Breakdown of scaling in droplet ﬁssion at high Reynolds …

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Breakdown of scaling in droplet fission at high Reynolds numberMichael P. BrennerDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Jens EggersUniversitat Gesamthochschule Essen, Fachbereich Physik, 45117 Essen, Germany

Kathy Joseph, Sidney R. Nagel, and X. D. ShiJames Franck Institute, University of Chicago, Chicago, Illinois 60637

~Received 8 July 1996; accepted 30 January 1997!

In this paper we address the shape of a low-viscosity fluid interface near the breaking point.Experiments show that the shape varies dramatically as a function of fluid viscosity. At lowviscosities, the interface develops a region with an extremely sharp slope, with the steepness of theslope diverging with vanishing viscosity. Numerical simulations demonstrate that this tip forms asa result of a convective instability in the fluid; in the absence of viscosity this instability results ina finite time singularity of the interface far before rupture~in which the interfacial curvaturediverges!. The dynamics before the instability roughly follow the scaling laws consistent withpredictions based on dimensional analysis, though these scaling laws are violated at the instability.Since the dynamics after rupture is completely determined by the shape at the breaking point, thetime dependences of recoiling do not follow a simple scaling law. In the process of demonstratingthese results, we present detailed comparisons between numerical simulations and experimentaldrop shapes with excellent agreement. ©1997 American Institute of Physics.@S1070-6631~97!00306-1#

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I. INTRODUCTION

Droplet breakup has been a subject of scientific scrufor over 100 years. Early contributions include Plateau’sscription of the instability mechanism,1 Lord Rayleigh’s cal-culation of the most unstable wavelength,2 Worthington’spictures of splashes,3 and Edgerton’s high-speed strobscopic photographs,4–6 which first revealed the intricateshapes during rupture.

In the 1970’s, there was a resurgence of interest inbreakup,7–11 mainly driven by its technological relevanc~e.g., to ink jet printing!. Detailed experiments studied thdynamics of both high- and low-viscosity fluid jets emaning from a nozzle, focusing on the early stages of pinchiA principal goal of this work was to understand the contof the size distribution of satellite droplets. More recentnumerical methods have been developed to study jet debased on the assumption of inviscid, irrotational flow12 orhighly viscous Stokes flow.13–15

From a mathematical point of view droplet breakup pvides a simple example of singularity formation.16–19 Start-ing from a smooth initial shape with finite fluid velocities,breaking drop develops singularities in a finite amounttime in which physical quantities, such as the curvaturethe interface or the fluid velocity, diverge. A fundamenquestion is to determine the nature of the flow in the neiborhood of the singularity.

For a liquid drop breaking in vacuum, the shape nearbreaking point depends critically on the viscous length scfirst identified by Peregrineet al.,20 defined as

l n5n2r

g,

wheren is the kinematic viscosity,g the surface tension, an

Phys. Fluids 9 (6), June 1997 1070-6631/97/9(6)/1573/18/

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r the fluid density. Figure 1 shows several shapes for thdifferent fluid viscosities: the radius of the water [email protected]~a!# is much larger than the viscous length scale, whichonly 140 Å. Increasing the viscosity by a factor of [email protected]~b!# leads to a 104 increase inl n . At such a high viscosity,the shape is slender near the breaking point. For even higviscosity fluids like glycerol (l n51 cm) the slenderness imore pronounced@Fig. 1~c!#, with a much more gradual teardrop shape.

The goal of the present paper is to understand the qutative differences between the shapes in these picturehigh viscosities, the connection of the fluid thread to the dis gradual@Fig. 1~c!#, whereas at low viscosities the shapehighly asymmetric, with an abrupt transition from the threto the drop separated by a region of extremely steep s@Fig. 1~a!#. In fact, as will be shown below through experments and numerical simulations, the steepness of the sactually diverges with vanishing viscosity. Through a cobination of numerical simulations and experiments, we wattempt to answer the following questions: What setsscale of the asymmetry? How does the asymmetry vary wfluid viscosity? And how does this asymmetry arise from tdynamics leading to droplet fission?

In addressing these questions, our study will focus ofundamental idea about how liquid drops break, originaintroduced by Keller and Miksis:21 Arbitrarily close to thebreaking point there are no external length scales descrithe interface, so the dynamics should beself-similar:22 theshape of the interface when the minimum radius is 1mmshould be the same as the shape when the minimum radi10 mm, modulo rescalings of the axes. Peregrineet al. re-fined this idea20 and applied it to their experiments on fallinwater drops: They point out that for low-viscosity fluid

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FIG. 1. Shape of the fluid interface immediately before rupture, for three different viscosities.~a! is waternH2O50.01 cm2/s; ~b! is a 85 wt. % glycerol watermixture with a viscosityn5100nH2O

; ~c! is pure glycerol with viscosityn51200nH2O.

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there is a range of scales where the thickness of the flneck is much larger than the viscous length scalel n , butmuch smaller than the length scale, where energy is fedthe system. Over this range of scales, the self-similaritypothesis might be expected to hold. For high-viscosity fluithe thickness of the thread is usually much smaller thanl n ,so that self-similarity will also hold.

These considerations suggest that a breaking fluid inface should be described by a similarity solution to the gerning hydrodynamic equations. The first study to oknowledge that succeeded in constructing a similarity sotion for a breaking fluid thread was for the rupture of a drin the two-dimensional Hele–Shaw cell.16,17This initial suc-cess was followed up with the identification of further simlarity solutions for the Hele–Shaw cell,19,23 as well as thediscovery of a similarity solution for three-dimensional drolet fission for fluid threads with thickness much smaller thl n .

24–26 In this case the similarity solution is unstablefinite-amplitude perturbations,27 with the critical amplitudefor instability approaching zero at the singularity.28 At thehigh viscosities where this similarity solution is relevant, tdroplet shape is long and slender, as apparent in Fig. 1~c!.

The question addressed in this paper is whether simscaling ideas can explain the interfacial shapes duringbreakup of low-viscosity fluids, such as the water dropFig. 1~a!. Based on the above discussion, this translatesthe question of whether the self-similar singularities in tequations ofinviscid hydrodynamics describe experimenwhen the thread thickness is larger than the viscous lenscalel n? To answer this question, we will describe in detthe characteristics of a water drop falling from a nozzle bbefore and after a fission event. Through a combinationexperiments, numerical simulations, and theory, we arthat the scaling hypothesis fails for this problem. When

1574 Phys. Fluids, Vol. 9, No. 6, June 1997


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thread thickness is far greater than the viscous length~105

times greater for a water drop falling from a 4 mmnozzle!, adynamical instability produces another small length scawhich destabilizes the similarity solution. In the absenceviscosity, this instability actually causes a singularity in tcurvature of the interface before rupture occurs. This insbility of the scaling solution at low viscosities providesnatural explanation for the steep front in water [email protected]~a!#. Although there is no inviscid singularity after breakuthe scaling behavior before breakup is imprinted on the tidependences after breakup,29 since the solution at the time obreakoff provides an initial condition for the recoiling; thshape at the rupture point is not a perfect power law, whresults in a more complicated time dependence than polaw scaling.

