A Cascade of Structure in aDrop Falling from a Faucet

X. D. Shi, Michael P. Brenner, Sidney R. NagelA drop falling from a faucet is a common example of a mass fissioning into two or morepieces. The shape of the liquid in this situation has been investigated by both experimentand computer simulation. As the viscosity of the liquid is varied, the shape of the dropchanges dramatically. Near the point of breakup, viscous drops develop long necks thatthen spawn a series of smaller necks with ever thinner diameters. Simulations indicate thatthis repeated formation of necks can proceed ad infinitum whenever a small but finiteamount of noise is present in the experiment. In this situation, the dynamical singularityoccurring when a drop fissions is characterized by a rough interface.

What happens when liquid drips from afaucet.7 As it falls, its topology changes froma single mass of fluid into two or moredrops. This common phenomenon is one ofthe simplest hydrodynamic examples of asingularity (1) in which physical quantitiesbecome discontinuous in a finite time.Here, we investigate the shape of this singularity for fluids of varying viscosity dripping through air from a cylindrical nozzle.Scientific studies of this system date back toLord Rayleigh's stability analysis of a liquidcylinder (2) and Plateau's analysis of ahanging pendant droplet (3). More recentwork has begun to address the shape of theinterface near the singularity. Eggers andDupont (4) simulated falling droplets bymeans of the Navier-Stokes equations andshowed that their solutions agreed with theexperimental shapes of water photographedby Peregrine et al. (5).

Although our study focuses on a specificexperimental system, the dynamics near asingularity should be insensitive to changesin the initial conditions and external forcingwithin a wide range of parameters. Thus, theshape of the interface near the breakup pointshould not depend on whether the drop isfalling in a gravitational field or being pulledapart by shear forces (6). This expectationarises from the realization that as the interfile breaks, its thickness must eventuallybecome much smaller than any other lengthin the problem (until microscopic atomicscales are reached). In this regime, the onlylength (7) that can affect the dynamics is thethickness of the fluid itself, so that thesimplest assumption for the dynamics is self-similarity—that is, the shape near the sin-hilarity changes in time only by a change inscale. The mathematical definition of a sim-rarity solution is

Hz,t)=f(t)Hz - Z o/(t) f

(1)

where h describes the radius of the drop asa function of the vertical coordinate z andtime t; /(t) is an arbitrary function of time;P is a constant; and z0 is the positionwhere the droplet breaks. This scalinghypothesis has worked well in describingother types of singularities, ranging fromcritical phenomena (8) to droplet breakupin a two-dimensional Hele-Shaw cell (9,10). Also, for our problem here Eggers(1 I) constructed a similarity solution thatshowed good agreement with numericalsolutions to the Navier-Stokes equation at

bs Franck Institute and Department of Physics.Brsity of Chicago. 5640 South Ellis Avenue. Chi-J. IL 60637, USA.

low viscosities. On the other hand, Pumir,Siggia, and co-workers (12) have analyzedseveral mathematical models of singularities with a high Reynolds number thatshowed nonsteady corrections to Eq. 1.However, to the best of our knowledgethose models did not have an experimental realization. For the case of the dripping faucet, we will show that both viewshave some validity: the similarity solutionprovides a basis for the underlying structure, but there are time-dependent corrections to this solution that alter the shapedramatically.

There are three independent lengthscales that characterize the hydrodynamicsof the dripping faucet (4): (i) the diameterof the nozzle D; (ii) the capillary length

I -PS

which gives the balance between surfacetension, 7, and the gravitational force pg,where p is the fluid density and g is theacceleration of gravity; and (iii) the viscouslength scale

L .1n P Y

V Fig. 1. The shape close to thetime of breakup of five drops withdifferent viscosities. The liquids,with-rj = 10-2P(A), 10"' P(B), 1P(C),2P(D),and12P(E), wereallowed to drain slowly through aglass tube with a nozzle diameterD of 1.5 mm. (A) and (E) showpure water and pure glycerol, respectively. We used an 80-mmHasselblad lens attached to abellows and a still camera; thedrop was illuminated from behindby a fast (=5 »xs) flash from astrobe (EG&G model MVS 2601,Salem, Massachusetts) that wastriggered with a variable delayfrom the time the drop intersecteda laser beam incident on a photo-diode.

SCIENCE • VOL. 265 • 8 JULY 1994 219

,[where tj is the viscosity. For water (5), L^= 100 A. For that system, it is not possible to photograph the shape of the interface where the thickness is asymptoticallysmaller than L^ because current opticaltechniques cannot resolve objects muchsmaller than 1 u.m.

To find the asymptotic shape of thesingularity, we studied droplet snap-off as afunction of viscosity. By mixing glycerolinto water, we can increase the viscosity bya factor of 103 so that L^ can be increasedby 106 while the surface tension is notvaried by more than 15% (13). We thoroughly mixed the water and glycerol, takingcare to eliminate air bubbles. The shapesclose to the breakup point of five liquids

with different viscosities draining slowlyfrom a nozzle of diameter D = 1.5 mm varyas the viscosity is varied (14) (Fig. 1). Asthe value of t) increases, the neck of thedrop elongates and forms structures notobservable in dripping water. Herein, weconcentrate our attention on the structureof a drop of liquid with intermediate viscosity and report a combination of photographic studies of the singularity and numerical simulations of the Navier-Stokesequations.

As a drop of an 85% glycerol and 15%water mixture [tj = 1 poise (P), where 1 P= 1 dyne-s cm-2, and L^ = 0.13 mm]breaks (Fig. 2), it forms a long thin neck offluid connecting the nozzle exit plane (a flat

Fig. 2. A photograph of a drop of a glycerol in water mixture (85 weight %). (A) The initial stages ofthe breakup as the droplet separates from the nozzle by a long neck. A secondary neck is visiblejust above the main drop. (B) A magnified view of the region near the breakup point (obtained froma different photograph). The thickest region in this picture corresponds to the long neck of (A) Onlyone secondary neck is visible. (C) The same region as in (B) but at a slightly later time. The secondneck and a well-developed third neck are clearly visible in this picture. Using the Eggers similaritysolution, we estimate that (B) and (C) are 2 x lfj-3 and 2 x 10"4 s, respectively, before the timeof breakup. The photographs were taken as in Fig. 1, except with a 35-mm lens. Because theclearest demonstration of the structure of the singularity was obtained with the use of the largestdrop, we found it advantageous to replace the glass nozzle by a brass plate riddled with numeroussmall holes on which the liquid would slowly collect before falling. The diameter of the drop at theplate was approximately 20 mm.

brass plate) to the main part of the drop. Acloseup photograph (Fig. 2B) gives a clearerview of the region just above the main dropwhere the neck decreases in width, forminga secondary neck. At a slightly later time(Fig. 2C), this secondary neck suffers aninstability and forms a third neck. Weestimate the thickness of the smallest neckin Fig. 2C to be about 5 u.m. Throughhigh-speed photography, we have verifiedthat both the second neck and the thirdneck develop by a similar process: an initialthinning near the drop is followed by arapid extension of the neck upward awavfrom the drop. This leads us to suspect thatthe repeated formation of such necks couldhappen indefinitely. Our simulations described below can produce such a phenomenon ad infinitum (with as many as sevensuccessive necks at the resolution limit) aslong as a noise source is present. On theother hand, in the absence of noise, oursimulations indicate that near the singularity the interface is smooth. We argue thatnoise, which is always present in the experiment, invariably leads to a singularity withcomplex structure.

Numerical simulations of the Navier-Stokes equations have provided a well-developed tool for reproducing the large-scale structures and velocities of fluid flows.In addition, they can investigate structureon finer scales than can be seen in thefigures here. