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undulations later grow large enough to break the jet.Savarts research showed that (i) breakup always occursindependent of the direction of gravity, the type of uid,or the jet velocity and radius, and thus must be an in-trinsic property of the uid motion; (ii) the instability of the jet originates from tiny perturbations applied to the jet at the opening of the nozzle.

In spite of his fundamental insights, Savart did notrecognize surface tension, which had been discoveredsome years earlier (de Laplace, 1805; Young, 1805) asthe source of the instability. This discovery was left to(Plateau 1849), who showed that perturbations of longwavelength reduce the surface area and are thus favoredby surface tension. On a level of quasistatic motion itwould thus be desirable to collect all the uid into onesphere, corresponding to the smallest surface area. Evi-dently, as shown in Fig. 1, this does not happen. It wasRayleigh (1879a,1879b) who noticed that surface tensionhas to work against inertia, which opposes uid motionover long distances. By considering small sinusoidal per-turbations on a uid cylinder of radius r , Rayleigh foundthat there is an optimal wavelength, R 9r , at whichperturbations grow fastest, and which sets the typicalsize of drops. Analyzing data Savart had obtained al-most 50 years earlier, Rayleigh was able to conrm histheory to within 3%.

Accordingly, the time scale t 0 on which perturbationsgrow and eventually break the jet is given by a balanceof surface tension and inertia, and thus

t 0r 3

1/2

, (1)

where is the density and the coefcient of surfacetension of the uid. This tells us two important things:Substituting the values for the physical properties of wa-ter and r 1 mm, one nds that t 0 is 4 ms, meaning thatthe last stages of pinching happen very fast, far belowthe time resolution of the eye. Secondly, as pinchingprogresses and r gets smaller, the time scale becomes

shorter and pinching precipitates to form a drop in nitetime . At the pinch point, the radius of curvature goes tozero, and the small amount of uid left in the pinchregion is driven by increasingly strong forces. Thus thevelocity goes to innity, and the separation of a dropcorresponds to a singularity of the equations of motion,in which the velocity and gradients of the local radiusdiverge.

Even in the case of an innite-time singularity of theequations of hydrodynamics, the physical event of breaking may occur in nite time. That is, when the uidthread has become sufciently thin, it may break owing

FIG. 1. A dolphin in the New EnglandAquarium in Boston, Massachusetts; Edger-ton (1977). The Harold E. Edgerton 1992Trust, courtesy of Palm Press, Inc.

866 Jens Eggers: Nonlinear dynamics and breakup of free-surface ows

Rev. Mod. Phys., Vol. 69, No. 3, July 1997


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to microscopic effects that are outside the realm of hy-drodynamics. The crucial distinction from the nite-timesingularity, which results from surface tension, is thatthere is a chance to describe breaking in terms of con-tinuum mechanics alone without resorting to micro-scopic notions. The description of such singularities willform a substantial part of this review.

For one hundred years after Rayleighs original work,theoretical research focused on the extension of his re-sults on linear stability. For example, Rayleigh (1892)himself considered a highly viscous uid, but the generalNavier-Stokes case was only treated in 1961 by Chan-drasekhar. Tomotika (1935) took the surrounding uidinto account; Keller, Rubinow, and Tu (1973) looked atthe growth of a progressive wave rather than a uniformperturbation of a cylinder. To illustrate the power of Rayleighs ideas even in completely different elds wemention the application of linear stabilty to the breakupof nuclei (Brosa, Grossmann, and Mu ller, 1990), in

which case the equivalent of a surface tension has to becalculated from quantum mechanics. Another exampleis the instability observed on pinched tubular vesicles(Bar-Ziv and Moses, 1994), where entropic forces drivethe motion.

Meanwhile, experimental results had accumulatedthat probed the dynamics of free surfaces beyond thevalidity of linear theory. Early examples include Ray-

leighs photographs of jets (1891), Worthingtons studyof splashes (1908), and Edgerton, Hauser, and Tuckers(1937) photographic sequences of dripping faucets. Ex-perimental techniques have also become available morerecently with sufcient resolution in space and time tolook at the immediate vicinity of the point of breakup.Notable examples include the jet experiments of Rut-land and Jameson (1970) as well as those of Goedde andYuen (1970) for water jets and of Kowalewski (1996) for jets of high-viscosity uids. A momentous paper by Per-egrine, Shoker, and Symon (1990) not only helped tocrystallize some of the theoretical ideas, but also con-tained the rst high-resolution pictures of water falling

from a faucet. For higher viscosities, corresponding pic-tures were taken by Shi, Brenner, and Nagel (1994).By comparison, the development of computer codes

that would permit the calculation of free-surface owsfrom rst principles has been slow. Owing to the dif-culties involved in implementing both moving bound-aries and surface tension, resolution has not been pos-sible anywhere near the experimentally attainable limit,even with present-day computers. An important excep-tion is the highly damped case of the breakup of a vis-cous uid in another, which recently led to a detailedcomparison between experiment and numerical simula-tion (Tjahjadi, Stone, and Ottino, 1992).

Only gradually did the theoretical tools evolve thatallowed for an analytical description of the nonlineardynamics close to breakup. The rst was developed inthe theory of waves and often goes by the name of lu-brication theory or the shallow-water approximation(Peregrine, 1972). It captures nonlinear effects in thelimit of small depths compared with a typical wave-length. During the 1970s, lubrication approximationswere developed for the corresponding axisymmetricproblem, to study drop formation in ink-jet printers.This is of particular relevance since a jet does not breakup uniformly, as predicted by linear theory, but ratherinto main drops and much smaller satellite drops. Thesatellite drops fundamentally limit the print quality at-tainable with this technology, as drops of different sizesare deected differently by an electric eld, whichshould direct the stream of droplets to a given positionon the paper. Thus a fully nonlinear theory is needed tounderstand and to control satellite formation. The rstdynamical equation, based on lubrication ideas, was in-troduced by Lee (1974) for the inviscid case. His nonlin-ear simulations indeed showed the formation of satellitedrops. But it took two decades until systematic approxi-mations of the Navier-Stokes equation were found thatincluded viscosity (Bechtel, Forest, and Lin, 1992; Egg-ers and Dupont, 1994).

FIG. 2. Perturbations growing on a jet of water (Savart, 1833).

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Another important concept, which allows for the de-scription of nonlinear effects, is that of self-similarity(Barenblatt, 1996), which arises naturally in problemsthat lack a typical length scale. In the case of a singular-ity, the length scale of the solution will depend on time,reaching arbitrarily small values in the process. Thusself-similarity here means that the solution, observed atdifferent times, can be mapped onto itself by a rescalingof the axes. In the context of ows with surface tension,self-similarity was introduced by Keller and Miksis(1983). Kadanoff and his collaborators (Constantinet al. , 1993; Bertozzi et al. ., 1994) have looked at singu-larities in a Hele-Shaw cell, which is the two-dimensional analogue of the present problem, as asimple model for singularity formation. They inge-niously combined lubrication ideas and self-similarity toarrive at a detailed description of the pinchoff of abubble of uid.

In the wake of this success, Eggers (1993) and Eggersand Dupont (1994) applied the same idea to the three-dimensional case. As spelled out rst by Peregrine et al.(1990), the dynamics near breakup are independent of the particular setup such as jet decay, a dripping faucet,or even the complicated spraying shown in Fig. 1, butrather are characteristic of the nonlinear properties of the equations of motion. As the motion near a point of breakup gets faster, only uid very close to that point isable to follow, making the breakup localized both inspace and time. Thus one expects the motion to becomeindependent of initial conditions, and the type of experi-ment becomes irrelevant to the study of the singular mo-tion. This brings about two crucial simplications: (i) ina local description around the point of breakup, the mo-tion becomes universal, thus reducing the number of relevant parameters. The only parameter upon whichthe motion near the singularity still depends is the length

l 2

, (2)

which characterizes the internal properties of the uid(Peregrine et al. , 1990; Eggers and Dupont, 1994); (ii) anasymptotic analysis of the equations of motion revealsthat the motion close to the singularity is self-similar,with the radius shrinking at a faster rate than the longi-tudinal extension of the singularity. Near the pinchpoint, almost cylindrical necks develop, making the mo-tion effectively one-dimensional close to the singularity.

Using these ideas, a local solution of the Navier-Stokes equation was found, which contained no free pa-rameters (Eggers, 1993). To select a specic predictionof this theory, the minimum radius of a uid thread at agiven time t away from breakup found to be

h min 0.03

t . (3)

The surface proles calculated from theory have beencompared quantitatively and conrmed by experiment(Kowalewski, 1996). The columnar structure of the uidneck allows for a stability analysis of the ow close tothe breaking point, and is modeled closely on Rayleighs

analysis of a liquid cylinder (Brenner, Shi, and Nagel,1994; see also Brenner, Lister, and Stone, 1996). As theneck becomes sufciently thin, it is prone to a nite-amplitude instability, which may be driven by thermalnoise. This causes secondary necks to grow on the pri-mary neck, which again have a self-similar form. Thecorresponding complicated structure of nested singulari-ties has also been observed experimentally (Shi, Bren-

ner, and Nagel, 1994).At the same time stability analysis indicates that cy-lindrical symmetry is not just a matter of convenience,but rather a generic property near breakup. Rayleighsanalysis tells us that any azimuthal variation results inonly a relative increase in surface area and is thus unfa-vorable. The universality and stability of the solutionnear breakup therefore lead to answers of a muchgreater generality than could be hoped for by investigat-ing individual geometries and initial conditions. At thesame time, the singular motion is the natural startingpoint for the calculation of nonlinear properties awayfrom breakup, which controls phenomena such as satel-

lite formation. Another advantage of universality is thatonly one particular initial condition needs to be investi-gated to construct a unique continuation of the Navier-Stokes equation to times after the singularity (Eggers,1995a). This establishes that breaking is described bycontinuum mechanics alone, without resorting to a mi-croscopic description, as long as observations are re-stricted to macroscopic scales.