The paper is organized as follows: The second secintroduces the experimental and theoretical frameworkspresenting a detailed comparison between a numerical slation of a drop falling from a faucet and experimental phtographs. The simulations use a one-dimensional ‘‘lonwavelength’’ approximation to the full hydrodynamiequations developed previously.24,30,31 The agreement between simulations and experiment is remarkably good, wsimulations reproducing detailed features of the experimeven though the long-wavelength assumption does not huniformly throughout the breaking. The next two sectiostudy in detail the solution near the breaking point in tlow-viscosity limit. In Sec. III we address solutions befobreakup, while in Sec. IV we address the situation afbreakup. In both cases we combine available theoretical,perimental, and simulational evidence to provide a comppicture of the singularities.

Brenner et al.


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II. FORMULATION OF PROBLEM

The experimental configuration addressed in this pais the falling of a drop from a circular nozzle. We considthe limit of slow dripping: the drop is adiabatically fillewith fluid until the shape becomes unstable. Two dimensiless parameters characterize this situation: the Bond num

Bo5R2rg/g;

and the Reynolds number,

Re5~R/ l n!1/2.

Here R is the radius of the nozzle,g is the gravitationalacceleration, andr, l n , andg are the fluid parameters defined above.

Several groups have recently studied large-scale featof a drop dripping from a circular nozzle.20,27,32Here, we areprimarily concerned with the nature of singularities in tlimit of zero viscosity~infinite Reynolds number!. Near therupture, there is a large separation between the time scathe flow at the singularity and the time scales characterizflow fields far from the breaking point. Thus, it is expectthat properties of singularity formation are only weakly dpendent on the specific experimental setup—in this caseBond number Bo. The results of this paper should therefapply to droplet breakup at high Reynolds number in aexperimental configuration. Indeed, Edgerton’s initial obsvation of sharp interfaces in low viscosity fluids occurredseveral experimental setups, ranging from his famoustures of splashes4 to fluid falling from a nozzle.

The goal of this section is twofold: first we describesimplified model for understanding the drop dynamics.inviscid version of this model was first proposed by Lee33

viscosity was later included by Bechtel, Forest, and Lin30

Eggers and Dupont,24 and Sellens.31 Then, we present twosets of experiments and simulations and compare the resThe first set studies a water drop falling from a nozzle, whthe second set focuses separately on the thin neck reseparating the drop from the nozzle. Eggers and Dupo24

already presented such a comparison for Peregrine’s phgraph of a water drop immediately before it breaks. Herecontinue the simulations through the rupture of the initdrop and the satellite drops until the original drop is serated from the nozzle by several smaller satellite drops.simulations continue to describe the experiment, even fornearly spherical satellite drops, when the long wavelenapproximation clearly no longer holds.

The experiments consisted of photographing a loviscosity fluid, deionized water at room temperature, drping from a thin vertical nozzle. The rate of drop formatioat the nozzle was kept small so that the initial velocitcould be assumed to be negligible. This was determineddecreasing the rate of flow until the macroscopic shape ofdrop no longer varied with the rate. The diameter of tnozzle could be varied. The photographs were taken inways. In order to determine the shape of the drop as a fution of time, we used a HyCam 400 16 mm movie camewhich, with its quarter-frame attachment, could take a mamum of 44 000 frames per second. In order to measuremovement of the drop as a function of time, the frames fr

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the HyCam were transferred onto videotape and analywith a computer. For precision pictures of the droplet shawe used a medium format camera with an open shutter astrobe. When the drop began to fall, it interrupted a labeam incident upon a photodiode. After a variable timelay, this triggered a fast 5 MS EG and G MVS-2601 StrobIn both cases, the lighting was from the rear so thatliquid/air interface would be plainly visible.~Since the dropacts like a small lens, the edges of the liquid appear bland bright spots appear along the axis.! For close-up photog-raphy, the lens was attached via a bellows to the camera

The mathematical problem assumes the dynamics oaxisymmetric column of fluid with kinematic viscosityn,densityr, and surface tensiong falling in vacuum. The fluidmoves via the incompressible Navier–Stokes equations, amented with the two boundary conditions that~i! normalstresses are proportional to the mean curvature and~ii ! tan-gential stresses are zero. The final condition is that the inface moves with the fluid.

The full hydrodynamic equations are difficult to undestand theoretically or simulate numerically. We instead csider a long-wavelength approximation of the fuequations24,30,31 by systematically expanding the full equations in powers of a slenderness parameter. Another dertion of the long-wavelength equations, is presented in Apendix A, which elucidates the physical constraints: Tform of the equations is fixed by seeking the simplest nlinear equations that preserve the conservation laws~massand momentum! of the flow, while preserving Galilean invariance, an idea that is quite old.34,35 The only nontrivialterm describes viscous dissipation: this term follows bymanding that the equations also generate the same disperelation of long-wavelength disturbances as the full hydronamic equations. The evolution equations for the interfaradiush(z,t) and the velocity fieldv(z,t) are

~h2! t52~h2v !z , ~1!

v t1vvz53

Reh2~h2vz!z2kz1Bo, ~2!

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hA11hz22

hzz

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where we have nondimensionalized all lengths by the nozradiusR, and all times by the basic time scale

t5ArR3

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Note that the equations as written above violateasymptotic derivation in powers of the slenderneparameter,36 since they selectively include higher-ordterms inhz . Specifically, it would be asymptotically correcto approximateA11hz

2'1, and also to neglect thehzz termentirely.36 The reason for including these terms is twofolfirst, only by keeping the higher-order terms in the pressdo the equations have the correct equilibrium shapes, whis important for capturing the flow away from the breakinpoint.24 Second, even starting from a cylinder as an initcondition, the asymptotic equations show exponential gro

1575Brenner et al.


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at arbitrarily short wavelengths. In simulations, this shoup as rapidly growing oscillations on the scale of the coputational grid, which would have to be damped out by soad hoc method without the presence of the higher-ordterms.

The numerical simulations solve a finite difference vsion of the lubrication approximation discussed above. Tdetails of the finite difference method have previously bediscussed in several places.16,24,19,27We briefly summarizethe major points of the simulations; technical details areserved for Appendix B. The finite difference equations asolved implicitly, resulting in a system of nonlinear equtions at each time step, which are solved using Newtomethod. The time step is dynamically adjusted to conseveral different indicators of the numerical error. The coemploys a dynamically evolving mesh, which is essentialobtaining the level of resolution necessary to resolvebreaking adequately. The mesh is adjusted wheneverspecified conditions on the solution are satisfied. Whenminimum thickness of the droplet drops below a threshothe code ‘‘breaks’’ the drop, by dividing the computationdomain into two pieces and then interpolating the shapethe interface around the breaking point. Once the dropartificially broken, the equations are solved at each mpoint. Numerical tests demonstrate that the large-scalesults are independent of the method of cutting the threadparticular, decreasing the threshold thickness for ‘‘breakindoes not affect any of the results. The details associatedimplementing this procedure efficiently are discussed in Apendix C. A typical simulation lasts a few hours on a SUSparc 20.

