constructed and used to iniize time dependent calculations. In addition, full twdimensional calculations of the Euler equations are pformed using the vortex method~see references therein! inorder to evaluate the accuracy and range of validity ofone-dimensional models. The models are found to worktremely well in most situations and to provide significaphysical insight. Mehring and Sirignano consider the staity of a semi-infinite sheet also, which is forced by a perioddisturbance at the nozzle. Using the one-dimensional modit is shown that for dilational waves at least, a breakup coccur if the Weber number is sufficiently large~cf. Fig. 13 inRef. 10!. This singularity structure is not quantified but apears to have a very similar form to that computed firstPugh and Shelley11 in a different context~see below!. It isworth pointing out, however, that the singular solutionsMehring and Sirignano arising from nonperiodic boundaconditions, are strikingly similar to those obtained for peodic ones. This fact is used in the present work, to motivperiodic calculations.~We also note that the two-dimensionthin sheet models have now been extended to thdimensional ones by Kim and Sirignano12 using a similarexpansion scheme to Ref. 10.!

As already mentioned, the inviscid thin sheet equatiohave been used in a related but different context by PughShelley11 in their study of the rollup of vortex sheets witsurface tension. Numerical computations of vortex shevolution indicate that when surface tension is presentsheet rolls up and generates regions where the flow calocally described as a thin two-dimensional jet with surfa


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tension flowing through a surrounding fluid of the same dsity. Kelvin–Helmholtz instability acts but short waves adispersively stabilized by surface tension. Starting fromBirkhoff–Rott equation, evolution equations for thin symmetric horizontal jets are derived by Pugh and Shelley. Thequations are found to be nonlocal due to the surroundfluid, and to aid the understanding of the dynamics, a lomodel is proposed in Ref. 11. This local model is exactly tderived elsewhere for inviscid liquid sheets. The KelvinHelmholtz instability is built into the model system by peturbing about an undisturbed state which has a velocity juacross the interface. It is found that solutions encountefinite time singularity with the jet rupturing at a point. Simlarity solutions are constructed guided by the numericallutions and it is found that there is remarkable agreembetween the local and nonlocal models. We note that calations and quantification of singular solutions is quite a decate but important endeavor and Pugh and Shelley werefirst to study this problem extensively for the inviscid thin jequations.

Rather than addressing the full Euler and Maxwell eqtions we derive reduced long-wave models which are capaof describing pinching and other nonlinear behavior. Thenonical system of evolution equations derived and studhere is

St1~uS!x50,

ut1uux5l fSxxx1leH~Sxx!,

whereH is the Hilbert transform operator andl f , le arenondimensional parameters which measure the effects offace tension and the electric field respectively~see later!. Thesystem is reminiscent of a combination of a Kortweg dVries ~KdV! term ~the third derivative Sxxx! and aBenjamin–Ono~BO! term ~the Hilbert transform!. One ofour objectives is to address the effects of the electric fieldboth traveling wave solutions as well as finite-time singulaties. There have been many studies of the KdV and BO eqtions and the reader is referred to Albert, Bona, aRestrepo,13 and Abdelouhabet al.14 and references thereifor the analysis of such nonlinear equations. The latter reence is particularly interesting because it presents a thefor equations which are not completely integrable but canviewed as perturbations of integrable systems.

The paper is organized as follows. In Sec. II, the coupset of partial differential equations is derived starting frothe Euler and electric field equations and boundary contions. The fluid is nonconducting and the coupling betwethe fluid dynamics and the electric field arises throughMaxwell stresses that enter the normal stress balance cotion. We present a perturbation scheme with the average flthickness to wavelength ratio being the small parameMatched asymptotic expansions are set up that enable maing with the electric field~equivalently voltage! far from theliquid layer where the slenderness assumption breaks doThis matching is the source of the nonlocal term~the Hilberttransform! shown above. In Sec. III we study the system fnonlinear traveling waves and in particular consider thefect of the electric field on traveling waves found by oth


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3549Phys. Fluids, Vol. 13, No. 12, December 2001 Dynamics and rupture of planar electrified liquid sheets

investigators in its absence~see Refs. 9, 10!. In Sec. IV weconsider the initial value problem of the evolution systeOur main concern is the careful numerical and analyticwhere possible, study of rupture which is a finite-time singlarity of the equations. We present results in the absencelectric fields initially and recover previous work by othinvestigators~see Ref. 11!. With the electric field present wepresent three sets of results depending on initial conditio~i! conditions that do not lead to pinching and which arfrom relatively low levels of input energy~equivalently rela-tively small Weber numbers!—these are found to be quasperiodic in time; ~ii ! relatively large initial perturbations~equivalently sufficiently large Weber numbers!, which leadto finite-time singularities—these are found to retain thecal form of the zero electric field singularities;~iii ! travelingwave solutions of different periods perturbed by very loamplitude~0.01%! long-wave perturbations to study modlational instabilities—these lead to pinching finite-time sgularities as in~ii !. Finally, in Sec. V we make some concluding remarks and point to future directions.

II. DERIVATION

A. Problem formulation

Consider the irrotational flow of an incompressible, iviscid, nonconducting liquid between two free surfacescated aty56S and bounded on either side by a gas in whfluid motions are negligible, as is depicted in Fig. 1. Therenonzero surface tension at the interface with constant cocient s. Gravity is ignored and a uniform electric fieldE5V0 /(2L) i is acting, whereV0 is the characteristic voltagdrop over a typical dimensional length 2L. The average dis-tance between the free surfaces is given by 2d, where weshall assume later thatd!L. Introducing dimensional fluidand voltage potentialsf and V(1,2), respectively, the fieldequations are Laplace’s equation

¹2f~x,y;t !50, ~1!

¹2V~1,2!~x,y;t !50, ~2!

where in what follows superscripts 1, 2 denote the film apassive regions, respectively~see Fig. 1 also!. The electricfields areE(1,2)52¹V(1,2).

FIG. 1. Physical problem under consideration: inviscid, irrotational, incopressible fluid jet in a passive medium with surface tension in an appelectric field.


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-

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We consider symmetric disturbances about the centery50, which yields the boundary conditionsfy5Vy

(1)50; asuyu→`, Vx

(2)→V0 /(2L). The conditions at the free surfacy5S are the kinematic boundary condition, continuitynormal stresses, and the continuity of normal displacemand tangential electric field components, i.e., aty5S(x,t):

St1fxSx2fy50, ~3!

@n•T•n#215s div~n!, ~4!

@~eE!•n#2150, ~5!

@E•t#2150. ~6!

The notation@•#21 denotes a jump in the quantity as the i

terface is crossed from the fluid region. The vectorsn, t arethe outward pointing normal and tangent to the interfarespectively, while the stress tensorT is given by

Ti j 52pd i j 1Ei j ,~7!Ei j 5e~EiEj2

12uEu2d i j !.

The first contribution in Eq.~7! is the usual hydrodynamicinviscid part and the second contribution comes frostresses exerted by the electric fields at the interface as gby the Maxwell stress tensor~see Jackson,15 Chapter 6!. Theparametere is the dielectric constant which takes differevalues in regions 1 and 2.

The momentum equations can be integrated as usuayield a Bernoulli equation at the interface. The fluid pressin region 2 is ambient and equal to a constant. Eliminationthe pressure jump across the interface from Eq.~4!, yieldsthe following Bernoulli boundary condition:

r1S f t11

2u¹fu2D1

1

11Sx2 $Sx

2@E11#2122Sx@E12#2

11@E22#21%

5sSxx

~11Sx2!3/21Kp , ~8!

where the Maxwell stresses are given by

E115e

2~Vx

22Vy2!, E125eVxVy , E225

e

2~Vy

22Vx2!,

~9!

andKp is a function of time at most, which is related to thpolarization pressure induced on fluid interfaces by the etric field ~see Melcher and Schwarz2!. As an example of thisforce consider the case of a quiescent uniform flat film stwhich hasS5d andf5constant. It can be seen that this issteady exact solution of Eq.~8! as long asKp5(V0

2/8L2)3(e22e1). Note, however, thatKp can be eliminated bydifferentiation of Eq.~8! with respect tox ~see later!.

We nondimensionalize the variables as follows: the flupotential byAsd/r1, the interfacial height and the verticacoordinatey by d, the independent variablesx by L, t byL2Ar1 /(sd) and the voltageV by V0 . ~The same symbolsare used for dimensional and nondimensional variables.! Thechoice of different horizontal and vertical length scalesmade in order to utilize the smallness of the ratiod/L in arational asymptotic expansion scheme. Physically, a defl

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tion of the interface of orderd will induce a voltage pertur-bation of orderd. In dimensionless terms, then, we can esmate the voltage perturbation to be of orderd and write

V~1,2!~x,y,t !5x

21dv ~1,2!~x,y,t !, ~10!

where V(1,2)(x,y,t) is the total voltage andv (1,2)(x,y,t) isthe perturbation to it in regions 1 and 2, respectively. Tdimensionless system to be addressed is given below@inwhat follows we have substituted expression~10! for thetotal voltage#. The field equations are

d2fxx1fyy50, ~11!

d2vxx~1,2!1vyy

~1,2!50. ~12!

The dimensionless kinematic, Bernoulli and continuitynormal displacement and tangential electric field componboundary conditions are evaluated aty5S(x,t) and aregiven by

d2~St1fxSx!2fy50, ~13!

d2S f t11

2fx

2D11

2fy

21Eb

11d2Sx2

3H 1

2d2@M11#2

122d2@M12#211

1

2@M22#2

1J5

d2Sxx

~11d2Sx2!3/21

1

8d2Eb~12e!; ~14!

e~vy~1!2dSx@

121dvx

~1!# !5vy~2!2dSx@

121dvx

~2!#, ~15!

vx~1!1Sxvy

~1!5vx~2!1Sxvy

~2! , ~16!

where the parameterEb5(e2V02)/(sd) is the ratio of the

electrically induced pressure compared to capillary forcand we call this an electric capillary number. The stress coponents in Eq.~14! are given by

M11~1,2!5Sx

2 H e

e2Fd2S 1

21dvxD 2

2d2vy2G J ~1,2!

, ~17!

M12~1,2!5Sx H e

e2S 1

21dvxD dvyJ ~1,2!

, ~18!

M22~1,2!5H e

e2F2d2S 1

21dvxD 2

1d2vy2G J ~1,2!

. ~19!

The problem has been formulated on a bounded horiztal domain. In the interest of brevity, however, it is easiertreat the problem for the perturbations on infinite domaand to use Fourier transforms rather than Fourier series.connection between one and the other is then readily avable.

The above formulation is exact and in particular tchoiced51 provides the full problem with no scale disparitOur interest is in the nonlinear development of the systwhen d is small. As is seen below, in this limit the electrfield competes with surface tension forces and fluid inertiathe distinguished limitdEb5O(1). Wewrite, then,


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Eb5Eb

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whereEb is an order one parameter.Before proceeding with the derivation of nonlinear lon

wave equations, we give the linear stability results in geeral, and their long wave limit in particular.

B. Linear stability

In this section we carry out a temporal linear stabilanalysis of the problem in order to guide the nonlinear dnamics which is the central focus of this work. An exasolution of the dimensionless problem isS51, u50. Linear-ization of the system~11!–~16! about this state~the Bernoulliequation is differentiated with respect tox first!, and appli-cation of the symmetry conditions aty50 gives the follow-ing solutions for the fluid and voltage potentials in Fourspace:

f~y,t;k!5C~k!cosh~kdy!, ~21!

v ~1!5A~k!cosh~kdy!, ~22!

v ~2!5B~k!exp~2ukudy!, ~23!

where f5*2`` f(x,y,t)exp(2ikx)dx, etc. Using solutions

~21!–~23! into the kinematic, Bernoulli, and electric fielboundary conditions at the unperturbed levelS51, and look-ing for solutions proportional to exp(vt), gives four homo-geneous equations for the unknownsA(k), B(k), C(k),S(k), whereS(k) is the Fourier transform of the interfaciaperturbation. This leads to the eigenrelation

d

tanh~kd!v21

dEb~e21!2

4

k3 cosh~kd!

ek sinh~kd!1ukucosh~kd!1k3

50. ~24!

In the nonlinear long wave theory developed next, we csider solutions asd→0. This limit, along with expression~20! for the electric capillary number, gives the long-wadispersion relation

v21Eb~e21!2

4k3 sgn~k!1k450. ~25!

This is the appropriate dispersion relation for the linear sbility of the nonlinear long-wave system derived next. Itworth noting that the system is linearly stable and respodispersively to perturbations. It is useful, therefore, to cosider nonlinear effects.

We note in passing, that the linear stability is affectdiffusively in a stable way when viscosity is included in thfilm fluid, and the electric field acts in a dispersive mannershown above. This remains true in the long-wave limit whethe growth rate is found to be negative and proportionak2. A band of unstable waves, however, enters in the prence of Van der Waals forces~see Ref. 4 for more details inthe case of no electric field!. The combined effect of viscosity and electric fields on the nonlinear dynamics is the sject of current research.


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C. Small d limit: Inner problem

Solutions need to be found in two regions of verticextenty5O(1) ~regions I and II fory,S andy.S, respec-tively! and y5O(1/d) ~region III with matching providingthe leading order evolution equations. The reason for thithe voltage boundary condition asy→`, which is notachievable by extension of the solution local to the [email protected]., y5O(1)# to infinity.

We begin with regions I and II, then, and expand tsolution in powers ofd

f5f0~x,y,t !1df1~x,y,t !1d2f2~x,y,t !1¯, ~26!

v ~1,2!5v0~1,2!~x,y,t !1dv1

~1,2!~x,y,t !1¯, ~27!

S5S0~x,t !1dS1~x,t !1¯ . ~28!

Substitution of Eqs.~26!–~28! into the field equations~11!and ~12! gives the following leading order solutions:

f0[f0~x,t !, ~29!

f1[f1~x,t !, ~30!

f252 12y

2f0xx1x~x,t !, ~31!

v0~1![v0~x,t !, ~32!

v1~1![a~x,t !, ~33!

v2~1!52 1

2y2v0xx1c~x,t !, ~34!

v0~2![ v0~x,t !, ~35!

v1~2![A~x,t !1yB~x,t !, ~36!

v2~2!52 1

2y2v0xx1yC2~x,t !1D~x,t !. ~37!

It turns out that the orderd2 contributions to the voltage arnot required for the derivation of the leading order evolutiequations but as seen above, they can be generatedeasily.

Using these solutions into the continuity of normal dplacement and tangential electric field components aty5S0

@Eqs.~15! and ~16!# gives

B~x,t !5 12~12e!S0x , ~38!

v0x5 v0x⇒v05 v0 , ~39!

ax5Ax1~BS0!x⇒a5A1 12~12e!S0S0x . ~40!

We note that the functions of time that can possibly ariseintegration of Eqs.~39! and ~40! can be chosen to be zersince in the undisturbed state att50, v (1,2)50 whenS51@this state in turn impliesB[0, from which Eq.~40! fol-lows#.

The Bernoulli equation~14! is identically satisfied atO(d0) and atO(d2) we find ~after differentiation with re-spect tox!

f0xt1f0xf0xx1Eb

2~12e!v0xx5S0xxx , ~41!

while the kinematic condition~13! at order d2 yields theconservation law


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e

uite

n

S0t1~f0xS0!x50. ~42!

Equations ~41! and ~42! are two nonlinear evolutionequations for the three leading order unknowns,f0 , S0 , andv0 ; the latter is determined by matching requirements witsolution valid in region III as described next.

D. Small d limit: Outer

To close the problem, we need to solve the outer prlem away from the interface and include the conditionv (2)

→0 asy→`. The voltage in region II has been found to bthen,

v ~2!~x,y,t !5v0~x,t !1d~A~x,t !1 12~12e!yS0x!

1O~d2!, ~43!

with A(x,t) given previously; it follows, then, that the expansion breaks down wheny;d21. Introducing new outervariables

x5X, y5Y

d, ~44!

along with the expansion for the perturbation voltage ingion III, v (3) say,

v ~3!5V01dV11d2V21¯, ~45!

gives the Laplace equationViXX1ViYY50 for each correc-tion in Eq.~45!, with boundary conditionVi(x,`,t)50. Tak-ing Fourier transforms as defined earlier, the two orderslution satisfying the conditions at infinity is

v ~3!5a~k,t !exp~2ukuY!1db~k,t !exp~2ukuY!

1O~d2!, ~46!

where functionsa(k,t) and b(k,t) are to be found bymatching with the inner solutionv (2). This can be done bymatching Eq.~43! as y→` with Eq. ~46! as Y→0 and ismost easily achieved in Fourier space since matched teare linear. The following conditions are found:

a~k!5 v0~k,t !, ~47!

2ukua~k,t !5 12~12e!ikS0 . ~48!

It follows from Eqs. ~47! and ~48! that v052 i /2(12e)sgn(k)S0, and so

v0xx512~12e!ik2 sgn~k!S0 .

Utilizing the Hilbert transform operator defined by

H~ f ~x!!51

pPVE

2`

` f ~y!

x2ydy, ~49!

where PV denotes the principal value of the integral, ausing the fact that differentiation commutes with the Hilbetransform operator, we can write

v0xx52 12~12e!H~S0xx!. ~50!

Letting u5f0x , S5S0 , the evolution equations found become

St1~uS!x50, ~51!


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ut

nn

f


ut1uux5Sxxx1Eb

4~12e!2H~Sxx!, ~52!

where

H~Sxx!51

pPVE

2`

` Syy

x2ydy

on unbounded domains and

Hp~Sxx!521

2pPVE

2p

p

cotS x2y

2 DSyy dy

on periodic domains. Some further properties of the Hilbtransform will be given as their need arises, but to recothe linear stability results we note the property~for m anypositive integer!,

HS ]mS

]xmD 5~ ik !mi sgn~k!S.

The linear stability of the system~51! and~52! about theexact solutionS51, u50, using the notation of Sec. II Bgives the dispersion relation Eq.~25! as expected. The system developed, then, is an appropriate long wave modelcan capture strongly nonlinear effects such as rupturenonlinear propagating pulses. We consider the initial vaproblem in a later section but first we turn our attentionpossible nonlinear traveling wave solutions. In what followe cast Eqs.~51! and~52! into canonical form which is moreappropriate for analysis and computations. Making the traformations x15p/L, t15(p/L)t, S15S, and u15L/p,along with the re-introduction of a parameter in front of tsurface tension term, gives the canonical system~subscripts1 have been dropped!,

St1~uS!x50, ~53!

ut1uux5l fSxxx1leH~Sxx!, ~54!

wherel f andle are positive constants. The parameterle isrelated to the scaled electric capillary number introducedlier by the equation

le5Eb~12e!2

4. ~55!

The numerical calculations of the initial value problem reto different values ofEb and the above expression is notfor future reference.

In most of the work that follows we fixl f51/2 andconsider solutions parametrized by the electric field paraeterle , for the case of periodic boundary conditions

S~x12p,t !5S~x,t !, u~x12p,t !5u~x,t !.

III. NONLINEAR TRAVELING WAVES

A. Equations and numerical methods

Traveling waves can be constructed by consideringlutions of the form

S~x,t !5S~j!, u~x,t !5u~j!, where j5x2c0t,

with c being the speed of the waves, yielding


rtr

atnde

s-

r-

r

-

-

2c0Sj1~Su!j50, ~56!

2c0uj1uuj5l fSjjj1leH~Sjj!. ~57!

Such solutions of Eqs.~53!, ~54! in the absence of electric fields (le50) and l f51 have already been found bMatsuuchi9 and more recently by Mehring and Sirignano10

Following Matsuuchi withle50, then, the resulting equations can be integrated twice resulting in a single equatfor S in the form

S dS

dj D 2

5c0

2~S2c1!~c22S!

S, ~58!

where the constantsc1 , c2 satisfy 0,c1<c2 with equalityresulting in a uniform~trivial! solution sincec1 is the waveminimum andc2 the wave maximum. The wavelengthL,say, can be written in terms of the complete elliptic integof the second kind@E(a)# ~see Abramowitz and Stegun,16 p.587!

L54

uc0uAc2E~m!, where m25

c22c1

c2. ~59!

We have used these special solutions to validate limitinglutions of our numerical work and to initialize time depedent calculations. This is achieved by noting that givensolution as in Eq.~59!, a 2p-periodic one follows by makingthe transformations j5(p/L)j, u5(p/L)u and c5(p/L) c. @The special functionsE(a) are available inMAT-

LAB.# We also note that the form Eq.~58! can be used toprove that no solitary waves are possible.

When leÞ0, the nonlocal term does not allow thsimple phase plane description provided by Eq.~58!. Theproblem is addressed numerically in this section. Witholoss of generality we can study solutions of Eqs.~56!–~57!which have zero mean @defined as ^S&5(1/2p)*2p

p S(j)dj, etc.#—this property becomes useful ianalyzing stability which is the subject of future work. Givea solution S, u of Eqs. ~56!–~57! which has^S&51, ^u&5K, the substitutionsS511h, u5K1W, along with theredefinition of the speedc5c02K provide the system

2chj1~hW!j50, ~60!

2cWj1WWj5l fhjjj1leH~hjj!, ~61!

where now

^h&5^W&50. ~62!

It is clear from the form of Eq.~58! that traveling wavesin the absence of electric fields (le50) are even functions oj ~note that there is an algebraic relationship betweenu andSand sou is even wheneverS is even!. This important sym-metry persists for the nonlocal Eqs.~60!–~61!. To see this wenote the following fact:

H~ f ~2x!!52H~ f ~x!! when f ~2x!5 f ~x!. ~63!

@The above can be shown starting from Eq.~49!, changingvariables and using the symmetry off (x)—an identicalanalysis shows Eq.~63! for the operatorHp also.# We have,therefore


-


FIG. 2. Interfacial traveling waves. Electric field effects: l f51/2 andle50,1,...,10. The wave minimumdecreases as the electric field is increased.

breo

le

t.e

m.areredesol-

sian

ul-aus-ur

und

ingentre.

tric

s-y ised

rolu-

e.,

h~2j;c!5h~j;c!, W~2j;c!5W~j;c!,

which in turn provides the Fourier series representation

h~j!5 (n51

`

hn cos~nj!, W~j!5 (n51

`

Wn cos~nj!.

~64!

In practice the infinite series are truncated after a suitalarge number,N say, since the functions computed asmooth and so we can expect exponential decay of the Frier coefficients. It is easy to show, using complex variabmethods, thatHp@cos(nj)#52sin(nx) for n.0. Substitutionof the series~64! into Eqs. ~60!–~61! gives the followingnonlinear system of 2N equations.

For n51,2,...,N:

chn2Wn21

2 (m51,nÞ1

n21

hmWn2m

21

2 (m51,nÞN

N2n

~hmWn1m1hn1mWm!50, ~65!

cWn2n~nl f1le!hn21

4 (m51,nÞ1

n21

WmWn2m

21

2 (m51,nÞN

N2n

WmWn1m50. ~66!

These are 2N equations for 2N11 unknowns, the addi-tional unknown being the wave speedc. An additional equa-tion is required to fix the solutions and we have chosenprescribe the total energy of the interfacial perturbation, i

(m51

N

hm2 5E2. ~67!


ly

u-s

o.,

Note that this is equivalent to choosing the wave minimuand maximum~c1 andc2! in the zero electric field case Eq~58!. The nonlinear system is now closed and solutionsobtained by a Newton iteration. All the results that follow aresolved in the sense that a doubling of the number of moN leaves the solution unchanged to within the iteration terance. The latter was set to between 1026 and 1028. Thematrices obtained are not sparse and a full-pivoting Gauselimination method is used.

B. Numerical results

The problem is nonlinear and it is possible to have mtiple states existing at the same parameter values. An exhtive search of all possible bifurcation branches is not omain interest here. Such bifurcation diagrams can be foby using continuation codes such asAUTO, for example. Weare mainly concerned with constructing nonlinear travelwaves and using them as initial conditions in time dependcalculations to study stability and possible routes to ruptuSuch results are described in a later section.

1. Fully modal traveling waves

The first set of results considers the effect of the elecfield on the traveling waves found by Matsuuchi9 and Me-hring and Sirignano.10 Starting with le50, then, andl f

51/2, we generated one of the ‘‘exact’’ traveling waves uing the iteration procedure described above. The energtaken to beE250.1 and convergence was easily obtainwith N540 starting with the initial conditionsh15h2

5W15W250.01, all other Fourier coefficients set to zeinitially. The converged solution was used to generate sotions at increasingly higher values ofle . Note that the classof solutions reported in this subsection are fully modal, i.


c


FIG. 3. Horizontal velocity traveling waves. Electrifield effects:l f51/2 andle50,1,...,10. The minimumvelocity decreases as the electric field is increased.

-

uen

an

n

thei-

file-

ed

the solutions are of basic period 2p and, except for the expected exponential decay at high wave numbers,hm , Wm arenonzero. Typical results are given in Figs. 2 and 3 for valof le ranging from 0 to 10 in unit increments. The variatioof the wave speedc for values ofle as large as 300, isshown in Fig. 4 which contains also the result ofasymptotic analysis valid for largele given by solution ofEqs. ~69!–~70! below, that predictsc50.8896le

1/2. ~Notethat the lowest value ofle included in Fig. 4 is 1.0, andagreement with the asymptotic solutions is excellent eve


s

at

such low parameter values; as expected, this is lost atlower values 0,le,1.! These results provide strong evdence of an asymptotic regime valid in the limitle@1 whichis analyzed next.

The numerical results suggest that the interfacial proreaches an asymptotic state~see Fig. 1 where profiles become indistinguishable at values ofle larger than about 10!,while the film velocity as well as the traveling wave spescale asle

1/2 ~see Fig. 4!. This limit can be analyzed bywriting

FIG. 4. Variation of the wavespeedc with electric fieldcapillary numberle . Circles—numerical solution forl f51/2; solid line, asymptotic result.


y


FIG. 5. Asymptotic behavior for large electric capillarnumbers. Numerical results scaled withle

1/2 for differ-ent le : dotted line,le55; dash-dot,le510; dash-dash,le5100; solid, asymptotic.

g

p

ticess

n

sic-

a

e

m

ng

e

pa-

es.l

rves

a

ing

ter-

h~j!5h~j!1O~le21!, W~j!5le

1/2W~j!1O~le21/2!,

~68!c5le

1/2c1O~le21/2!,

which on substitution into the system Eqs.~60!–~61! yieldsthe leading order problem

2 chj1Wj1~ hW!j50, ~69!

2 cWj1WWj5H~ hjj!. ~70!

The system~69!–~70! must be solved numerically also usinthe same methods given above withl f50, le51 ~the valueof E250.1 is the same!. We find c50.8896; this is theasymptotic result included in Fig. 4. The asymptotic aproach ofh~j! to its limiting form asle becomes large isessentially given in Fig. 2. A confirmation of the asymptostate with the numerical solutions for the velocity profilW(j;le) is provided in Fig. 5 which compareW(j;le)/Ale with W(j). Solutions forle55, 10, 100 aregiven ~computed with l f50.5, E250.1! along with theasymptotic solutionW(j), and agreement is excellent eveat moderate values ofle .

2. Multimodal traveling waves

By multimodal solutions we mean solutions of a baperiod 2p/k, k52,3,... . The only nonzero Fourier coefficients of such solutions are thehk , h2k , h3k ,..., andsimi-larly for Wk , etc.; this section is devoted to the numericconstruction of such traveling states.

The following scaling property of the traveling wavsystem~60!–~61! is very useful in providing initial condi-tions for the calculations. Suppose

h~j; c,le!, W~j; c,le!,


-

l

is a solution of Eqs.~60!–~61! of minimum period 2p ~allsolutions in the previous subsection are of this type!. Then,given any integerM,

h~Mj;Mc,M le!, MW~Mj;Mc,M le!, ~71!

is also a solution. This property can be verified directly frothe traveling wave equations~60!–~61!. It is the Galileantranslation version of the following more general scaliproperty of the unsteady equations~53!–~54!: If$x,t,u(x,t;l f ,le),S(x,t;l f ,le)%, then $Mx,M2t,Mu(Mx,M2t;l f ,Mle),S(Mx,M2t;l f ,Mle)% is also a solution.Note that theO(M2) time scale is exactly analogous to thtraveling wave form given in Eq.~71!. If the electric field isabsent (le50), the scaling is exact.

We use Eq.~71! to generate initial conditions.~This isnot completely necessary, however: for example with therametersh250.01,W250.02,W450.01,c51, and all otherFourier modes zero, withl f51/2, le50, E250.1, conver-gence with 80 modes to a tolerance of 1029 was achievedwith 13 iterations.! Such shorter multimodal waves can bconstructed, forle50, from the phase plane description Eq~58! and~59! by lettingc0 become large. Once a multimodawave is found, the behavior as the electric field parametele

increases is along the same lines as for fully modal wadescribed in the previous subsection. In Fig. 6 we followbimodal branch asle increases. Profiles are shown forle

50, 5, 10, 100 withl f51/2, E250.1. The lower part of thefigure shows the variation of the wavespeed withle and iscompared with the asymptotic value obtained by solvEqs. ~69!–~70! for bimodal waves; this calculation givesc51.2581 implying the solid line curvec51.2581l1/2 of Fig.6. We have also confirmed the asymptotic behavior of infacial and velocity profiles as before.


c

e


FIG. 6. Bimodal traveling waves at different electricapillary numbersle . Upper left, interfacial ampli-tudes; upper right, horizontal velocity~values of le

given in the figure!. Lower graph; wave speedc vsle—circles are numerical calculations and the solid linis the asymptotic result.

iovns

ls

of

peed

isitaraveby

s.ityall,-

ns

eakal

iesthed.

he-

-nte

Waves can be constructed with different modal behavIt has been found that convergence is quite easily achieby using the scalings given above to guide initial conditioA typical set of multimodal waves having basic periodsp/2,p/4 and p/6 ~that is M54, 8, 12! all at the electric fieldparameterle510 ~also l f51/2 andE250.1 as before!, isgiven in Fig. 7. The wavespeed of the traveling waves is agiven in the figure and the behavior at larger values ofle canbe surmised from the previous discussion~see Fig. 6, forexample!. We see that asM increases and the basic period

FIG. 7. Multimodal traveling waves forle510. Figures on the left represent interfacial amplitudes and on the right the corresponding horizovelocity; the value ofM gives the modal value of the waves and thwavespeed is given on the figure.


r.ed.

o

the waves decreases, the corresponding traveling wave sincreases linearly withM as expected.

The construction of such multimodal traveling wavesuseful in analyzing the stability asymptotically; in the limM@1 the modulational stability can be studied using linehomogenization theory, for instance. Such approaches hbeen applied to the Kuramoto–Sivashinsky equationsFrisch, She, and Thual17 and by an all-orders solvabilitytheory by Papageorgiou, Papanicolaou, and Smyrli18

Analogous multimodal waves also arising from a similarproperty of the equations, have been found by Dando, Hand Papageorgiou19 in their study of the modulational instabilities of Taylor vortices in curved channel flows.

IV. THE INITIAL VALUE PROBLEM: QUASI-PERIODICSOLUTIONS AND FINITE-TIME SINGULARITIES

In this section the evolution equations~53!–~54! are ad-dressed numerically subject to 2p-periodic boundary condi-tions. We study the system for a variety of initial conditioincluding:~i! 2p-periodic initial conditions of varying ampli-tude which are even and odd forSandu, respectively, aboutthe domain center which in our calculations isx5p; in ad-dition the spatial mean ofu is taken to be zero;~ii ! travelingwave states constructed in Sec. III with superposed wperturbations of different long wavelength. Typical initiconditions used for case~i! are

S~x,0!51, u~x,0!5u0 sin~x!.

Such conditions are useful for singularity formation studbecause the even and odd symmetry is preserved duringevolution and the pinch points are symmetrically placeThis choice is not necessary since pinching is a local pnomenon~see Ref. 11 also!.

al


dotarp

eic

rgto

le

intri

oth

itiast

of

roow

nt

inud

iorap-o a

-

ent


Our numerical calculations are carried out using pseuspectral methods in space and a four-stage Runge–Kuttime. The time step is adaptive depending on the numbespatial modes whose amplitudes are greater than somescribed criterion, typically between 1029 and 10212. Further,the solution is spectrally interpolated whenN/225 modeshave an amplitude larger than this prescribed criterion, whN is the number of collocation points. Since the systemdispersive, the accuracy requirement can be severe. Thisbe seen from the linear dispersion relation which for lawave numbersk, say, provides solutions proportionalexp(ikx6ik2lf

1/2t) irrespective of the value ofle5O(1).These solutions rotate the Fourier modes in the compplane and to control such rotations we requirek2Dt,n,wheren is sufficiently small and is between 0.1 and 0.8our calculations. The code maintains this accuracy restion. As a check on the code, we investigate Eqs.~53!–~54!for with le50, that is, with no electric field. This check alsprovides a reference which can be used to quantify howapplied electric field affects the fluid flow.

A. No electric field

1. Quasi-periodic solutions

For moderate values of the initial amplitudeu0 , the sys-tem develops a quasi-periodic in time response. As the inamplitude increases, the multiple frequencies of the quperiodic solutions are better seen as is evidenced fromPoincare´ plots of Fig. 8; the figure shows the evolutioniS21i2 vs iui2 ~where we definei f i25*0

2p f 2 dx! for u0

50.2, 0.3, 0.4, 0.5, 0.6. There is an exchange in energy fthe maximum deflection of the interface and stationary flto that of minimum deflection and the fastest fluid flow.

The smaller initial conditions yield a nearly constaline, suggesting thatiui21iS21i25C, whereC is constantindependent of time, with a simple energy exchange takplace between kinetic and surface energy. As the amplit

FIG. 8. Plot of theL2 norms of the velocityu(x,t) vs the interfacial dis-placementS(x,t)21 parameterized by timet.


-in

ofre-

resane

x

c-

e

ali-

he

m

ge

of the initial condition becomes larger, this simple behavno longer holds, and the effects of nonlinearity becomeparent. Increasing the value of the initial condition leads tdifferent evolution of the system~53!–~54!. In particular, wetake u(x,0)50.8 sinx, S(x,0)51 and introduce the quanti

FIG. 9. Time evolution ofumax(t) ~upper figure!, ux,max(t) ~middle figure!,and Smin(t) ~lower figure! for u(x,0)50.8 sinx, S(x,0)51. Note that theamplitude of the velocity is bounded by 1.2, while the maximum gradican become quite large.


-

elu

buhe0rse

tiobarva

d

als

is

aofto

u-

del

inde

m

ityi-

ty

inch-

i-tme.


ties Smin(t)5min$S(x,t)%, umax(t)5max$u(x,t)%, and ux,max(t)5max$ux(x,t)% to facilitate comparison with the singular solutions of Pugh and Shelley.11

The valueu050.8 is just below the threshold amplitudrequired to produce sheet rupture. Figure 9 shows the vaof umax(t), ux,max(t), and Smin(t) over the duration of thesimulation. It is seen that the sheet is trying to rupturedoes not quite make it. There are short interval bursts wthe minimum sheet thickness reaches a value just abovewhile the corresponding velocity and velocity gradient bualso in a near-singular fashion. There is not enough momtum to carry the sheet to rupture and a quasi-periodic moresults with almost periodic near-singular events. A notafeature of these solutions is their spatial structure particulclose to the near-singular events. The magnitude of thelocity u remains bounded but its gradient develops large vues; the sheet thickness decreases to just above zero anspatial structure emerges~locally in time! which is differentfrom the canonical singular one described next—seeRef. 11.~For brevity, these results are not included.!

2. Singular solutions

When u050.85, the solution becomes singular. Thisshown in Fig. 10, where we plotumax(t) andSmin(t). In thissituation, the maximum of the velocity grows unboundedt→tc'1.8502. This simulation was run until the valueSmin(t)50.001. The singular behavior found is identicalthat reported by Pugh and Shelley11 ~see also Fig. 11!. Pughand Shelley11 were the first to document the singular soltions of Eqs.~53!–~54!. It is useful to briefly review thiswork for two reasons: First validate our code, and, seconset the stage for the electric field calculations. For a wresolved calculation, one can determine the exponentsa, b, cof the similarity solution in the form

Smin;~ tc2t !a,

umax;~ tc2t !b,

FIG. 10. Time evolution ofumax(t) and Smin(t) for the initial conditionsu(x,0)50.85 sinx, S(x,0)51. We see that the magnitude of the velocibecomes singular ast→1.8502.


es

tn

.02tn-n

lelye-l-so a

o

s

tol-

ux2xcu;~ tc2t !c,

wheretc is the critical time~rupture time!, with the relationsb5(a/4)2(1/2), and c5(a/4)1(1/2) following fromorder-of-magnitude arguments; the constanta is not chosenby the system and so a similarity solution of the second kis to be expected.~The reader is referred to Ref. 11 for thdata fitting methods used to determine exponents.! A typicalterminal solution is shown in Fig. 11 when the minimuinterfacial height has reachedSmin50.01.

We have performed data fitting to extract the similarscaling given above for a large set of different initial cond

FIG. 11. Interfacial and velocity profiles att50.136 709 91 from the initialconditionsu(x,0)59 sinx, S(x,0)5110.5 cosx. We see that the velocitybecomes singular as the interfacial height approaches zero at the two ping points.

TABLE I. Comparison of similarity exponents for different initial condtionsu(x,0)5u0 sinx, S(x,0)5110.5 cosx. Note that the exponents do nodepend, to this degree of accuracy, on the initial conditions in this regiThe simulations were terminated whenSmin(t)50.01.

u0 a b b2(a/4)1(1/2) tc

9.0 0.74 20.33 20.0156 0.13757.0 0.74 20.33 20.0145 0.18245.0 0.74 20.32 20.0067 0.26903.0 0.73 20.34 20.0216 0.4983


tye

in--


FIG. 12. Comparison of the magnitude of the velocifield iui2 with the interfacial disturbance amplitudiS21i2 for values of the electric bond numberEb

50.5, Eb51, Eb53, Eb56, for u050.5. Notice thatthe scale of the interfacial deflection decreases withcreasingEb , although the velocity scale remains relatively constant.

n

lue

al3

averuchades

-hee-a-

si-

tions ~in Ref. 11 only one singular run is presented!. We areinterested to see if this similarity form persists for differeinitial conditions.

We summarize our results in Table I which lists the vaof u0 , the exponent forSmin , the exponent forumax, the errorin the calculation from the self-similar form, and the critictime tc . In the least-squares solutions, we always tookpoints from the end of the run which was defined bySmin

,0.01. We see from the table a very similar terminal behior irrespective of initial conditions. As expected, the lowamplitudes provide larger rupture times, but the local strtures are the same. Finally, note that the interfaceevolved so that its minimum is two orders of magnitusmaller than its initial state, analogous to the refined runPugh and Shelley.11


t

0

-

-s

of

B. Dynamics in the presence of electric fields

In the presence of an electric field (leÞ0), the evolu-tion equations~53!–~54! are solved by the methods described earlier with only a minor modification that casts tHilbert transform term into Fourier space. In the results blow, we outline how the strength of the electric field, mesured through the scaled electric capillary numberEb @seeEq. ~55!#, affects the dominant frequencies of the quaperiodic oscillations and the singular solutions of theEb

50 case.

1. Quasi-periodic solutions and finite-timesingularities

As seen already, a linear stability analysis of Eqs.~53!–~52! about the uniform stateu50, S51, gives the dispersion

FIG. 13. ~a! Disturbance voltage profilev(x,t) ~solid line! and interfacial deflectionS(x,t) ~dashed line! at t50.146 832 33 withEb50.5, e53, andu0

58 over 0,x,2p. Notice that the voltage across the singularity nearx52.54 increases significantly.~b! Detailed plot of the boxed region in~a!.



FIG. 14. ~a! Critical time tc as a function ofEb for e53 andu058. ~b! Location of singularityxc as a functionEb for e53 andu058.

s-

ol

ryi-nde

ith

nns

th

thela-

ty

ig.u-aci-

hehis

ltsrmd-cei-

relation v56 i ukuAl fk21leuku. Thus, linear wavespeed

are increased sinceEb.0. Nonlinear effects are tackled numerically, and Fig. 12 shows plots ofiui2 vs iS21i2 overtimes 0,t,200 for different values ofEb with u050.5.These results bring out the quasi-periodic nature of the stions ~the diagrams are essentially Poincare´ maps!. The over-all effect of Eb is a tendency to regularize the oscillatosolutions. The spread in the Poincare´ maps due to the quasperiodicity is reduced significantly by the electric field, aalthough the velocity magnitude remains virtually unchangas compared to theEb50 results~see Fig. 8!, the corre-sponding interfacial magnitude is reduced significantly wincreasing electric field strength.

It is also found that the electric field does not preverupture but instead delays its formation. In the calculatiopresented next, we investigate the singularity in this cabeginning with the results foru058 and Eb51/2 given inFig. 13 at a time very close to rupture. The figure shows


u-

d

tse,

e

final computed values of the sheet thickness along withcorresponding voltage distribution calculated through retion ~50!. As can be seen from Fig. 13~a!, the disturbancevoltage ~solid curve! remains bounded as the singulariforms and the interface~the dashed line! pinches. Figure13~b! shows an enlargement of the boxed region in F13~a!, indicating that the voltage gradient is becoming singlar as the sheet ruptures. Physically, this shows that a captance is developing in the vicinity of the rupture points. Tstructure of the singularity was also investigated for trun. The exponents that were calculated area'0.73 andb520.33, which are in line with the zero electric field resusummarized in Table I. This means that the electric field tedoes not affect the local structure of the singularity, to leaing order at least. This is not surprising given the evidenthat the singularity is of the focusing type with local horzontal extent of orderuxc2xu;(tc2t)c, c.0; this in turnestimates the size ofSxxx to be asymptotically larger than

e
FIG. 15. Comparison of the rupture criterionuc
50.229Eb10.81 found by using a linear fit and thnumerical calculations.


hea

hec

thhes

s

ionsins

eteri-iteser-in

ults

si-on-ues

nsof

ba-

lin


that of the electric field termH(Sxx) by a factor (tc2t)2c ast→tc2.

Having shown typical results for a single value of telectric capillary number, we consider next the behaviorEb is increased. It is found that foru058, which producesrupture for the zero electric field problem, the sheet pincfor higher values ofEb , also. Results are presented colletively in Figs. 14~a! and 14~b!, for Eb ranging from 0 to 50.The figures show the dependence of the rupture time,tc , andthe rupture location,xc , on Eb . Notice that the electric fieldhas a stabilizing influence on the dynamics, prolongingtime to rupture and moving the rupture points closer to eitx50 or x52p. The latter finding implies that the volume

FIG. 16. Final stages of the modulational instability of hexamodal travewaves. The initial condition isu(x,0)5W6(x)10.0001 sin(x). The times aregiven in the figure.


s

s-

er

of the cylindrical filaments that form after pinching takeplace increase with electric capillary number.

So far we have presented results that describe solutas Eb varies for the distinct case when the solution remabounded~at relatively low values ofu0! or the sheet rupturesafter a finite time~relatively high values ofu0!. We describenext, a phase diagram that identifies regions in the paramspace of Eb2u0 which support either bounded quasperiodic in time solutions, or, rupture behavior after a fintime. This is constructed numerically for a relatively coargrid resolution of 256 modes, which is sufficient for the puposes of the diagram. The maximum time of integrationthe case of quasi-periodic solutions is 500 units. The resare shown in Fig. 15 for the range 0<Eb<2. The upperregion denotes pairs of values of (Eb ,u0) which producepinching, and the lower region those that result in the quaperiodic bounded oscillations of the sheet. The diagram ctains also a linear fit to the data produced by the two valof Eb50, 1, which is seen to be quite reasonable.

2. Modulational instabilities of periodic travelingwaves

Next we consider the stability of the periodic wave traifound in Sec. III. Denoting a multimodal traveling wavebasic period 2p/M by @hM(j),WM(j)#, we introduce initialperturbations as follows:

S~x,0!511hM~j!1ds sin~px!,~72!

u~x,0!5WM~j!1du sin~px!,

where the amplitudes of the disturbance areds anddu and,more importantly, the disturbance wavelength is 2p/p wherep is an integer satisfyingp<M . In most runs we takep51 which corresponds to the longest wavelength perturtion in our setup.

g

ero
FIG. 17. Stable dodecamodal wavetrains under zinitial perturbation.

ayi-

ngthInovbis

s,unthric

ta

viil

nsp

a

editsealsngis

on-larf

srter

thethe

meod

rba-


The first set of results consider the stability of hexmodal (M56) traveling waves calculated in Sec. III. A verlow level of perturbation is introduced and the initial condtions are Eq.~72! with

M56, ds50, du50.0001, p51.

If no disturbance is introduced, it is found that the traveliwaves are stable and continue to propagate preservingform, at least for the relatively long times of integration.the presence of a long wavelength perturbation as abhowever, the traveling waves are modulationally unstaand in fact encounter a pinching finite-time singularity. This consistent with Matsuuchi’s weakly nonlinear analysi8

and a computation of a hexamodal wave perturbed by amodal wave of amplitude 0.001. Our results indicate thatmodulational instability is active in the presence of electfields also. Results are shown in Fig. 16 which hasl f51/2andle51. The figure shows the last stages of the compution before the sheet ruptures. The behavior is seen todifferent from the pinching solutions presented in the preous subsection in that the wave is traveling to the right whthinning; this results in sizeable oscillations ofSmin(t) beforepinching, as seen from the profiles in the figure.

Similar results are found for other multimodal solutioalso. We conclude with a case that has a large electric calary numberle510 and a basic wavelength ofp/6 ~i.e., M512!. In Fig. 17 we show the evolution of the dodecamodtraveling wave without any initial perturbations~i.e., ds

5du50! up to times of seven time units. Profiles are plottevery 0.1 time units and are shifted vertically by 1.2 unThe wave is seen to travel to the right with its shape esstially unchanged~a detailed analysis of the data indicates ththere is a very weak quasi-periodic oscillation; this was aconfirmed in runs which begin with traveling waves havile50 and calculating their evolution when the operatorperturbed by a small nonzerole!.


-

eir

e,le

i-e

-be-e

il-

l

.n-to

The same traveling waves are found to be modulatially unstable, however. This has been found for the particucasesds50 anddu50.01, 0.05, with both disturbances owavelength 2p (p51) ~see Fig. 18!. The flow results into apinching singularity and we show the solutions att50 and att52.2 and 1.6, respectively~note that the singularity occurvery rapidly soon after these times!. As expected, the higheamplitude initial condition leads to a singularity at a shortime.

In order to emphasize the modulational character ofinstability, we present results from a run which perturbsdodecamodal traveling waves with an amplitudedu50.01,but with the wavelength of the perturbation being the saas that of the underlying traveling waves, that is of peri

FIG. 18. Unstable dodecamodal wavetrains under small initial pertutions. Top figures: perturbationdu50.01. Lower figures: perturbationdu

50.05.

er
FIG. 19. Stability of dodecamodal wavetrains undsmall initial perturbations of the form 0.01 sin(12x).

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p/6 or p512. The integration was carried tot512.8 andthere was no sign of an instability. As before, the responsthe weak quasi-periodic modulation of the traveling wawith no singular behavior found. The results are shownFig. 19 which plots the interfacial amplitude every 0.1 timunits along with a vertical shift of one unit for each profilefacilitate the graphics.

V. DISCUSSION

We derived a system of evolution equations that descthe nonlinear stability of an inviscid, nonconducting thin liuid sheet. The system of equations retains surface teninertia, and polarization for varicose solutions and is capaof describing a variety of nonlinear phenomena includsheet rupture.

The dynamics elucidated in this study can be dividinto solutions which do not lead to rupture and those thatNonrupturing states evolve generically from sufficiensmall amplitude initial conditions~see below for a violationof this when modulational instabilities are present, howev!.The flow response in this case is a quasi-periodic oscillain time; we have calculated a phase diagram~see Fig. 15, forexample! which delineates bounded from unbounded sotions at least for the initial conditionsS(x,0)51, u(x,0)5u0 sin(x) considered here. This behavior is seen whetthe electric field is present or not. In its absence, therereduction in the amplitudes of oscillation, even thoughvelocities are not reduced significantly.

If the initial amplitudeu0 is sufficiently large, the solu-tions terminate into finite-time singularities. Different initiaconditions are studied and it is found that the structure ofsingularity is independent of both initial conditions and tintensity of the electric field as measured through the eleccapillary number parameterle . The effect of the electricfield is in delaying the formation of the singularity. We hanot encountered a case where the singularity is removethe electric field. This possibility may arise at asymptoticalarge values ofle where the scalings of Sec. III B 1 are petinent. The canonical system to be solved is as before E~53!–~54! but with l f50, le51. Due to the change of thoperator, we expect different singularity structures. Tproblem is not addressed in the present work.

The present work has also considered the effect ofelectric field on traveling wave solutions initially studied bMatsuuchi9 and Mehring and Sirignano.10 The effect of theelectric field on these solutions is threefold:~i! The minimumthickness of the fluid sheet is decreased;~ii ! the local hori-zontal velocity in the sheet attains much higher magnitu~in fact these scale asle

1/2; see Sec. III!; ~iii ! the Galileanspeed of the traveling waves also increases as in~ii !. Thestability of these traveling waves is studied numericallyimposing initial perturbations of different amplitudes awavelengths~hence, our stability study is nonlinear!. Amodulational instability is found: perturbations of very smamplitude~about 0.01% of the undisturbed sheet thickne!


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and wavelengths longer than the basic traveling wave lenlead to initially exponential growth which causes nonlineamplification and eventually leads to finite-time singularitiand sheet rupture. The dynamics in this case are quite cplicated due to the interaction between the forming singuity and the momentum of the traveling wave. Even thouwe did not analyze the singularity structures in this casetensively, we expect a similar behavior to that found fromore generic initial conditions.

ACKNOWLEDGMENTS

B.S.T. was supported by a grant from the New JersInstitute of Technology, Grant No. 421610, and a grant frothe National Science Foundation, DMS-9971383. P.G.P.supported by a Grant from the Air Force Office of ScientiResearch, AFOSR Grant No. F49620-99-1-0072. D.T.P.supported by the National Science Foundation, Grants NDMS-9904793 and DMS-0072228.

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Dynamics and rupture of planar electrified liquid sheets

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