The scope of this review is limited mostly to the dy-namics in the immediate vicinity of the point of breakup.This is motivated by the expectation that pinching is uni-versal under quite general circumstances, even if themotion farther away from the singularity is more com-plicated. In the nonasymptotic regime, our focus is onthe axisymmetric case of a jet with or without gravity.This excludes many important examples of nonlinearfree-surface motion, such as drop oscillations, the dy-namics of uid sheets, and in particular the vast eld of surface waves.

We begin with an overview of the experimental basisof the subject. Here and in the rest of this review, weconne ourselves to cylindrical symmetry. In the case of free surfaces, this is representative of the majority of experimental work in the physics literature. But it ex-cludes important effects like bending (Entov and Yarin,1984; Yarin, 1993), branching (Lin and Webb, 1994), andspraying (Yang, 1992) of jets. There is also substantialwork on splashes, i.e., the impact of drops on liquid(Og uz and Prosperetti, 1990) or on solid surfaces (Yarinand Weiss, 1995) surfaces. Mixing processes can also notbe expected to respect cylindrical symmetry. The outlineof experimental work in the second section is comple-mented by a review of numerical work in the third sec-tion. As indicated above, numerical simulations of thefull hydrodynamic equations are only slowly catching upwith the resolution possible in experiments. On theother hand, important information on the velocity eldis not available experimentally. This and the superiorvariability of simulations, for example, in the choice of




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uid parameters and of initial conditions, is bound tomake simulations an important source of information.

In the fourth section we give a detailed account of linear stability theory, which is the classical approach tothe problem, but which remains an area of research tothe present day. Some nonlinear effects can be includedin perturbation theory, but the expansion quickly breaksdown near pinching.

The groundwork for the description of nonlinear ef-fects is laid by the development of one-dimensionalmodels. We spell out two different approaches to theproblem and explain some of the properties of the re-sulting models in Sec. V. In Sec. VI we study in detailthe universal self-similar solution leading up to breakup.The solution can be continued uniquely to a new solu-tion valid after breakup, which now consists of twoparts. The nonlinear stability theory of the asymptoticsolution explains the complicated structure seen in thepresence of noise.

Section VII explores the dynamics away from theasymptotic regime, but where nonlinear effects are stilldominant. The area best understood is the case of highlyviscous jets, in which the pinching has not yet becomesufciently fast for inertial effects to become important.In the opposite limit of very low viscosity, the smoothingeffect of viscosity is missing, and gradients of the oweld become large at a nite time away from breakup.This makes the problem a hard one, and the understand-ing of this regime is only in its early stages. However,this subject is bound to remain an interesting and fre-quently studied topic for the years to come, since lowviscosity uids like water are the most common. From atheoretical point of view, the understanding of the sin-gularities of the Euler equation is one of the major un-solved problems in hydrodynamics, and uid pinchingserves as a particularly simple model system. To con-clude Sec. VII, we describe some research on satelliteformation and present a numerical simulation of the sta-tionary state of a decaying jet.

So far we have dealt only with free surfaces, with sur-face tension being the only driving force. We relieve thisrestriction in the nal section, where we explore someexamples of related topics. First we look at two-uidsystems, which are particularly important for the theoryof mixing and the hydrodynamics of emulsions. Anasymptotic theory for breakup in the presence of anouter uid has not yet been developed. Electric or mag-netic elds represent another possible external driving

force. They force the uid into sharp tips, where theelds are strong, out of which tiny jets are ejected. Thisallows for the production of very ne sprays. In chemicalprocessing, macromolecules are often present in solu-tion. They result in non-Newtonian properties of theuid, to which we give a brief introduction in the contextof free-surface ows.

II. EXPERIMENTS

Historically, research on drop formation was moti-vated mostly by engineering applications, hence the

three most common experimental setups, which are de-scribed in more detail below: (1) Jets are produced whena uid leaves a nozzle at high speeds; (2) slow drippingunder gravity has been used for the measurement of sur-face tension; and (3) liquid bridges are used to suspenduid in the absence of gravity. For a review on dropformation in the context of engineering applications of spraying, see Walzel (1988). Early work focused either

on the early stages of drop formation, characterized bythe growth of linear disturbances, or on the size andnumber of the resulting drops. Either aspect of drop for-mation is relatively easy to observe, but is highly depen-dent on the experimental setup and on parameters likenozzle diameter or jet speed.

Only slowly, as experimental techniques becameavailable for observing the actual evolution of the owduring drop formation, did common features emergefrom the seemingly disparate results of individual ex-periments. The last stages of the evolution are domi-nated by the properties of the pinch singularity, which isthe same for all cases. This idea was rst enunciated

clearly by Peregrine et al. (1990). The appearance of themotion depends only on the scale of observation l obsrelative to the size of the internal length l [see Eq. (2)]of the uid. If l obs / l is large, which is typical for owsof low viscosity like water, the shapes near the pinchpoint are cones attached to a spherical shell. Afterbreakup, as the uid neck recoils, capillary waves areexcited. If on the other hand l obs / l is small, as foruids of high viscosity like glycerol, long and skinnythreads are observed, which rapidly contract into tinydrops after breakup. At the highest viscosities, thethreads tend to break at random places. It is from thisuniversality that much of the physical interest of the sub-

ject of drop formation derives, and we try to emphasizecommon features in discussing different kinds of experi-ments.

A. Jet

By far the most widely used experimental setup in thestudy of drop formation is that of a jet of uid leaving anorice at high speeds. The earliest jet experiments wereperformed with uid being driven out of holes near thebottom of a container (Bidone, 1823). The focus of theearly research was on the shape of jets produced by ori-ces of different forms. It was Savart (1833) who dis-tinctly noticed the inevitability of a decay into drops andcarefully investigated the laws governing it. By deliber-ately disturbing the jet periodically at the nozzle, he pro-duced disturbances on the surface of the jet with thesame frequency. Many other 19th-century researchersrepeated these experiments, notably Hagen (1849),Magnus (1855), and Rayleigh (1879b,1882). Both Pla-teau (1873) and Rayleigh were able to perform somequantitative tests of their theories, but without photog-raphy it was impossible to record the shapes of jets indetail. Photographic methods were introduced by Ray-leigh (1891), but these observations were only qualita-tive in nature. The rst quantitative experiments were




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those of Haenlein (1931), Donelly and Glaberson(1966), and Goedde and Yuen (1970). Their main con-

cern was to test the linear theories of Rayleigh (1879a,1892) and Chandrasekhar (1961) for different viscosities.Goedde and Yuen also recorded shapes characterizingthe nonlinear behavior near breakup. Experiments fo-cusing on the measurement of satellites were performedby Rutland and Jameson (1970), Chaudhary and Max-worthy (1980a,1980b), and Vasallo and Ashgriz (1991).Becker, Hiller, and Kowalewski (1991, 1994) studied thenonlinear oscillations of drops produced in the breakupprocess. Their experimental setup was used by Kow-alewski (1996) to record surface shapes near breakupwith a spatial and temporal resolution far superior toprevious work. We shall describe this experiment insome detail, since it demonstrates the degree of sophis-tication the experimental technique has acquired overthe last few decades.

Figure 3 gives an overall ow chart of the apparatus(Becker, Hiller, and Kowalewski, 1991). Fluid is forcedout of a nozzle directed vertically downward, so the axisof symmetry is preserved. Typical jet speeds are of theorder of m/s, so the air drag can be neglected. On theother hand, the breakup takes place far from the nozzle(approximately 1000 jet diameters), so the entire regionof interest can be regarded as being in free fall. Nozzlediameters vary between 1/10 mm and 2 mm. In an ante-chamber of the nozzle a piezoceramic transducer pro-duces pressure oscillations, which translate into sinu-soidal disturbances of the jet speed at the nozzle exit. Asa result, the jet breaks into drops in a perfectly periodicfashion. This is an advantage of this experimental setup,as it is amenable to observations by the stroboscopicmethod pioneered by Rayleigh (1882).

To this end the region of interest is illuminated withthe light of a pulsed light-emitting diode (LED), thetypical duration of the ash being 0.2 s. The jet is thenviewed through a microscope and images are recordedwith a charge-couple device (CCD) camera. The opticsis set up in bright eld illumination, exposing parallellight to the camera, so that the image is bright. If a piece

of uid is in the path of the light, the light is diffractedaway from the cameras view, so it appears black, apartfrom a bright center, where light passes straight throughthe uid. This allows for spatial resolution of about 1

m. To make observations of a given stage in jet decay,one arranges for the ash to illuminate the jet with thesame frequency with which it is excited. If the light pulseis ensured to be in a xed phase with the current driving

the jet modulator, a stationary image is seen. By chang-ing the phase, a different time within the evolution of one period is illuminated. The whole process is so stableand reproducible that a time resolution of 10 s isreached. By scanning an entire range of phases, moviesof the breakup process with this time resolution can beproduced, some examples of which are presented in Sec.VI.

We now discuss the dimensionless parameters con-trolling jet decay. We shall assume that the driving ispurely sinusoidal, so that the speed at the nozzle is

v nozzle v j

r 0

1/2

sin 2 ft . (4)

Here v j is the speed of the jet and r 0 its unperturbedradius. The dimensionless perturbation amplitude ismultiplied by the capillary velocity u 0 ( /( r 0) )1/2. Byproperly adjusting the driving frequency f , we canchoose a wavelength v j / f . In typical experimentalsituations the frequency is several kHz.

The parameter most signicant to jet decay is the re-duced wave number

x 2 r 0 / . (5)

If x 1, the jet is unstable to the corresponding pertur-bations, and at small viscosities, x R 0.697 is the mostdangerousor the Rayleigh mode for which pertur-bations grow fastest. For x 1 irregular breakup is ob-served, as the jet responds to tiny random perturbationswith components x 1. The temporal perturbation istranslated into space by convection with velocity v j . Thetime scale on which surface perturbations grow ist 0 ( r 0

3/ ) 1/2, since surface-tension forces will be bal-anced by inertia, at least at low viscosities. Thus the We-ber number

2 r 0v j 2/ (6)

measures how much a disturbance can grow from oneswell to the next. Typical values of are O (100). Thelast parameter is the Reynolds number 1

Re u 0r 0 / r 0 / l 1/2, (7)

1It should be noted that the nomenclature in Eq. (7) is opento criticism, since at high viscosities the velocity scale is nolonger set by surface tension and initial radius alone, but alsodepends on viscosity. A better name, used for example in Mc-Carthy and Molloy (1974), is Ohnesorge number. An Ohne-sorge number conventionally is a number that depends on ge-ometry and uid properties alone. However, since Reynoldsnumber is so much more likely to carry a meaning for physi-cists, we decided to be cavalier about these subtleties.

FIG. 3. Experimental setup for the stroboscopical observationof a decaying jet as developed by Becker, Hiller, and Kow-alewski (1991).




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constructed from the jet radius r 0 and the capillary ve-locity u 0, divided by the kinematic viscosity / . Itmeasures the damping effects of viscosity on the motion

caused by surface tension. For water and a jet diameterof 1 mm, Re 200, but technologically relevant uidscover a wide range of different viscosities. For example,in the case of glycerol the Reynolds number is reducedto 0.5, and by mixing water and glycerol a wide range of Reynolds numbers can be explored.

So, in the case of purely sinusoidal driving, there arefour dimensionless parameters governing jet decay: thedriving amplitude , the reduced wave number x , theWeber number 2, and the Reynolds number Re. Therange of possible dynamics in this large parameter spacehas never been fully explored, so we shall focus on thedependence on the two most important parameters, thereduced wave number and the Reynolds number.

Figure 4 shows typical pictures of a decaying jet of water for three different wavelengths. The bottom pic-ture is for the mode of maximum instability. It is easilyfound by tuning the frequency, since it makes thebreakup point move closest to the nozzle. This situationis the least sensitive to noise, since all other frequencieshave lower growth rates and therefore decay relative tothe Rayleigh mode. Up to one wavelength from thepoint where a drop rst separates, the disturbances lookfairly sinusoidal. (There are signicant higher-order cor-rections, though, which will be discussed in Sec. IV.) Butthe last neck pinches off almost simultaneously at bothends, causing signicant deviations from linear, sinu-soidal growth. This nal, localized pinching producescharacteristic forms, namely, a sharp conical tip attachedto a at cap. In recoiling, the tip excites capillary waveson its surface, which give it the appearance of a string of pearls. While the tip is still recoiling, it breaks on its rearside and starts recoiling on the other side as well. Thus asmall satellite drop is produced, which is a remnant of the neck. In general it will receive momentum from therecoiling process and therefore has a velocity slightlydifferent from the main drops. This will make it mergeeither with the preceding or the following drop a fewwavelengths downstream.

If the wavelength is long enough, the growth rate of the second harmonic will be larger than that of the pri-mary disturbance. Since higher harmonics are always ex-

cited at the nozzle or through the nonlinear interaction,a swell develops in the middle between the drops. If ithas a chance to grow sufciently large, the jet will breakin this /2 mode and produce drops and satellites attwice the fundamental frequency of excitation. This isshown in the middle picture of Fig. 4 for a driving fre-quency that is a factor of 0.36 below the frequency cor-responding to the Rayleigh mode. The appearance of such a double stream of droplets thus depends sensi-tively on the amplitude of the second harmonic pro-duced by the driving. Plateau (1857) used a cello to ex-cite the jet, and always found a double stream at half thefrequency of the Rayleigh mode. Later Rayleigh (1882)showed that this was due to harmonics inconvenientlyproduced by a musical instrument. Using tuning forksinstead, he still observed breakup with the principal fre-quency at a third of the Rayleigh frequency. For evenlonger wavelengths (top picture), the satellite drop be-comes substantial, owing to the much longer neck. Thiscauses the recoil patterns to be even more pronounced,since there is more time for capillary waves to develop.As a result, the satellite drop is subject to very compli-cated secondary breakings.

Decreasing the Reynolds number signicantly, for ex-ample, by increasing the viscosity using a glycerol-watermixture, causes the breakup process to change substan-tially. After the initial sinusoidal growth, a region devel-ops where almost spherical drops are connected by thinthreads of almost constant thickness, which take quite along time to break (see Fig. 5). In general, the threadwill still break close to the swells. If the viscosity is in-creased further, the threads become so tenuous beforethey break at the end, that they break at several placesin between, in what seems to be a random breakup pro-cess.

Jet experiments are particularly useful for studyingthe universal motion near breakup with extremely highprecision, as we shall see in Sec. VI. On the other hand,it is hard to design an experiment to make the full evo-

FIG. 4. Photographs of a decaying jet (Rut-land and Jameson, 1971) for three differentfrequencies of excitation. The bottom picturecorresponds to x 0.683, which is close to theRayleigh mode. At longer wavelengths sec-ondary swellings develop (middle picture,

x 0.25), which cause the jet to break up attwice the frequency of excitation. At the long-est wavelength (top picture, x 0.075) mainand secondary swellings have become virtu-ally indistinguishable. Reprinted with permis-sion of Cambridge University Press.




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lution of the jet leading up to breakup completely repro-ducible. This is because the driving is never purely sinu-soidal, but contains harmonics that depend on the nozzledesign and that signicantly inuence jet evolution.Thus it is hard to make exact quantitative comparisonswith numerical simulations, comparisons that areneeded to validate numerical codes. The great merit of the two other experiments to be described in the follow-ing two subsections is that they are both simple, and allof the dynamics are completely reproducible.

B. Dripping faucet

This experimental setup involves the opposite limit tothat of a jet: Fluid is released very slowly from a nozzle,so that at rst surface-tension forces are in balance withgravitational forces. One can easily reach a limit wherethe ow plays no role and the hanging drop goesthrough a sequence of equilibrium shapes. These se-quences have been studied very carefully by Worthing-ton (1881). The theory of their stability was begun byPlateau (1873) and Maxwell (1875) and is summarizedby Michael and Williams (1976). Instability will set insooner or later as the drop becomes heavier and gravityovercomes surface tension. All these static aspects canonly depend on the dimensionless number

Bo gr 02/ , (8)

which is called the Bond number and expresses the ratioof gravitational forces to surface-tension forces.

After the initial instability, the uid begins to fall andeventually a drop separates. This familiar phenomenonhas been the subject of some early work (Guthrie, 1864;Tate, 1864). Tate measured the drop weight W and no-ticed a proportionality to the nozzle radius. Rayleigh(1899) analyzed the problem in terms of dimensionlessgroups and arrived at the simple empirical relationW 3.8 r 0. To produce an accurate method for themeasurement of surface tension, Harkins and Brown(1919) performed a careful experimental study of therelation between the dimensionless drop weightW /( r 0) and the Bond number. They found a compli-cated functional form, which deviated considerably from

the proportionality proposed by Tate. In fact, it waspointed out by Hauser et al. (1936), as well as Edgerton,Hauser, and Tucker (1937), that it was unlikely that asimple theoretical derivation for such a relation existed.In fact, their high-speed motion pictures revealed a verycomplicated structure produced by the neck between themain drop and the nozzle. As in the jet experiments, thisstructure resulted in the production of one or more sat-

ellite drops. They also noticed a signicant dependenceon viscosity, with a long and thin neck forming at highviscosities. The beauty of the experiment lies in the factthat the only other dimensionless parameter needed tospecify the entire evolution is the Reynolds numberRe ( r 0 / 2 ) 1/2.

But it was only the pioneering work of Peregrine et al.(1990) that focused on the dynamics immediately beforeand after the bifurcation. The whole sequence of eventscontains no free parameters, which makes it an idealtesting ground for theory. Later, this work was extendedto higher viscosities by Shi, Brenner, and Nagel (1994).Figure 6 shows a sequence of single ash photographs of

a water drop. Shortly after the lower part of the hangingdrop begins to fall, it produces a neck on which surfacetension acts, making it thinner. At a certain stage, pinch-ing sets in and a drop separates. Again the pinch point isvery localized and the shapes remarkably resemble Fig.4. Recoil produces capillary waves, but before the tipcan completely retreat the neck breaks at the other end,the at part now being on top of the cone. The wholesequence of pictures is completely reproducible. Quali-tatively, this process is quite similar for different Bondnumbers, but the size of the main drop and the length of the neck increase with Bond number. At high viscosities,a transition to long and thin necks is observed. A par-ticular example is shown in Fig. 7, with a uid 100 timesmore viscous than water. Note the emergence of a tinythread coming out of the neck just above the drop. Justas in the jet experiments, at the highest viscositiesthreads form which are thin enough to be prone to ran-dom breakup.

In addition to the single ash photographs describedabove, Zhang and Basaran (1995) and Brenner et al.(1997) produced time-resolved motion pictures of thebifurcation. From those, the temporal dependence of neck radius and neck length could be measured. Zhangand Basaran not only varied the radius of the nozzle andthe viscosity of the uid, but also the ow rate, to obtaina detailed phase diagram of main drop and satellitesizes, nal neck lengths, etc. The dependence of dropsize on the ow rate has also been measured and com-pared with a simple one-dimensional theory by Wilson(1988). Singular properties like the temporal evolutionof the radius of the neck close to pinchoff were found tobe insensitive to the ow rate (Zhang and Basaran,1995). On the other hand, viscosity had a profound in-uence, not only on the nal length of the neck, but alsoon the time dependence of the radius. This is in accordwith the expectation that singular properties dependonly on the internal length l . To measure the neckradius well below the time scale t 0 ( r 0

3 / ) 1/2, which is

FIG. 5. A photograph of a viscous jet (Donnelly and Glaber-son, 1966) for a reduced wave number of x 0.268 and a Rey-nolds number of 2. A thread connecting two main drops has just broken at the ends and is contracting into a satellite drop.Reprinted with permission of the Royal Society.




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with which the experimental parameters can becontrolled. To begin with, the problem of static stabilityof the bridge becomes purely geometrical: it dependsonly on the radii of the disks and the volume of the uid.This stability has been investigated theoretically in anumber of circumstances (Gillette and Dyson, 1971; Da-Riva, 1981).

If one wants to observe breaking, the bridge has to bemade unstable. This can be achieved either by suckinguid from the bridge (Meseguer and Sanz, 1985) or bypulling the disks apart (Spiegelberg, Gaudet, and McK-inley, 1994). The latter method is illustrated in Fig. 8.The initial state was that of a cylinder with aspect ratio

r 0 /L 0.77, where 2 L is the distance between thedisks. Over a period of a few minutes, it was pulled apartto an aspect ratio of 1.58, which is an unstable con-guration. The gure shows the collapse after the disks

stopped moving. As expected for the extremely lowReynolds number of 3.7 10 3, a thin thread forms be-fore the bridge nally breaks.

III. SIMULATIONS

In many areas of uid mechanics, ow simulationshave become standard procedure. If the Reynolds num-ber is not too high, carefully executed simulations canvirtually replace experiments, since the validity of theequations of motion is not a source of concern. In thepresence of a free surface, however, the situation is dif-

ferent. The ow geometry is essentially determined bythe evolution itself or may change its topology alto-gether due to breakup events. Thus computations needto be tailored to each initial condition. If breakup oc-curs, the validity of continuum mechanics, underlyingthe equations of motion, is itself called into question.This concern must be addressed separately through amore careful study of the singularities involved, orthrough complementing simulations of the underlyingmolecular dynamics (Greenspan, 1993; Koplik and Ba-navar, 1993).

Free-surface ows are also very sensitive to the for-mation of cusp singularities (Joseph et al. , 1991) even inseemingly innocuous ow situations. It seems as if sur-face tension should make the surface more regular, thussimplifying simulations. But instead it offers very littleresistance to the formation of singularities (Jeong andMoffatt, 1992) and makes the system more sensitive tonoise and prone to numerical instabilities, whose nonlin-ear origins are poorly understood. Indeed, surface ten-sion introduces a complicated coupling between the owthat advances the interface and the interfaces shape,which through the Young-Laplace equation determinesthe pressure driving the uid.

For that reason, so-called boundary integral formula-tions are very attractive, since they involve only infor-mation about the surface of the uid. They are possiblewhenever the ow is governed by a linear equation, forwhich the Greens function is known. This is the case for

FIG. 8. Liquid-bridge evolution starting froman unstable conguration. The disk diameteris 3.8 cm, the Reynolds number is 3.7 10 3.The outer uid, which eliminates buoyancyforces, has a viscosity approximately 1000times smaller than the inner uid. (Spiegel-berg, Gaudet, and McKinley, 1994).




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inviscid, irrotational ow and highly viscous ow gov-erned by the Stokes equation. Then the only informa-tion needed apart from the position of the interface is itsvelocity. Thus one is relieved from calculating the subtleinterplay of internal motion and the boundary shape,since all the information is contained on the boundary.In the more general case of Navier-Stokes ow, the in-terior and the boundary have to be dealt with sepa-rately. Both the surface tracking and the ow computa-tions are highly nontrivial problems in themselves andare now coupled in a complicated fashion. Such compu-tations have been performed only fairly recently and arenot sufciently accurate to resolve the universal behav-ior close to the singularity. Boundary integral methodsare more accurate, but neglect either viscous or inertialforces, both of which become important asymptotically.Nevertheless, simulations are an indispensable tool forpredicting the nonuniversal dynamics away from

breakup. One hopes that numerical codes will soon be-come sufciently reliable to yield an alternative to ex-periments. A useful overview on computational methodsof free-surface ows is provided by Tsai and Yue (1996),who draw most of their examples from problems relatedto free-surface waves.

A. Inviscid, irrotational ow

We rst turn to the problem of a low-viscosity dropwith a free surface (see Fig. 9). Most studies assumeaxisymmetry, so the boundary is given by a curve in the

r z plane, but this is not essential to our discussion. If the velocity eld is irrotational initially, and viscosity canbe neglected, it will remain so for the rest of its evolu-tion (Landau and Lifshitz, 1984a). Thus the velocity po-tential ,

v r ,z r ,z , (9)

obeys the Laplace equation

0. (10)

The evolution of follows from the Bernoulli equation

t v2/2 p / 0,

where p is the pressure. On the boundary, and in ab-sence of viscous forces, the pressure is given by theYoung-Laplace formula, hence

t v2/2

1R 1

1R 2

on , (11)

where R 1 ,R 2 are the principal radii of curvature.If x( ,t ) is the position of the surface as a function of

some marker (or grid point) , the surface moves ac-cording to

t x ,t v . (12)

Note that the denition of as a surface marker makesthe left-hand side a convective time derivative. The po-tential can be written as a function of the marker, aswell, according to

,t (x , t , t ) ,

and the resulting Lagrangian evolution equation is

t ,t v2/2

on . (13)

Here we abbreviated (1/ R 1 1/R 2), which is twice themean curvature, to . Thus Eqs. (12) and (13) give theevolution for both the surface and the value of the po-tential on it, provided the velocity v on the surface isknown. Decomposed into tangential and normal compo-nents, v can be written as

v s n n t , (14)

where n and t are unit vectors normal and tangential tothe surface, respectively. The tangential derivative s can be evaluated from the knowledge of on the sur-face alone, but to compute n one must bring theLaplace equation into play. That is, given Eq. (10), itfollows from Greens second theorem that

2 r P.V.

r n1

r r n rr r

d 2 s ,

(15)

where both r and r lie on the surface. This is an integralequation that can be solved for n , once is known on

.The procedure outlined was developed by Longuet-

Higgins and Cokelet (1976) for the study of water wavesin two dimensions. It has been adapted and used for thestudy of drops by Dommermuth and Yue (1987) andSchulkes (1994a, 1994b). Most of the computing time isspent on solving the integral equation (15), which is amatrix equation in discrete form. If the number of gridpoints is N , then the effort in computing the matrix ele-ments is N 2, and to invert the matrix N 3. ForN 100, a typical value for current computations, mostof the time is spent evaluating the matrix coefcients,which contain elliptic integrals. Clearly, if much higherresolution is to be attained, matrix inversion becomesthe limiting factor. Therefore other authors have used amethod originally developed for vortex simulations byBaker, Meiron, and Orszag (1980). The idea is rst tocalculate a distribution of dipoles on the surface,

FIG. 9. Sketch of a typical ow geometry. Rotational symme-try around the z axis is assumed. The velocity eld inside theuid is v( r ,z ) v z ( r ,z ) e z v r ( r ,z ) e r .




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which produce the potential . This can be done ef-ciently by iteration. The dipole distribution is then usedto calculate the normal component of the velocity eld,which is an N 2 process. For the present problem, thisprocedure has been adopted by Og uz and Prosperetti(1990) and Mansour and Lundgren (1990).

To formulate the system comprised by Eqs. (12)(15)in discrete numerical form, a number of different proce-

dures were adopted, some of which are compared byPelekasis, Tsamopoulos, and Manolis (1992). Usuallythe position of the interface x and the potential istaken at a discrete set of nodes i , 1 i N . Using suit-able interpolation formulas, one evaluates the curvatureat the interface and the integral in Eq. (15). An explicitRunge-Kutta step is then used to advance the position of the interface xi and the potential i . However, a prob-lem that appears, regardless of the numerical implemen-tation, is nonlinear instability on the scale of the gridspacing, giving the interface a saw-tooth appearance.The origin of this instability has not yet been found.Moore (1983), investigating a model problem, has shown

that such short-wavelength instabilities may be a resultof spatial discretization. But since the instability was re-ported in all work on the problem, using many differentnumerical techniques, there is a distinct possibility thatthe evolution equations (10), (12), and (13) do not pos-sess regular solutions, but rather have unphysical singu-larities before breakup occurs. One has to bear in mindthat the underlying equations are inviscid, so there is nonatural way for momentum to be diffused, and cata-strophically sharp gradients can occur, just as in thethree-dimensional Euler equation with large-scale driv-ing (Majda, 1991). This analogy is of course a very roughone, since singularities in the three-dimensional Eulerequation come about through the growth of vorticity,which is constrained to be zero in the present case of irrotational ow.

A number of different procedures have been pro-posed to get rid of the instability. Grid points were re-distributed after every time step to ensure equal gridspacing (Dommermuth and Yue, 1987; Og uz and Pros-peretti, 1990; Schulkes, 1994a). This step removes short-wavelength components and results in some numericalenergy dissipation (Schulkes, 1994a). Other possibilitiesare the inclusion of numerical diffusion (Og uz and Pros-peretti, 1990; Pelekasis, Tsamopoulos, and Manolis,1992), or simply ltering of the short-wavelength com-ponents (Dold, 1992).

Only a few papers have focused on drop formation,namely those of Mansour and Lundgren (1990), whosimulated a liquid cylinder with periodic boundary con-ditions; Spangler, Hilbing, and Heister (1995), who in-cluded the effect of a surrounding gas; and Schulkes(1994b), who considered a dripping faucet. The spatiallyuniform liquid cylinder is only a rough approximation of the steady state of a decaying liquid jet, but the simula-tions qualitatively produce the correct features. In par-ticular, Mansour and Lundgren predict satellite drops atall wave numbers, whose volumes agree well with ex-periment.

Schulkes (1994b) directly compares his simulationwith the experiment by Peregrine et al. (1990) and ndsgood agreement, except in the immediate neighborhoodof the bifurcation point (see Fig. 10). Instead of formingan almost at interface at the side of the drop, the pro-le turns over in the inviscid simulation. This seems tobe a universal feature of this approximation, as a similaroverturning is observed for the initial condition consid-ered by Mansour and Lundgren. In experiments, a verylarge, but still nite, slope is observed. The steepening of

the interface has been the subject of a recent experimen-tal and numerical study (Brenner et al. , 1997).Overturning thus seems to be an artifact of the invis-

cid theory, since the viscosity does become important asthe interface becomes steeper, as will be discussed inmore detail in Sec. VII.B. The inviscid approximation isthus invalidated even on scales much larger than l .However, overturning is observed after breaking, whenthe small remnant of the neck on the drop side recoils.The momentum it acquires is large enough to produce adent on the at surface of the drop. In experimentalpictures, which usually show a projection perpendicularto the axis of symmetry, it turns up as a perfectly straight

interface, as seen in the sixth frame of Fig. 6. It would beworthwhile to take pictures at an angle as well, to fur-ther investigate the dented region.

B. Stokes ow

The other case that can be treated by boundary inte-gral methods is that of a highly viscous uid, describedby the Stokes equation

v p . (16)

FIG. 10. The shape of a drop of water falling from a nozzle atthe rst bifurcation. The Bond number is Bo=1.02 and theReynolds number Re=452. The comparison with theory (solidline) is taken from Schulkes (1994b). Reprinted with permis-sion of Cambridge University Press.




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Here inertial terms have been dropped, so the uid den-sity does not appear. For Eq. (16) Greens functions areknown (Ladyzhenskaya, 1969), so that an integral equa-tion for the interfacial velocity can be derived in thesame vein as in the inviscid case. This was rst accom-plished by Youngren and Acrivos (1975) for a gasbubble in a very viscous uid. The method has beengeneralized by Rallison and Acrivos (1978) to the case

of drops of arbitrary viscosity in another viscous uid.However, the early numerical work on viscous dropswas limited to the calculation of stationary shapes (Ral-lison, 1984).

Only some years later did Stone and Leal (1989a,1989b) develop codes of sufcient accuracy to study thedynamics of extended drops. Because of the ample vis-cous damping, this problem is much less sensitive to per-turbations, and no numerical instabilities have been re-ported. Most of the work concerns a drop of uid inanother uid, which is quiescent at innity. However,the viscosity of the outer uid can be taken to be zero,so the drop is isolated.

Some pictures of extended drops breaking up are con-tained in Stone and Leal (1989b). The shape of the in-terface near breakup and the velocity eld is studied insome detail. But only Tjahjadi, Stone, and Ottino (1992)demonstrated the full power of viscous boundary inte-gral methods by performing a quantitative comparisonwith experiments on viscous drops in extensional ows.This study showed extraordinary agreement betweensimulation and experiment, which continued through anumber of breakups (see Fig. 11), and revealed an as-tonishingly ne structure of satellite drops. Breakup wassimulated by cutting the interface and smoothing theends once some critical radius was reached. As the uid

moves faster and faster near breakup, the Stokes ap-proximation eventually will break down, but on thescales observed it seems to remain remarkably good. Asimilar Stokes code has recently been applied to astretching liquid bridge (Gaudet, McKinley, and Stone,1996). The results are in excellent agreement with ex-perimental data, and in addition supply a wealth of in-formation about the dynamics of the bridge at differentviscosity ratios and stretching rates.

C. Navier-Stokes simulations

The boundary integral methods described in the rstsubsection are valid only for very low or very high vis-cosity. As we we are going to see in Sec. VII, eitherapproximation breaks down close to the pinch point,where viscous or inertial effects become important, atleast in the case that the outer uid can be neglected. Inthose cases, the full Navier-Stokes equation has to besolved in the uid domain, subject to a singular forcingat the boundary. Even in a xed domain this is not asimple problem, so it is not surprising that Navier-Stokessimulations in the nonlinear regime are few. Still, resultsfrom rst-principles calculations are indispensable, asthey provide the full information on the ow eld, whichis not available from experiment. Johnson, Marschall,

and Esdorn (1985) and Marschall (1985) provide someexperimental measurements of the ow eld, but the re-gion near the point of breakup is particularly hard tovisualize, so the ow eld is not known here.

The interior of the ow is described by the Navier-Stokes equation

t v v v1

p v (17)

for incompressible ow

v 0. (18)

Here v( r ,t ) is the three-dimensional velocity eld and p ( r ,t ) the pressure. On the free boundary, pressure andviscous forces are balanced by capillary forces:

n n . (19)

In this formula, is the stress tensor and n the normalvector pointing out of the uid. With the velocityknown, the interface is moved according to Eq. (12).

The main technical problem, as compared to otherNavier-Stokes simulations, involves the coupling to amovable interface and the treatment of a varying com-putational domain. The earliest work is that by Fromm(1984), who uses a square grid with a rened grid super-imposed on it to track the surface. The paper focuses on

FIG. 11. Time evolution of a highly extended uid suspendedin another uid. The viscosity ratio is 0.067, and the dimen-sionless wave number is 0.5. The times the snapshots weretaken are shown in the middle (Tjahjadi, Stone,and Ottino,1992). Reprinted with permission of Cambridge UniversityPress.




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the specic problem of ink-jet technology and does notstudy pinching systematically. In the work by Shokoohiand Elrod (1987, 1990), the computational domain ismapped onto a cylinder with radius 1, making surfacetracking superuous. The price one has to pay is that theequations in the interior get very complicated. In par-ticular, the earlier paper (Shokoohi and Elrod, 1987)contains a number of surface proles for different initial

disturbances at a xed Reynolds number of Re 50.The technique most appropriate for the problem isprobably the one adopted by Keunings (1986), whichuses a deformable grid to accommodate the moving in-terface. A similar technique was recently used in themost extensive study of breakup to date (Ashgriz andMashayek, 1995). Here the uid is bounded by a heightfunction h ( z ,t ); see Fig. 9. Thus one is limited to situa-tions in which the prole does not overturn. The com-putational domain is divided into quadrangles in the r z plane, four of which constitute a column in the radialdirection. On each of the quadrangles, four shape func-tions are dened, into which the ow eld is expanded.

Using a Galerkin method, one can derive a discrete setof equations for the amplitudes of the velocity eld andthe boundary points h i . To follow the motion of theinterface, the mesh points are allowed to move in theradial direction. This is done in such away as to ensuremass conservation in each element. From this the freesurface is reconstructed. Simulations are reported over awide range of Reynolds numbers up to Re=200, whichcorresponds to a water jet of 1 mm in diameter. No nu-merical instabilities are reported even at the highestReynolds number.

Simulations were performed with periodic boundaryconditions in the axial direction and for initial sinusoidaldisturbances of different wave numbers. The resultswere tested against the predictions of linearized theory(Rayleigh, 1879a; Chandrasekhar, 1961), which are dis-cussed in greater detail in the next section. The overallagreement with predicted growth rates is good. The larg-est deviations occur for the highest Reynolds number,for which a maximum error of 10% is reported. Particu-larly interesting is a sequence of pictures testing the non-linear evolution leading to breakup for a variety of wavenumbers and Reynolds numbers. Although the wavenumber affects the overall shape, the appearance of theinterface near the point of breakup is very similar fordifferent wave numbers. However, viscosity affects thebreakup signicantly: just as seen in experiment, the in-terface looks like a cone attached to a steep front forlow viscosities and gets increasingly at for higher vis-cosities. We shall study the interfacial shape in greaterdetail when we compare it with the results of one-dimensional models (see Sec. V). One very importantpoint to notice right away is that overturning of the pro-le does not occur, even at the highest Reynolds num-bers, which are comparable with experimental ones withwater.

Almost all the work described so far deals with theproblem up to the point of rst breakup. To continuethrough the singularity, some ad hoc prescription for a

continuation has to be invented for each particular case,as was done in Tjahjadi, Stone, and Ottino (1992) orSchulkes (1994b). Apart from the theoretical questionsinvolved, it would be desirable to develop a general phe-nomenological scheme to describe breakup and mergingin an arbitrary geometry. Instead of tracking the inter-face with surface markers (front tracking), this is mosteasily accomplished by describing the interface by a sca-lar function C ( r ,t ), which is dened everywhere (frontcapturing). This function is 0 in one uid and 1 in theother. The crossover region represents the interface,which is maintained to have some nite thickness bynumerical diffusion. The function C is advanced every-where according to

t C v C 0. (20)

Obviously this description does not rely on specic as-sumptions about the connectivity of the uid domains.On a microscopic scale set by numerical diffusion, athin sliver of uid is dissolved and thus broken. Con-versely, if two pieces of uid come sufciently close,they are joined. The most widely used variant of thisidea is the volume-of-uid method developed by Hirtand Nichols (1981), which is designed to conserve vol-ume exactly. Surface tension is included in two recentpapers by Lafaurie et al. (1994), and by Richards, Len-hoff, and Beris (1994) by distributing surface-tensionforces continuously across the interface. The formergroup emphasizes collision and merging of drops, thelatter studies breakup of jets. The computational do-main, on which the Navier-Stokes solver operates, is sta-tionary. Figure 12, showing two drops colliding at highspeeds and their subsequent breakup, nicely illustratesthe generality of this method.

On the other hand, the resolution available near

breakup is not very high, so the singularity cannot bestudied in detail. Lafaurie et al. (1994) also report somenumerical instabilities at large Reynolds numbers. Sodespite some remarkable features, the development of general-purpose codes capable of faithfully resolvingbreakup in some detail remains a challenge for the fu-ture.

As a last possibility, we mention the work of Becker,Hiller, and Kowalewski (1994). It is concerned withlarge-amplitude droplet oscillations. The Navier-Stokesequation is here analyzed by expanding the velocity eldinto a set of eigenmodes of the linear problem. Thismethod is very efcient when surface deformations are

not too large, so that they are representable by a reason-able number of eigenmodes. However, one expects it tobecome inefcient close to rupture, where typical owelds are dissimilar to eigenmodes of the linear problem.

IV. SMALL PERTURBATIONS

A. Linear stability

For a piece of uid to break up, it must be unstableagainst surface tension forces at some point during itsevolution. If the uid is at rest initially, or the time scale




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of the surface-tension-induced motion is much shorterthan other time scales, then the problem of stability ispurely geometrical; that is, instability means that thesurface area can be reduced by an innitesimal surfacedeformation.

The classical example studied by Plateau (1873) andRayleigh (1896) is that of an innitely long cylinder of radius r 0. To a good approximation such a uid cylinderwill be produced by a jet emanating from a nozzle athigh speed. Now one considers a sinusoidal perturbationof wavelength on the cylinder. For example, any ran-dom perturbation on the jet may be decomposed into alinear superposition of such Fourier modes. In a linearapproximation all the modes evolve independently, andone may consider just a single sinusoidal perturbation.In most practical applications, however, perturbations of a given wavelength are produced at the nozzle and areconvected down the jet. As argued in Sec. II, for largeWeber numbers 2 r 0v j

2/ this results in periodic dis-turbances, which are practically uniform in space.

In cylindrical coordinates r , ,z the surface shape canbe written as r ( ,z ). It is convenient to measure theaxial coordinate z and deviations from the cylindricalshape r ( ,z ) r 0 in units of r 0:

z /r 0 (21)

and

r ,z /r 0 1 , . (22)

In addition to the sinusoidal perturbation of the radius,we also allow for small departures from the circularcross section. This is taken care of by an additional azi-muthal dependence of the initial perturbation:

, cos n cos x ,

x kr 0 .

The dynamical constraint is that the volume

V /r 03 1

2 0 2 / x1 2d d

2 2

x

1 2/2 n 0

1 2/4 n 1,2, . . .

per wavelength is kept constant as evolves in time.This means must also contain another contributionwhich ensures conservation of mass, and thus

, cos n cos x 2 1 n 0 /8 O 4 .(23)

On the other hand, the surface area is

FIG. 12. Frontal collision of two drops at We-ber number 2 20 and Reynolds number Re=250, based on the relative velocity of thedrops (Gueyfer and Zaleski, 1997). First atoroidal structure is formed, which then col-lapses and forms a cylinder. This cylinderbreaks up as required by the Rayleigh insta-bility.




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A /r 02

0 2 / x1 2

1

2 1/2

1 d d

2 2

x1

2

8 1 n 0 x

2 n 2 1 . (24)

Hence with growing disturbance amplitude the surfacearea decreases only if disturbances are axisymmetric( n 0) and x 1. In other words, the wavelength of aperturbation must be greater than times the diameterof the jet for it to be unstable. This famous result is dueto Plateau (1849). From this it may seem as if distur-bances of very long wavelength, i.e., small x , are themost rapidly growing ones, as they reduce the surfacearea the most. But this leaves out the dynamics of theproblem, which we have not considered yet. In fact, sur-face tension has to overcome both inertia and viscousdissipation. The problem rst treated by Rayleigh(1879a) is that of a low-viscosity uid like water, forwhich inviscid theory applies.

Following Rayleigh, we shall calculate the optimal orfastest growing mode xR , 0 xR 1. If a jet is disturbedrandomly by external noise, this will usually be themode that sets the size of drops, as it will soon dominateall the other excited modes. We shall consider only axi-symmetric perturbations here, since all nonaxisymmetricperturbations only create more surface area and aretherefore stable. For most of this subsection, we shall belooking at inviscid irrotational ow, so the velocity eldhas a potential ( r ,z ,t ), and Eqs. (10), (12), and (13) foran ideal uid apply.

The velocity potential is nondimensionalized using theinitial jet radius r 0 and the time scale t 0 ( r 0

3/ ) 1/2,

which implies a balance of surface tension and inertialforces. Hence

, , t 0r 0

2 r ,z , t , (25)

where

r /r 0 , z /r 0 , and t /t 0 . (26)

No confusion of the variable with the density shouldarise here. The nondimensional equations of motion arethus

0,

1 , (27)

112

2

2

1

1 21

1 2

1 2 1 .

The initial conditions are

,0 cos x 2/4 O 4 , (28)

,0 0.

Since the initial surface displacement is assumed to beproportional to some small parameter , the velocity willalso be small, so we try the expansion

, , 1 , , O 2 , (29)

, , 1 , , O 2 .

To rst order in , the equations of motion (27) become

2

2 1 0, (30a)

1 1 1 , (30b)

1 1 2 1 1 . (30c)

Looking for solutions of the form

1 A 1 cos x , (31)

1 B 1 f cos x ,

we nd from Eq. (30a)

f f / x2 f 0.

The only solution that is regular for 0 is f ( ) I 0( x ), where I 0 is a modied Bessel function of the rst kind. Substituting this into the boundary condi-tions (30b) and (30c) gives

A 1 B 1 xI 1 x and B 1 I 0 x A 1 1 x2 .

The solution of this set of equations with the initial con-ditions (28), namely, A 1(0) 1, A 1(0 ) 0, is

A 1 cosh , B 1

xI 1 xsinh , (32)

with

2 xI 1 x I 0 x 1 x2 . (33)Frequencies are measured in units of the inverse timescale 0 ( /( r 0

3) )1/2. The dimensionless growth rate / 0 is real for x 1, so perturbations grow expo-

nentially, making the jet unstable against arbitrarilysmall perturbations. For x 1, on the other hand, theinterface performs oscillations, which will eventually bedamped by viscosity. The inviscid dispersion relation(33) is plotted in Fig. 13. The most unstable mode, cor-responding to the largest , occurs at x R 0.697. This isthe famous Rayleigh mode, which has a wavelength

R 9.01r 0.This result can be checked directly against the obser-

vations by Savart (1833). If a jet is excited at the nozzleof a reservoir, it will break with the same frequency withwhich it is excited. If the impact of the resulting dropson another container is fed back into the reservoirthrough a mechanical coupling, the original perturbationwill be amplied. This amplication is greatest for themost unstable wave numbers, and strong enough for amusical note to sound. Savart measured the pitch of thesound, and from this observation Plateau (1849) inferreda wavelength of 4.38 times the diameter of the jet. Usinghis concept of maximum instability, Rayleigh (1879b)




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was nally able to interpret this result and found it to beremarkably close to the prediction of his theory.

The rst accurate measurement of the complete dis-persion relation was done by Donelly and Glaberson(1966). They excited a water jet with a loudspeaker andobserved photographically how perturbations grow fromone wave crest to the next. Knowing the jet velocity,they could obtain the temporal growth rate of the per-turbations. Instead of measuring the height of the crestswith respect to r 0, it is more convenient to take the difference between nodes and crests, which Donelly andGlaberson found to exhibit exponential growth up toone wavelength from the breakup point. This may bedue in part to a cancellation of nonlinear effects, asclaimed by Yuen (1968) and conrmed by Goedde andYuen (1970). That is, when one takes the difference be-tween crests and valleys, the contribution from second-order perturbation theory drops out. Both Donelly andGlaberson (1966) and Goedde and Yuen (1970) foundexcellent agreement with Rayleighs dispersion relationto within experimental scatter. The experimental pointshave been included in Fig. 13. It should be mentionedthat an earlier investigation (Crane, Birch, and McCor-mack, 1964) had found deviations from linear theorybased on both the breakup length and direct analysis of photographs. The authors attributed this to the large dis-turbance amplitude used in their experiment, makinglinear theory inapplicable.

It was rst pointed out by Plateau (1873) that viscositymight considerably alter the above results. As onecomes closer and closer to a quasistatic Stokes descrip-tion, inertia becomes less and less important, and themost unstable wavelength becomes longer, correspond-ing to the greatest reduction in surface area. Rayleighhimself analyzed the case of very large viscosities (Ray-leigh, 1892), assuming that the uid is described by theStokes equation. The full problem, using the Navier-Stokes equation, was rst treated in the monograph byChandrasekhar (1961). The result for is a complicatedimplicit equation, which offers little insight in itself, so it

is not reproduced here. Instead, we give the result thatcomes from expanding Chandrasekhars formula forsmall x kr 0. It is

012

x 2 1 x294

Re 2 x41/2 3

2Re 1 x2 .

(34)

This equation was rst obtained by Weber (1931), whoanalyzed the motion of thin slices of uid. Surprisingly,it turns out to be a uniformly good approximation for allwavelengths and viscosities. For zero viscosity, it is off by 7% at most, and becomes increasingly accurate forhigher viscosities. According to Eq. (34), the fastest-growing mode is

xR2 1

2 18/Re, (35)

which gets smaller with growing viscosity.As the Reynolds number decreases, the growth rate is

eventually determined by a balance of surface tensionand viscous forces alone, rather than inertial forces. Thedispersion relation (34) gives, in the limit of small Rey-nolds number,

16 1

x2 , (36)

where /( r 0 ) is a viscous growth rate. The vis-cous dispersion relation has been tested by Donelly andGlaberson (1966) and Goedde and Yuen (1970), whoboth found good agreement with Chandrasekhars(1961) results. A typical dispersion relation is shown inFig. 14 for Re 1.73.

Since Rayleighs pioneering work, linear stability of liquid cylinders has been applied to many other physicalsituations. Tomotika (1935) generalized Rayleighs(1892) analysis of a highly viscous column to include anouter uid of nite viscosity and density. Both uids aredescribed by Stokes equation. It was found that themost unstable wavelength varied with the ratio of thetwo viscosities. A simple formula describing the case of two equal viscosities has been given by Stone and Bren-ner (1996). We shall come back to the problem of breakup of one uid in another in Sec. VIII.A.

The inuence of a surrounding gas of low density wasstudied by Weber (1931), Lin and Kang (1987), and

FIG. 13. The dimensionless growth rate of sinusoidal pertur-bations on a cylinder as a function of the dimensionless wavenumber. The solid line represents Rayleighs theory for invis-cid ow, the squares are data from Donnelly and Glaberson(1966), triangles are from Goedde and Yuen (1970).

FIG. 14. Growth rates for the decay of a viscous cylinder,comparing data from Goedde and Yuen (1970) with Chan-drasekhars theory. The Reynolds number is Re=1.73.




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Yang (1992). This is of interest for understanding atomi-zation of liquid jets. The inuence of surface charges isincluded in the work of Schneider et al. (1967) and of Grossmann and Mu ller (1984). The latter authors alsolooked at the case of a compressible liquid (Mu ller andGrossmann, 1985). In an extending liquid column, theaxial velocity eld is not uniform, but the basic ow haslinearly increasing velocity. This case was treated for aninviscid uid by Frankel and Weihs (1985). Finally,Berger (1988) noted that a general disturbance on a jethas a continuous spectrum of Fourier modes, rather then just a superposition of discrete modes. This alters theevolution of a perturbation for early times, making itdeviate from a purely exponential time dependence.

B. Spatial instability

As we have seen, the temporal instability analyzed inthe previous subsection is different from the spatial in-stability occurring in real jet experiments. So far the per-turbation had been assumed to be uniform in space

while its temporal growth was observed. In a jet experi-ment, there is a pointwise disturbance at the nozzle,which grows downstream as it is convected by the meanow. Thus at a given point in space the size of the per-turbation will remain nite, if the convection is fastenough to sweep disturbances away before they cangrow. If, however, one considers the frame of referencein which the jet is stationary, it is convected with thedisturbances and one observes unlimited growth. This iscalled convective instability, as opposed to the case of absolute instability, where perturbations even grow ata xed point in space (Landau and Lifshitz, 1984a). Ab-solute instability implies that no stationary state existswithin the realm of linear theory, as any perturbationeventually will grow so much as to make the linear ap-proximation inapplicable.

There are two different physical effects at work whichdifferentiate the spatial from the temporal instability.First, the spatial structure far away from the nozzle isdifferent, as in the former case swells grow from onewave crest to the next. Thus perturbations are describedby a superposition of plane waves,

z ,t exp i kz t ,

where the wave number k is now allowed to be imagi-nary, while in Rayleighs theory it was prescribed to be

real. The second difference is that the spatial theory isformulated in the half-space z 0, with proper boundaryconditions at the nozzle opening z 0. Usually the jetradius is prescribed to be constant, while the velocity isv j with a small sinusoidal perturbation superimposed onit.

The problem was rst studied in a doubly innite do-main by Keller, Rubinow, and Tu (1973). It can besolved by transforming the problem into a coordinatesystem that moves with the uid. The plane-wave ansatzin a coordinate system tied to the nozzle is

, Ae i x , (37)

, , Be i x I 0 x ,

where , dened by Eq. (6), is the dimensionless jetvelocity. The transformation * moves the coor-dinate system such that the jet is at rest, which is thephysical situation of the previous subsection. Hence if we write

e i x e i x

* * ,

* x has to replace in the dispersion relation(33).

This means that the new dispersion relation, trans-formed back to the stationary coordinate system, is

2 x 2 I 0 x xI 1 x x2 1 , (38)

where / is the experimentally prescribed pertur-bation frequency at the nozzle, multiplied by 2 t 0.Hence Eq. (38) has to be solved for the complex wavenumber x , which represents the resulting spatial struc-ture.

To relate to the temporal analysis, we analyze Eq.(38) for large . Thus we put x 1 x1 O ( 2),where the rst term describes convection with velocity

. Comparing powers of one nds

x1

I 1 2 1 / I 0 1/2 O 2 . (39)

Hence up to terms of higher order in 1 we recover anexpression analogous to Rayleighs dispersion relation(33), which was to be expected on the grounds of ourearlier arguments. For 1, the second term becomescomplex, corresponding to an exponentially growingspatial instability. In typical experiments, is largerthan 10, so that terms of higher order in are negligibleand Rayleighs formula can safely be applied. For lower jet velocities corrections eventually become important,but then gravity also becomes relevant and has to beincluded in the description.

Apart from the roots of Eq. (39) the dispersion rela-tion (38) has an innite sequence of other roots, alreadynoted by Keller, Rubinow, and Tu (1973), which do notcorrespond to Rayleighs solution. To understand theirsignicance, one has to consider the full problem in half-space, with boundary conditions at the nozzle. This wasattempted in a series of papers (Pimbley, 1976; Bogy1978a, 1978b, 1979b), which correctly analyzed the prob-lem for large Weber numbers. However, the authorsfailed to notice the absolute instability later found byLeib and Goldstein (1986) for small .

Apart from the precise form of the radial velocity pro-le, which is irrelevant to the qualitative features of theproblem, there are two boundary conditions to be satis-ed at the nozzle. These are the constant jet diameterand a harmonic time dependence of the velocity eld,which excites the jet. Both can be satised by superim-posing just two traveling waves (37), corresponding totwo solutions of Eq. (38). At the other end of the jet, noboundary conditions are available, because the linear




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approximation will eventually break down as distur-bances grow. They have to be replaced by physical con-ditions on the propagation properties of outgoing solu-tions. One condition is that for a particular branch of solutions the wave number must go to zero as 0,which is obviously the case for the solutions of Eq. (39).This requirement simply means that in a steady state atime-independent boundary condition cannot produce aspatial structure. Expansion of Eq. (38) for smallshows that precisely two such solutions exist for 0,so one ends up with the correct number of boundaryconditions at the nozzle.

However, following the two physical branches to large, one nds that they correspond to disturbances with

diverging speeds of propagation / Re( x ) as .This would mean that energy is transported at arbitrarilyhigh speeds, which is also excluded on physical grounds.Only for c 1.77 does each solution intersect withanother branch of solutions of Eq. (38), which has thecorrect physical behavior for . Thus steady-statesolutions of the linear problem exist for c . On theother hand, no steady state solution exists for c ,the region of absolute instability found by Leib andGoldstein (1986).

Bogy (1978a) advanced similar arguments, but usedthe existence of a positive group velocity as a criterionto select physical branches. However, Leib and Gold-stein (1986) have shown that negative group velocities infact do exist even in the stationary state. These authorssolved the linear problem directly by studying the nozzleproblem in much greater detail. In particular, theylooked at the inuence of different velocity proles atthe nozzle lip, which typically is a Hagen-Poiseuille pro-le, as opposed to the plug prole analyzed above. Forthe Hagen-Poiseuille prole the critical jet velocity iseven lower, so the issue of absolute instability in jet de-cay will probably remain an interesting, albeit somewhatacademic problem, as gravity will always interfere.

C. Higher-order perturbative analysis

Starting from the result of linear theory, the simplestway to explore nonlinear effects is to extend Rayleighsanalysis to higher order in the small-amplitude . Therst such analysis was due to Bohr (1909), who studiedthe stable oscillations a jet performs around its equilib-rium shape if it is released from an orice with a noncir-cular cross section. In the following, we are going toconcentrate on the axisymmetric case, which is muchmore difcult, for both stable and unstable ranges of wave numbers exist. As a result, the expansions cannotbe expected to be uniformly valid for all wave numbers.

For short-wavelength perturbations, against which theinterface is stable, perturbation theory is the obviousway to include nonlinear effects, as the perturbationsremain bounded. Hence one can expect to obtain solu-tions that are uniformly valid in time. For unstable per-turbations, however, the perturbation solutions are validonly for short times, because higher-order terms haveincreasingly fast growth rates. This is not very surprising,

as the time scale t 0 ( r 03/ ) 1/2, on which the linear so-

lution is evolving is of the same order as the breakuptime. But obviously the dynamics close to the equilib-rium solution do not know about the behavior nearbreakup, so the solution breaks down when the pertur-bation is of the same order as the radius itself. So for adescription of drop formation other methods areneeded. However, there is some hope that as long as the

perturbations are small, the nonlinear analysis might de-scribe the growth of higher-order harmonics, which areautomatically excited through the nonlinear interaction.If the perturbation expansion is carried out to third or-der, it will give some indication of the nonuniformbreakup of a liquid jet, which produces smaller satellitedrops in between the main drops.

The rst such theory was attempted by Yuen (1968).The primary prediction of his work is that the boundarybetween stable and unstable wave numbers is shifted tolarger values, as compared to Plateaus result kr 0 1, if the perturbations have nite amplitude. Thus perturba-tions that are linearly stable may be unstable because of

nonlinear effects. To nd this stability boundary, it issufcient to treat the case of stable oscillations. In theregion of unstable wave numbers, the perturbation ex-pansions are far from rigorous. In particular, the coef-cients have singularities at resonant wave numbers of kr 0 1/2 and possibly others, which are not understood.For that reason, we shall focus on the issue of the non-linear stability boundary, which is a highly nontrivialproblem in itself and has been the subject of some con-troversy.

Nayfeh (1970) showed that Yuens treatment of thestability boundary was incorrect, predicting an evenlower cutoff wave number. Subsequently, Lafrance

(1975) published a theory that predicted no shift in thecutoff at all. This apparent discrepancy was later tracedback to an error in his algebra (Chaudhary and Rede-kopp, 1980). Since the origin of these discrepancies havenever been satisfactorily cleared up, we shall give anexposition of the essential features of the theory. Belowwe show that Nayfehs (1970) result x c 1 3 2/4 for thestability boundary is recovered by modifying Yuens(1968) original analysis. In the next section this resultwill also be tested by computer simulation on a one-dimensional simplied model (Eggers, 1995b).

To proceed, one writes the velocity potential and thesurface deformation as a power series in :

, , i 1

i i , , ,(40)

, i 1

i i , .

Plugging this into the equations of motion (27) and com-paring powers in , one generates a closed system of equations at each order. The boundary terms are evalu-ated by expanding and around 1. Thus the equa-tions at n th order have the structure




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2

2 n 0,

n n F 1 , 1 , . . . , n 1 , n 1 1 , (41)

n n 2 n G 1 , 1 , . . . , n 1 , n 1 1 ,

where F and G are complicated, but calculable, nonlin-ear functions of the lower-order terms. This allows one

to calculate i and i recursively to any order.Owing to the nonlinear interactions between modes,increasingly higher harmonics of cos( x ) will be gener-ated, up to cos( nx ) at n th order. Following Yuen(1968), we shall pursue this procedure to third order,which is where the rst corrections to the linear stabilityboundary come in. Corrections of typographical errorsin Yuens original work are given in Rutland and Jame-son (1971) and Taub (1976).

The structure of the second- and third-order terms is

2 , A 22 cos 2 x A 20 , (42)

2 , , B 22 I 0 2 x cos 2 x B 20 ,

and

3 , A 33 cos 3 x A 31 cos x , (43)

3 , , B 33 I 0 3 x cos 3 x

B 31 I 0 x cos x ,

where the coefcients A ij and B ij are complicated func-tions of time and the reduced wave number x (Yuen,1968; Rutland and Jameson, 1971; Taub, 1976). At n thorder, they contain terms cosh( n ) and sinh( n ), sothe corresponding perturbation is proportional to

n exp( n ). Thus for unstable wave numbers x 1 theexpansion is valid only for times log( / ). If theleading-order perturbation is of order one, which is thecase near breakup, all other contributions will be of or-der one as well. Note also that the coefcients containsingularities at subharmonic values of the cutoff wavenumber. For example, A 22( ) has a pole at x 1/2.

But even for x 1 the expansion is not uniformly validin time, even though for 0 the solution should bestable and all coefcients should be bounded. In fact, A 31( ) contains so-called secular terms, which grow lin-early in time and thus invalidate the solution for t .The reason is that the third-order solution is convectedand stretched by the rst-order contribution. This con-dition cannot be represented by harmonics of nite or-der, but rather one needs contributions from all ordersof the perturbation expansion.

One way around this problem is to reparametrize timeand space coordinates according to

, (44) k c ,

where 1 2 2 and k c 1 2k 2 bothhave expansions in itself. This procedure, called themethod of strained coordinates (van Dyke, 1975), wasrst used by Yuen (1968) in the context of the present

problem. Organizing the perturbation expansion in thenew variables and means one has to replace the dif-ferential operators and by (1 2 2 ) and k c (1 2k 2 ) . By adjusting the free con-stants 2 and k 2, one hopes to remove secular terms atthird order. Effectively, one is resumming certain contri-butions from all orders. It is, however, far from trivialthat this resummation procedure works at all orders. Wehave to be content with the third-order results, but shallcheck them against numerical simulations later.

If k 2 is nonzero, this implies a shift in the cutoff wavenumber, as cos( x ) cos( x ), with x x/k c . Thus thegrowth rate is now

1 x I 1 x

I 0 x 1 x 2

1/2

,

which is real for x 1 or x 1 2k 2 . (45)

In the strained coordinates the equation for A 31 ,which contains the secular term, is

2 A 31 12 A 31 P 1 x cosh ( 1 2 )P 2 x cosh ( 1 2 )P 3 x cosh 3 1 P 4 x 12P 5 x 2

( 12P 6 x x 4)k 2 cosh 1 x , (46where

22 2 I 1 2 x I 0 2 x

x 1 4 x 2 .

The particular solution coming from the last contribu-tion contains the secular term

A 31

2 1P 4 x 12P 5 x 2

( 12P 6 x x 4)k 2 sinh 1 x ,which grows with unless the terms in the angularbrackets cancel. Yuen used precisely this condition todetermine 2 and k 2, setting

21

12P 5 x P 4 x ( 12P 6 x x 4)k 2 . (47

Near the cutoff wave number x 1, this expression has asingularity, since 1 0. To cancel this singularity oneputs

k 2 P 4 1916 ,

which together with Eq. (45) is Yuens result for thecutoff wave number.

Yuen overlooked the fact that the termP 3( x )cosh(3 1 ) in Eq. (46) produces additional secularterms as 1 0. That is, since P 3(1 ) 3/16, Eq. (46) be-comes at x 1




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2 A 31 (P 1 1 P 2 1 )cosh 2

34

k 2 .

This means A 31 (3/4 k 2) 2 up to oscillatory terms.

Thus to cancel the secular terms in the cutoff region onehas to put k 2 3/4, and the correct cutoff wave numberbecomes

xc

13

4 2. (48)

This result was rst found by Nayfeh (1970), using themethod of multiple time scales (Bender and Orszag,1978).

It should be mentioned that the shift predicted by Eq.(48) is extremely small, so that an experimental verica-tion has not yet been possible. If the perturbation am-plitude is 10% of the radius, the wave number must beknown to within one percent. Even simulations have notled to a signicant conrmation (Ashgriz and Mashayek,1995). Therefore we shall test Eq. (48) in the next sec-tion within the framework of one-dimensional models,for which the perturbation theory is virtually identical,

except for some differences in the coefcients. The cut-off wave number comes out to be the same.Away from the cutoff wave number in the unst

free surface dynamics

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