As an illustration of how closely simulations match thexperiment, in the following figures we focus on a wadrop falling from aR51.5 mm nozzle~Bo50.3, Re5330!.In both experiments and simulations, the drop is slowly fillwith fluid, passing through a sequence of equilibrium shapat a critical volume, the drop falls from the nozzle. Figureshows both experimental photographs and simulations atferent stages of the breaking: Initially the drop hangs frthe nozzle in equilibrium. Then the drop falls, pulling outthin neck that separates it from the nozzle@Fig. 2~a!#. Afterbreaking@which occurs between the times of Fig. 2~b! andFig. 2~c!#, the drop continues to fall, while the thin necrecoils toward the nozzle@Fig. 2~c!#. As the thin neck recoilscapillary waves are generated and propagate towardsnozzle. These waves are excited since the wave speecapillary waves coincides with the retraction speed offluid neck~see Appendix C!. Then the recoiling neck breakat the top of the thin neck, near the nozzle@Fig. 2~d!#.

It should be emphasized that the simulation containsfree parameters: given the dimensionless Bond numberand Reynolds number Re the shapes are uniquely determby the dynamics. We find it remarkable that such a simmodel can capture so many details of the breaking proce

It should be noted, however, that these simulations cnot capture all qualitative aspects of the experiments: in pticular, simulations of the long-wavelength equations arecapable ofoverturning, where the thicknessh(z) becomesdouble valued. Close examination of movies of water dro



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shows that overturning never occurs before the initial fissevent, but does tend to occur during the oscillations ofsatellite drops. Typically, overturning in satellite drop osclations occurs for experiments with larger nozzles~Bondnumbers! than Fig. 2. The long-wavelength equations avooverturning~which would result in a singularity! by produc-ing a large amount of dissipation where overturning is abto occur. This feature allows the simulations to be continuto arbitrarily long times; however, the agreement betwesimulations and experiments degrades after an~experimen-tal! overturning event.

Recently, Schulkes37 performed a boundary integrasimulation of aninviscid, irrotational fluid dripping from anozzle, and presented a detailed comparison of his simtions with the Peregrineet al.20 experiments on water dropsAn interesting difference between the present simulatiand Schulkes’ inviscid simulations is that the latter demostrate overturningbeforethe initial fission event, whereas thformer do not. The reason for this difference will be elabrated in detail below: Based on comparisons of experimeand simulations, we argue that this difference is becauselong-wavelength equations more accurately describe thenamics than the assumption that the flow is inviscid airrotational. Our simulations show that viscosity is a singuperturbation to the dynamics in the long-wavelength eqtions; the inclusion of an~arbitrarily small! amount of vis-cosity stops overturning during rupture. The agreementtween these simulations and experiments leads usconjecture that the full equations also contain this singuviscosity dependence.

Now we summarize another set of simulations andperiments to be used in the remainder of the paper in alyzing the dynamics of the rupture. Immediately after ruture, the interface~when viewed on a length scale larger ththe viscous scale! consists of a sharp conical tip attached tospherical shell. Figure 3 shows a sequence of the recoil fboth simulations and experiments, for a water drop fallifrom a R53.5 mm pipette~Bo56.5, Re5500!. The initialstate of the drop was prepared exactly in the manner asthe previous simulation~i.e., by passing through a sequenof equilibrium shapes!. We show only the thin neck regionseparating the drop of fluid from the nozzle, since this isrelevant regime for the analysis in the following sectionhowever, we emphasize that the agreement between simtions and experiments persists throughout the entire dropdemonstrated in the previous figure. The speed of the ptography was enhanced by a factor of 4 by printing fosuccessive pictures~separated in time by 2.531025 s! on asingle frame of film. In each frame, the earliest timeprinted at the bottom, and latest frame at the top. The fofold increase in the timing gained allows a more precanalysis of the singularity reported in Sec. IV.

III. SHAPES NEAR RUPTURE

In the remainder of this paper, we analyze the pictuaiming at a complete description of the interfacial shapesfluid velocities close to a breaking event. For water drothe viscous length scalel n'0.01mm is much smaller thanthe length scales visible in the photographs~larger than 1

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FIG. 2. Left panel: Photographs of a water drop falling from a nozzle with radiusR51.5 mm ~Bo50.3, Re5330!. The nozzle is visible at the top of thepicture. Right panel: Sequence of shapes from numerical simulations of Eqs.~1!–~3! for the same configuration. The times were chosen so that the drop shmatch the experiments. Length scales are in units of the droplet radius. In both simulations and experiments a pendant drop is filled adiabaticabecomes unstable and falls.

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FIG. 3. Comparison of numerical simulations~left! and experiments~right!for the recoiling fluid neck of a drop falling from nozzle with radiuR53.5 mm. The characteristic timet52.431022 s. The lengths of thesuccessive drops~in units of the radius! in the simulations are 1.8, 1.5, 1.451.27, and 1.16, respectively. The times between the successive photog~in units of t! are 0.092, 0.01, 0.03, and 0.02, respectively. The timestween the successive frames from simulations are 0.15, 0.013, 0.03,0.02, respectively. Note that the neck in the first photograph is at a sliglater time than the first simulation frame. The absolute length of the velovectors is normalized by the maximum velocity. Sections IV and V gidetailed comparisons of the time dependences in simulations and exments near the breaking point.

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mm!. This separation of scales leads to the expectationthe viscous stresses shouldalwaysbe much smaller than thother forces when the drop thickness is much larger tl n . If this is true, then the viscous term 3(Reh2)21(vzh

2)z inEq. ~2! can simply be dropped when analyzing the flow fiethus, inviscid hydrodynamics might be expected to be sucient for describing rupture.

In their classical paper,21 Keller and Miksis advanced adimensional argument suggesting the time dependencesrecoiling inviscid wedge. Their argument is sufficiently geeral that it applies to the inviscid dynamics of any surfatension driven flow near a singularity.38 The application todroplet breakup was emphasized by Peregrine.20 When vis-cosity is unimportant, the only relevant dimensional paraeters near the singularity are surface tensiong and the fluiddensityr, since the external forcing and initial conditions aassumed to be unimportant. Using these parameters andimensional time intervalDt from the singularity~either be-fore or after!, only one length scale can be formed, name

S g

r D 1/3~Dt !2/3. ~5!

In dimensionless units, witht85Dt/t the dimensionless timeinterval to the singularity, this length isl (t8)5t81/3. If l isindeed the only length scale governing the dynamics, tthe shape and velocity field should obey a similarity solutof the form

h~z,t !5 l ~ t8!HS z8

l ~ t8! D , ~6!

v~z,t !5l ~ t8!

t8VS z8

l ~ t8! D , ~7!

where z85z2z0 is the position relative to the point obreakupz0 . These equations are exact solutions of the eqtions ~1!–~3! at zero viscosity.

The question is whether this dimensional argument sfices to explain the experiments and simulations preseabove. There are two assumptions hidden within the arment, which could cause it to fail in practice: First, itassumed that when the interfacial thickness is larger thanviscous length scale the viscous stresses are irrelevant;might be dynamical mechanisms that make viscosity imptant on a length scale different from the viscous length scl n . Second, it is assumed that the singular solution is noall affected by the outer length scales, i.e., those setboundary conditions at the nozzle or initial conditions. Athough it is generally true that scaling exponents are inpendent of outer scales, it is often the case that prefactordepend on outer length scales. The mechanism for this isthe singular solution must bematchedto the solution farfrom the singularity, which generically introduces constain the singular region depending on the outer scales.

We will demonstrate below thatbeforebreakup the di-mensional scaling law fails because viscosity becomesportant when the drop radius is much larger than the viscscale. This fact was already indicated by Eggers aDupont,24 and is a consequence of an intrinsic instabilitythe inviscid equations: without viscosity, the momentu



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equation resembles a forced kinematic wave equationwhich high-velocity regions move faster than low-velociregions. This leads to an instability that forms discontinuitin velocity gradients in finite time. The continuity equatioimplies that a discontinuity in a velocity gradient also causa discontinuity in the slope of the drop. A finite but arbtrarily small viscosity changes the situation drastically: tsteepening of the gradients is stopped by viscous strebefore the interfacial curvature diverges. The viscosity sthe value of the maximum curvature, and thus the maximslope, and therefore has macroscopic consequences on sof the order of the drop radius,evenwhenl n is infinitesimal.This instability provides a natural explanation for the strikidifference between the interfacial shapes near rupture forids of different viscosity~see Fig. 1!. At low viscosity thisadditional singularity causes the sharp front observed inperiments.

The initial recoiling after breakup for fluid dripping froma nozzle does not manifest the inviscid convective instabijust described: We show in Appendix C that when startfrom a cone as the initial condition, simulations obey tdimensional scaling laws discussed above. However,cause of the breakdown of scaling before breakup, the shat the rupture point deviates from a cone on scales mlarger thanl n . Using mass and momentum conservatioKeller29 showed that the initial conditions directly affect thdynamics after breakup. Therefore, after breakup a simscaling will only be observed over a range of scales sethow closely the shape at the rupture point is a pure polaw.

A particularly sharp way of illustrating this problem ithe set of scaling solutions for the recoiling of an inviscdrop proposed by Ting and Keller.39 They pointed out thatthere are actually a continuous family of other possible silarity solutions to the lubrication equations other than~6! and~7!.39 The more general form,

h~z,t !5 l radial~ t8!HS z8

l axial~ t8! D , ~8!

is also a possible solution to the dynamical equationst8→0 as long as

l radial5R~ l axial/R!b. ~9!

These solutions are consistent with the long-wavelengthsumption wheneverb.1, so thatl radial goes to zero fastethan l axial. Note that these solutions require the introductiof an additional length scaleR. Correspondingly, the solutions~8! and~9! also have a different asymptoticsawayfromthe singularity: since the solution must be time independfar from the breaking point, the scaling functionH(j) mustobeyH(j);jb at largej, so thath(z);zb.

IV. SHAPES BEFORE BREAKUP

In this section we analyze the dynamics before ruptureestablish the claims made above. We begin by measuringcharacteristic thicknesshmin of the interface as a function otime of the figures shown above.

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FIG. 4. ~a! Measurements of the minimum thicknesshmin of the interfaceas a function of the timet0 2 t to rupture. Solid dots denote experimentalmeasurements; the solid line is a numerical simulation. The dotted linrepresents thet0 2 t2/3 scaling law.~b! The characteristic horizontal lengthscalej as a function of time before rupture. The scalej is defined bymeasuring the axial distance over which the thickness of the thread icreases by a factor of two from the point of minimum thickness. The soliline is a numerical solution to the partial differential equation, while thedotted line represents thet0 2 t2/3 law. ~c! The maximum velocity as afunction of time before rupture. The solid line is a numerical solutionto the partial differential equation, while the dotted line representvmax; t02 t21/3.

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Figure 4~a! showshmin(t) for the second breaking evenfor both theory and experiments shown in Fig. 4. The mimum thickness was measured in units of the drop radiusR,and the time was measured before to the singular timet0 . Insimulations the singular time could be measured exactly.perimentally it could be measured to within one frame ofmovie corresponding to a time error of6 2.531025 s. Notethat the time is measured in terms of the time scale give~4! to facilitate comparison between simulations and expments. As above,R denotes the radius of the nozzle. Thabsolute scale on the photographs was not directly measubut was instead deduced by comparison with simulatioThe simulations can also measure the characteristic lengthe interface and the maximum fluid velocity as a functiontime to rupture. These dependences are shown in Figs.~b!and 4~c!.

Note that the viscous length scale is 431026 in dimen-sionless units, far below the scales shown.

Does this numerical data agree with the dimensioscaling law? There are two separate points to make:experiments can resolve 2.531025 s before the rupture, cor



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responding tot02t'1023. Beforet02t;1022.5 simulationsshow a decrease of the minimum thickness consistent wthe t2/3 law; however, the simulation shows that a drasevent occurs near this timeDTE , causing a severe deviatiofrom the scaling law. This event can be seen in all simulaquantities. Examining the experiments with this resultmind, there also seems to be a slight deviation of the dfrom straight power-law behavior aroundt02t;531023,though it is difficult to establish this conclusion definitivewith current resolution.~We remark, however, that we werunable to get rid of the slight ‘‘glitch’’ in the experimentatime dependences near the breaking point by adjustingtime of rupturet0 . This is consistent with our conjecture ththe deviation from power-law behavior is a real effect.!

We will discuss the physical origin of this behavior blow. It is demonstrated that the timeDTE at which the de-viation from t2/3 occurs isindependentof the viscosity of thefluid; for a fluid of viscosity ten times smaller than thatwater the deviation also begins neart02t51023. Thus, thescaling range does not become larger as the Reynolds nber Re→`. This indicates an intrinsic failure of the dimen

Brenner et al.


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sional scaling argument, which suggests that the agreembetween thet2/3 law and experiments is somewhat artificiaAlso, below, we present further experimental evidencethe breakdown of the assumptions leading to the dimensiscaling law, based on the shapes of the profiles.

What is the reason for the complex time dependenceis counterintuitive that the minimum thickness does notcrease monotonically as the drop breaks, but actuallyin-creasesfor a short time. The origin of this can be understoby carefully examining the shape profiles: Figure 5 shothe maximum slope of the profile as a function oft02t: Theinviscid similarity solution predicts that the slope shouldconstant. Before the nonmonotonic glitch, during the tiperiod when the scaling laws are consistent with the dimsional scaling laws, the slope is constant@max(hz)'2#; how-ever, att02t'1022 the slope increases rapidly, saturatinga constant value@max(hz)'110#.

The major consequence of the time dependence ofslope is that when the profiles are rescaled according to~6!, the shapes donot collapse on the steep side of the prfile, as shown in Fig. 6~a!. Thus, even though the time dependences agree with the dimensional predictions, the share not self-similar in time, even beforet02t'1023, whenthe time dependence seems to agree with the dimensscaling laws.

This same behavior can be seen in [email protected]~b!#: Profiles were extracted from the images using theage processing package IDL, and then rescaled as outabove: both radii and horizontal length scales are scaledthe characteristic lengthl (t8). The experimental profiles alsdo not collapse onto a single curve, implying that the dnamical behavior deviates from the similarity solution eqution ~6!. Qualitatively, the experimentally collapsed profil@Fig. 7~b!# are very similar to the simulations@Fig. 7~a!#.

It is interesting to note that simulations of inviscid, irrotational flow12,37show an overturning of the profile at a finittime away from the singularity, rather than a gradual ste

FIG. 5. The maximum slope of the interface as a function of time to rupfrom simulations. The similarity solution equation~6! predicts that this slopeshould be constant in time.



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ening. A profile that has overturned looks perfectly flat whviewed from the side, as done in the experiments. Thethat the simulations and experiments agree so well bebreakup provides evidence that no such overturning occbefore breakup. As remarked before, the situation is differduring violent satellite drop oscillations, where flat portioappear at the end of a drop.

The interfacial shapes in both simulations and expements are inconsistent with the similarity solution~6!. Al-though the characteristic length scales do agree withsimilarity solution over a range of scales, the agreembreaks down in a regime where the similarity solution shostill hold. Again, we emphasize that simulations with evsmaller viscosity show that the deviation from the scali

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FIG. 6. ~a! Rescaled profiles from numerics before the curvature singula(t02t>1023). Both horizontal and vertical scales are rescaled byt02 t)2/3. The location of the final breaking pointz0 is subtracted from each othe profiles. The solid, dotted, dashed, and long dashed lines repret02t50.03, 0.01, 0.003, and 0.001, respectively.~b! Collapsed profilesfrom experiment. The location of the final breaking pointz0 is subtractedfrom each of the profiles. The solid, dotted, dashed, and long dashed curepresentt02t50.019, 0.011, 0.007, and 0.003, respectively. Note tthese experimental profiles show no evidence of overturning.

1581Brenner et al.


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solution occurs very near the time the deviation occabove, so that the deviation isnota consequence of the finitfluid viscosity.

A. The curvature singularity

The overall deviation from self-similarity occurs becauthe slope next to the breaking point grows by a factor ofon a time scale much faster than the pinching. This steeping is a remnant of a finite time singularity in theinviscidequations; without viscosity, steepening leads to a singulain which the local gradients in the velocity blowup in finitime. The mechanism of the steepening is a convective inbility: regions of high velocity are convected at higher vlocities, leading to a singularity in which the velocity gradent diverges with the velocity remaining finite. This pointsthe dynamical formation of a new length scalelEuler, whichapproaches zero in finite time. The mechanism is reminiscof shock formation in compressible hydrodynamics. For flids with finite viscosity, the convective instability is stoppebeforelEuler→0, at a scalel set by the viscosity. The lengtl controls the maximum steepness of the front. This mecnism therefore provides a natural explanation for why loviscosity fluids have very sharp fronts.

The maximum slope of the sharp front is controlledthe dynamics of the steepening. If the inviscid or Lee’s eqtions are simulated, rapid growth occurs at the highest wnumbers, and the solution is dominated by fluctuationsthe scale of the grid. Therefore, we performed a seriessimulations at increasing Reynolds numbers, which we laextrapolate to the inviscid case. Even for the largest Rnolds numbers, no high-wave number instability was oserved. Figure 7 plots the maximum slope for the Reynonumbers Re543, 427, and 4273 as a function of the timerupture. All of the plots demonstrate a rapid increase inslope up to a saturation level determined by the Reynonumber, similar to Fig. 5.

Finally, we note that the simulations depicted in Fig.

FIG. 7. Maximum slope during the course of a simulation as a functiontime for liquid bridge simulations, for various Reynolds numbers. A liqubridge is a cylinder of fluid held at constant radius at both ends.



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were done for a liquid bridge~a finite cylinder of fluidpinned at both ends! instead of drops falling from a faucethe major differences are the absence of gravity for the liqbridge, and the different initial conditions for the two caseAlthough the basic qualitative features of the formationthe curvature singularity are independent of experimenconfiguration note that there are several quantitative difences between the dynamics of Fig. 7 and Fig. 5: thesimulations have a different early time transient thandrop simulations. Also, at the highest Reynolds num(Re54273), the jet simulation shows a nonmonotonic bhavior. We do not know if this nonmonotonic behavior pesists when Re→`. There are two important qualitative features of Fig. 7: first, the maximum slopedivergesas theReynolds number Re→`. Figure 8 plots the maximum slopas a function of Reynolds number for the above simulatio

The points are numerical data, and the solid line repsents the scaling law

maxhz;Re1.25. ~10!

We cannot be sure, however, that this exponent represthe asymptotic scaling law. Note that Re54273 represents afluid with one-tenth the viscosity of water, falling from acm nozzle. At the point where the slope reaches its mamum, the smallest features resolved are of the size 1027,which is the best we can do with the present numerical simlations.

The second important point is that the time at which tsteepening begins is roughly independent of Reynolds nber. Therefore, we conclude that the steepening is a remof a finite time singularity in the inviscid equations, in whicthe interfacial curvature diverges. This ‘‘curvature singulaity’’ is distinct from the singularity at rupture, and introducea new length scale into the problem a finite time from trupture. This is the reason that the dimensional scaling labreak down before rupture. Moreover, the inviscid equatioare not able to describe the bifurcation of a liquid drop,the curvature singularity provokes a breakdown of the eqtions before breakup occurs.

f

FIG. 8. Maximum slope as a function of Reynolds number, extracted frthe liquid bridge simulations of Fig. 7.

Brenner et al.


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As a word of caution, we reiterate that our theoreticdescription was based on an equation that has not beenrived from the Navier–Stokes equations; moreover, themal motivation for the equations breaks down in the limitsharp slopes. The argument that we are making is that~1! thelubrication equations show this curvature singularity;~2! thesolutions of the lubrication equations agree quantitativwith experiments, and therefore~3! it appears that the fulequations must also show a curvature singularity in theviscid limit, of the type described here. The actual struct~e.g., time dependences! of the singularity in the full Eulerequations may be different than the structure in the lubrtion equations.

B. After the curvature singularity

After the curvature singularity is stopped by viscosiwhat is the resulting dynamics? This is a regime whereonly have simulational evidence. Since the curvature sinlarity occurs at a time independent of Re, in principleRe→`, there could be a large scaling region for anothscaling solution. One possible scenario is that viscosityonly important in a thin boundary layer region, where it sthe maximum slope. In the other parts of the solution, vcosity is not important until the minimum thickness crossthe viscous length scale. Depending on how strongly theparts of the solution affect each other, Keller–Miksis scalcould be observed.

Although the simulation above does seem to followt2/3 law in the maximum velocity after the curvature singlarity, the scaling of the other quantities behaves differenMoreover, the profiles in this regime do not collapse uprescaling. We take this as a hint that immediately aftercurvature singularity there is a transient regime in whichscaling occurs. Neither our numerical simulations nor oexperiments are able to resolve what happens after transsettle down, so that we are not able to determine the strucof the solution in this asymptotic regime.

We remark that Chen and Steen have recently carout an interesting study40 of the dynamics of a membranwith surface tension surrounded by an inviscid fluid. Thstudy shows a regime before rupture similar to that shohere, where the interfacial shape is single valued andt2/3 breaks down. However, in their example, the subsequdynamics is quite different than that described here: Notain their case, the interface overturns before pinch-off~be-comes double valued!. After overturning, the dynamics eventually returns to thet2/3 scaling. We believe that the difference between the two studies is that in the present studycurvature singularity is regularized by viscosity, which stothe steepening of the slope and prevents overturning. Hever, more work on the limit of infinite Reynolds numberthe full Navier–Stokes equation is necessary.

V. SHAPES AFTER RUPTURE

Immediately after rupture, the shape of the interfacethe length scales of the experimental photographs@Fig. 4~b!#looks like a sharp conical tip attached to a spherical shelconical tip has no intrinsic length scale, and thus should



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coil according to thet2/3 scaling law. Measurements of howthis sharp tip relaxes after the rupture can be extracted fboth experiments and simulations, in the same manneoutlined above. There are two dynamical length scalesinterest: the lengthL(t) that the tip has recoiled a timet afterthe rupture, and the characteristic thicknessw(t) of the neck.Simulations can also measure the velocityv tip of the recoil-ing tip. Figure 9 shows plots of these quantities from boexperiments@Fig. 9~a!# and simulations@Figs. 9~b! and 9~c!#.We remind the reader that the rupture time used for tfigure is exactly the same time as the one used for the angous plots before breakup. Thus, given the rupture timetermined before breakup, the plots after breakup containfree parameters.

Although the experimental data seems to be consiswith the dimensional scaling laws, in the simulations theare rather severe deviations, especially very close to theture time. It should be noted that the deviations from tdimensional scaling law occur when the droplet thicknesfar greater than the viscous length scale. The basic quesis whether the deviation of the simulations from the dimesional scaling law represents afundamentalor if it merelyreflects a breakdown of the approximations within the simlations.

The reason for the breakdown in the scaling laws forsimulations can be understood by examining the shape ointerface immediately at the point of rupture, before anycoiling occurs~Fig. 10!. Recall that the experimental shapeas shown in Fig. 4~b!, show a conical tip before the recoiling

The simulations can probe much closer to the breakpoint than the experiments. It is clear from Fig. 10~a! that theshape at the instant of rupture in the simulations is mcomplicated than a simple conical tip~though at large scalethe shape does look conical!. The region bracketed in thefigure between 0.83,z,0.91 is conical, though there artransitions to different shapes on both sides of this regiFigure 10~b! shows a closeup of the region on the left-haside of the conical tip, nearest to the breaking point.

The closeup revealsanother conical region between0.795, z, 0.8, and also a crossover to amore slender regnearz50.795. The complicated structure of the interfacethe rupture point is a remnant of the dynamics befobreakup. The crossover atz50.795 noted in Fig. 10~b! arisesfrom the curvature singularity before breakup, and corsponds roughly to the interfacial thickness (hmin'1023),where the curvature singularity saturates~see Fig. 6!. Thecrossover between the two conical regions nearz50.83 oc-curs when the recoiling tip interacts with the capillary wavcaused by the recoiling of the first rupture at the bottomthe neck@see Fig. 4~a!#.

The argument of Keller29 based on mass and momentuconservation of the recoiling tip~see Appendix C! shows thatthe time dynamics of recoiling reflects the detailed structof the initial shape. If the initial shape is a power law, ththe time dependences are power laws; Given the complicinitial shape in the present situation, we expect the timenamics after breakup to be more complicated than a simscaling law, with crossovers corresponding to the crossovin the initial shape.

1583Brenner et al.


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FIG. 9. ~a! Recoil length and width as a function of time measured froexperiments. Distances are measured in units ofR. The recoil length isshifted vertically by a factor of 10. The solid line represents the (t02t)2/3

scaling law.~b! Plot of the recoil length as a function of time from simulations. The solid line corresponds to the Keller–Miksis scaling la(t02t)2/3. ~c! Plot of velocity of the recoiling tip as a function of time fromrupture in simulation.

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Indeed, the two crossovers noted in the time depdences of Fig. 9~b! to the crossovers in the initial shape. Thfirst crossover, occurs when the recoil length is 1022, andcorresponds to the crossover noted in Fig. 10~b!. The secondcrossover occurs when the recoil length is around 231021,and corresponds to the large-scale crossover in Fig. 10~a!.These crossovers are also apparent in the scaling for thvelocity v tip . There does exist a region on the plot of trecoil length where the time dependence is well appromated by thet2/3 law. This region, betweent02t'1023 andt02t'231022 occurs nearly at the same time that the eperiments show time dependences consistent withKeller–Miksis laws ~Fig. 9!. The ‘‘wiggles’’ appearing inthe simulations at times larger thant02t.231022 do notseem to exist in the experimental data@Fig. 9~a!#; this seemsto represent a quantitative discrepency between the exments and the simulations. To substantiate that‘‘wiggles’’ arise from the interaction of the recoiling necwith capillary waves in the initial shape, we have studiedtime dynamics of recoiling at the bottom of the neck@Fig.



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3~a!#: in this case, the crossover due to the curvature sinlarity still occurs at the same characteristic thickness, whthe crossover at large scales is completely absent, as noillary modulations exist before the recoiling. It should albe noted that Fig. 9~c! is completely inconsistent with thedimensional scaling lawt21/3, even in the range where thlength is roughly consistent with the dimensional scalilaw.

As mentioned in our description of the dynamics befobreakup, a more precise test for whether the scaling hypesis is realized is to collapse the profiles by their characistic length scales. Because of the complications outlinabove we focus on only the regime where scaling occuFigure 11~a! shows the interfacial profiles after rupture frothe simulations, rescaled according to Eq.~6!, for severaldifferent times after the rupture.

A similar procedure can be applied to the experimendata, as shown in Fig. 11~b!. The data collapse reasonabwell, although only over a finite range in similarity variableThe reason for this is twofold: First, there are finite si

Brenner et al.


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effects associated with the neck having a finite lengthobservations being made a finite distance away from thegularity. The second reason is the initial shape is not a ppower law, and this causes deviations.

In Appendix C we show a simulation thatdoesstart outwith a conical initial shape, and show that it is perfecdescribed by the Keller–Miksis scaling theory. This vadates the argument of this section that the deviations fthe Keller–Miksis scaling law are not because of a dynacal instability but instead complications in the initial shap

VI. CONCLUSIONS

Experiments demonstrate that the shape of a fluidplays a dramatic dependence on fluid viscosity~cf. Fig. 1!.Whereas high-viscosity fluids are slender near the breapoint, water drops exhibit a sharp conical tip. In this papwe have presented detailed comparisons between exments and numerical simulations of the breaking of lo

FIG. 10. ~a! Shape of the interfacea the instant of rupture from numericasimulations. Because the shape is not a power law, the argument of K~see Appendix C! suggest that the time dependences will not be perpower laws.~b! Closeup of~a! of the initial shape of the interface beforrecoiling.



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viscosity fluid drops, in order to understand better the dynamics at low viscosity. The simulations are of one-dimensional evolution equations that were previouslyderived by several groups24,30using asymptotic analysis fromthe Navier–Stokes equations. In Appendix A we present aalternative way of looking at the equations emphasizingsymmetries and conservations laws. The version of the equtions employed here uses an approximation of the mean cuvature that captures the exact equilibrium shapes of pendadrops. Because of this, the numerical simulations and expements are in quantitative agreement well beyond the initiafission event. We present a series of two experiments ansimulations, demonstrating the excellent agreement betwesimulations and experiments throughout the breaking process.

Detailed comparisons between the dynamics of thsimulations and the experiments uncovered several intrigu

lert

FIG. 11. ~a! Collapsed solutions from simulations for the recoiling fluidneck.~b! Collapsed solutions from the experiment. Length scales are measured in pixels from the photographs.

1585Brenner et al.


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ing features of the dynamics of low-viscosity fluid drops.this study we demonstrate the presence of aconvective insta-bility of the drop shape before breakup. The dynamics ofinstability is reminiscent of shock formation in gasdynamiand occurs because regions of high fluid velocity are cvected faster than low-velocity regions. This instabilitystopped by viscous stresses; in this way viscosity setsscalefor the steep tip in water drops. The convective insbility reflects that the infinite Reynolds number limit is sigular: the convective instability forces equations wRe5` to develop a finite time singularity well before thdrop breaks. With very small viscosity, the sharp tip formon a very fast time scale well before rupture, though the fibonafide singularity occurs at rupture. An intriguing consquence of the formation of a sharp tip is that for fluids wviscosity much smaller than water, so that the viscous lenscale is smaller than the molecular length scale, the contive instability is actually regularized by molecular effecinstead of viscous dissipation: in this case, the hydrodynaequations break down well before the drop itself breaks.

In this study we provide a concrete example of the coplications that can occur in describing a singularity. Athough dimensional arguments and their corresponding slarity solutions are suggestive, in general whether theyactually realized depends on subtle dynamical issues, wat present are not entirely understood. Instabilities of scasolutions in the limit where dissipation approaches zmight be important in other contexts where singularitiescur in inviscid flows, such as the three-dimensional Euequation. An example of of a basic issue that we still dounderstand is why the system chooses the similarity soluin Eq. ~7! with b51 before breakup, corresponding to thKeller–Miksis scaling law. In principle, any value ofb couldhave occurred, and would have caused both different tdependences and different interfacial shapes away frombreakup point. Although our study confirms that theresomething special about the similarity solution suggesteddimensional analysis, we do not understand the dynammechanism that selects this solution. Perhaps selectionciples, analogous to those used previously in resolving slar issues for traveling wave solutions41 are relevant for theformation of singularities.

On a different level, this study provides another sttoward understanding the dynamics, leading to the breakof a fluid drop. The general picture emerging is that ruptis described by a number of different scaling regimes; Tparticular state of the drop is controlled by dimensionleparameters formed by combining fluid parameters wlength scales characterizing the drop. Since the latter aredependent, a drop typically passes through several diffescaling regimes during a single fission event. Understandthe details of each scaling regime individually, and whenvarious crossovers occur, yields a quantitative descriptiothe breaking dynamics.

ACKNOWLEDGMENTS

We thank Dan Mueth and David Grier for assistanwith writing the image processing programs used to extr



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the profiles from the experiments. We are also grateful toreferees, both for their insightful comments, and for thpatience during the review process.

This research was partially supported by the MRSProgram of the National Science Foundation under AwNo. DMR-9400379. In addition, M.P.B. acknowledges a Ntional Science Foundation postdoctoral fellowship, andSloan Fund of the School of Science at MIT. J.E. acknoedges support by the Deutsche Forschungsgemeinsthrough Sonderforschungsbereich 237.

APPENDIX A: DERIVATION OF LUBRICATIONEQUATIONS

In this appendix, we motivate the nonlinear evolutioequations by relying on conservation laws and symmetriethe full equations. Our principle aim is to emphasize thatlong wave equations are the simplest one-dimensional padifferential equations that could be expected to modelmotion of a droplet. The discussion is similar to the previowork of Green,35 whose theory was applied by Shieldet al.42

The first equation that the fluid must satisfy is mass consvation:

~h2! t52~h2v !z , ~A1!

whereh(z,t) is the thickness of the fluid neck an axial ditancez from the nozzle, andv(z,t) is the azimuthal compo-nent of the velocity. As in the main text, we nondimensioalized lengths by the nozzle radiusR, and times byArR3/g. Galilean invariance implies that the equationmotion is of the form

v t1vvz52kz1D, ~A2!

wherek is the curvature andD is the dissipation function. Ageneral form forD andp follows from the requirement thathe total energy,

E5E dz h2v21E dz hA11hz2,

satisfies] tE , 0. This implies thatp5dE/dh, and thatD isof the form

D~h,v !51

h2@2 f 1v1~ f 2vz!z

2~ f 3vzz!zz1~ f 4vzzz!zzz1•••#,

where thef i are positive definite functions ofv andh. Thevalues of thef i can be obtained to leading order bydemand-ing that the model equations exactly reproduce the lingrowth rate of the full hydrodynamic equations. For the caof present interest, liquid falling into vacuum, this impliethat f 150 and f 253/Re.

Thus, the equations are

~h2! t52~h2v !z , ~A3!

v t1vvz53

Reh2~h2vz!z2kz , ~A4!

Brenner et al.


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APPENDIX B: NUMERICAL CONSIDERATIONS

The numerical simulations utilize a second-order ctered finite difference scheme of the partial differential eqtions~1!–~3!, which has been previously described in seveplaces. The goal of this appendix is to mention the variopitfalls and tricks that were necessary to develop forsimulations reported in this paper.

There were two different numerical challenges: the fiinvolved issues associated with breaking the drop and silating the subsequent dynamics~Figs. 2 and 3!. A priori itwas unclear that these simulations would work becausethe concerns about overturning expressed in the mainand demonstrated in completely inviscid simulations. It walso necessary to develop algorithms to deal with the silation of many drops simultaneously.

The second numerical challenge was to resolve the vous singularities described in this paper. We emphasizein order to achieve enough numerical data to make accuassertions about singularity mechanisms, it is necessarhaveat leastseveral decades of power law behavior. Withoseveral decades of scaling, it is possible to be fooled inpossible ways: first convergence to similarity solutions dnot occur exponentially in time, but instead exponentially2 log(t02t). This means it typically takes a few decadesscaling for the transients to die out and reveal the true scabehavior. Without several decades of scaling measuremof exponents will be inaccurate; the latter can be sconsistently tested by determining if the solution converto the spatial structure of the similarity solution. Seconrecent work23 has demonstrated that there are concrete phcal examples of singularities which can destabilize at atrarily small time distances from singularities. Although thcan never be ruled out completely in a purely numeristudy, a sufficiently large number of decades of scalmakes this outcome less probable.

Before proceeding into specific issues that arise weach of these problems, we first summarize some of theeral procedures. The time step control was adjusted soonly one Newton iteration was required at each time stThe time step is adjusted by using a two step method:each time step, we first step byDt, and then redo the calculation with two time steps of sizeDt/2. The relative error inthe solution is then computed. Typically the time stepcontrolled by demanding that the relative error is less thafixed threshold~e.g., 1%!.



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High resolution is achieved by using a dynamically ajusting mesh. At low viscosities there are two different sgularities: one associated with the minimum height goingzero, the other with diverging gradients. Thus we monitoboth the minimum height and the gradient of the mass flh2v. If either changed by a specified percentage, new mpoints were added to the grid. The mesh is organizeddefining an underlying macrogrid, and adjusting the numof microgrid elements into which each macrogrid is divideAfter introducing new grid points, the solution is reinterplated onto the new grid. Typically linear interpolation or cbic splines are used.

A subtlety of this procedure is that at places wheremesh spacing changes by a factor of 2, we observed anstability of the discretized equations. This may be connecwith the fact that our spatial discretization was only firorder correct at those places~because of thehzzz term at themesh points!. This ‘‘numerical noise’’ does not have a largeffect on highly viscous fluids, though it is capable of iniating instabilities in viscous similarity solutions. However,Re→` these effects are quite substantial and make accunumerical simulations very difficult. In order to perform thsimulations at the highest Reynolds numbers, it was nesary to smooththe mesh. This was done by diffusing thmesh points until the ratio of neighboring mesh spacindiffered by less than a percent.

To break a drop, we used the following procedure.threshold thicknesshthreswas defined~typically 10

25!. Whenthe minimum thickness of the fluid neck passes belowhthresthe drop is manually broken into two pieces, by attachsmall spherical masses of fluid to each end. The radius ofspherical masses was determined by demanding that itconstant pressure. The interpolation of the spherical monto the fluid was done so thathzz is continuous. Extensivetests revealed that the macroscopic dynamics was insensto both the precise spatial location of the breaking pointwell ashthres.

Once the drop was broken, each of the drops remainwere evolved by the code. The thicknesshi , the velocityv i , and the positionszi of each point were stored and thequations of motion applied at each time step. Regriddproved to be crucial for following through many ruptures,order to always have enough mesh points.

APPENDIX C: IDEALIZED INITIAL CONDITIONS:CONES AND CUSPS

We have seen that simple theories based on a rescaof the pinch region are invalidated by the appearance ofother singularity, which introduces a second length scaledirectly associated with the pinch. This inviscid singularityconvective in character, and thus arises through the massacross the pinch region. After breakup, this mechanismabsent, because the flow across the pinch region is inrupted. Hence it seems as if self-similar solutions shoexist after breakup. However, it turns out that the solutafter breakup depends heavily on the shape of the interh0(z)5h(z,0) at breakup, which in turn is a reflection of th

1587Brenner et al.


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dynamicsbeforebreakup. If additional length scales playrole before breakup, they are thus also introduced into tdynamics after breakup.

This is immediately evident from the arguments advanced by Keller29 and worked out in more detail by Keller,King, and Ting,43 using a matched asymptotic expansionThe first assumption is that velocities at breakup are smcompared with the velocities produced by the violent recoso h0(z) is the only input needed. The second assumptiowhich was validated by the matching in Ref. 43, is that thfluid contained in the part of the neck that has already rcoiled is sucked up into a spherical head. Intuitively, thisbecause there is no flux across the head, so it is acceleruniformly by the recoil, and assumes a static shape. To a fiapproximation, one neglects the crossover region betwethe bulbous head and the static part of the neck, which hnot yet recoiled. Thus, apart fromh0(z), the geometry iscompletely specified by the recoil lengthL(t8) ~which de-fines the head’s center, witht85t2t0 the dimensionless timefrom the initial cone! and the width of the headw(t8). Fig-ure 12 shows the time dependences from a numerical simlation of a recoiling cone. The initial shape is conical withsmall spherical cap of sizeh0 , so thatL(0) andw(0) are oforder h0 . The time dependences agree quite well with thKeller–Miksis scaling law, except for a small transient period of orderAh03r/g ~in dimensional units!.

Mass conservation gives

4p

3w32pE

0

L

h02~z!dz50, ~C1!

and from conservation of energy we have

2p

3w3Lt

21H 4pw222pE0

L

h0A11h0z2 dzJ 50. ~C2!

FIG. 12. Time dependences for the recoiling lengthL as a function of thetime t2t0 from the initial cone. The dots represent numerical data and tsolid line is the Keller–Miksis scaling law. The crossover atL'1024 cor-responds to the small-scale cutoff in the initial condition.



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To exemplify the dependence of the solution onh0 , we as-sume that

h0~z!5bzb, b>1, ~C3!

as was done by Ting and Keller.39 We hasten to add thatself-similar solution withbÞ1 was never observed expermentally or in simulations. The caseb51 was observedover a limited scaling range, whose size does not expanddecreasing viscosity. To focus on scaling properties, wesume that

w~ t8!;t8a1, L~ t8!;t8a2. ~C4!

From ~C1! one has

w;L ~2b11!/3,

and combining that with~C2! one obtains

a152

3 S 2b11

b12 D , a252

b12. ~C5!

Two crucial observations are to be made here: First,solution explicitly depends onb, i.e., on the initial conditionh0 . If h0 does not scale like a power law, as it is observedsimulations, neither willw or L. Second, only ifb51,which corresponds to Keller–Miksis scaling, will the soltion be describable by a single rescalingj5z/L of the ab-scissa. Namely, forb51 w andL both scale liket82/3, as tobe expected from Keller–Miksis scaling, while forb.1 thetwo will scale differently. The time dependence of the nelengthL agrees with the scaling law proposed by Ting aKeller,39 which is based on the scaling ansatz~8!, ~9!. How-ever, the width of the tipw is in disagreement with thisscaling forb.1. This is not surprising because the undering assumption is that the slopeh8 is negligible, which evi-dently is not the case near the tip. Therefore, the solumust assume a more complicated structure, the neck andtip region being governed by different scaling laws.

Finally, we discuss the caseb51 in a little more detail,since it can be understood completely within the framewof our simplified model~1!, ~2!. It also gives the closesapproximation to existing experimental data. Namely, in tcase~6!, ~7! is an exact solution of the inviscid equations,H andV obey the similarity equations

2

3H2

2j

3H852VH82

1

2V8H,

~C6!

2V

322j

3V852VV81S 1

HA11H822

H9

~A11H82!3D 8.

Here the primes denote a derivative with respect to the slarity variablej. These equations have to be solved subjectwo sets of boundary conditions. At infinityH andV shouldbehave like

H5aj, V5bj21/2, ~C7!

and at the tip, located atj5j0 , one has

H~j0!50, H8~j0!5`, V~j0!52j03

. ~C8!

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Two free parameters, which are the tip position and thelocity gradientV8(j0) at the tip have to be adjusted to mathe solutions match the free constantsa andb in ~C7!. Thisprocedure is analogous to the one employed in Ref. 26viscoussolutions.

In Fig. 13 a solution of~C6! is compared with the resulof a simulation, starting from a perfect coneh0(z)5az andzero initial velocity. The valuea50.13 was chosen close tthe opening angles observed in Fig. 3, while in experimeunfortunatelyb is not available. Among other things, thexcellent agreement between simulation and the predicof similarity theory shows that the small amount of viscospresent in the simulation does not appreciably affect thelution on scalesL@ l n . Thus the viscous terms do not reprsent a singular perturbation, as was the case before brea

The similarity solutions also allow for an analytical uderstanding of the capillary waves excited on the surfawhich are the most characteristic feature of inviscid sotions. Namely, repeating the perturbative analysis of Tand Keller39 for the model~1!, ~2!, we find the amplitude andwavelength of capillary waves on the receding cone afunction of the parametersa and b. The wavelength justdepends ona, reflecting the weak dependence of the soluton the initial velocity field:

l52pS 98 a

~A11a2!3D 1/2j21/2. ~C9!

Thus the wavelength gets shorter farther away from theThe analytical formulas are confirmed beautifully by tsimulation, but are difficult to observe experimentally, bcause of the shortness of the scaling range.

The same observation was made by Schulkes,37 whosimulated inviscid, irrotational flow. While solutions startinfrom initial cones collapse well under Keller–Miksis scalinno quantitative collapse is found for initial shapes that cafrom his computations.

FIG. 13. Collapsed numerical data for the recoiling cones. Each ofsymbols refers to numerical data, corresponding to the times 1026 ~circles!,1025 ~squares!, 1024 ~diamonds!, 1023 ~triangles!, 1021 ~side triangles!,and 5021 (X) from the initial conical shape. The solid line is a similarisolution of the inviscid equations.



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Brenner et al.


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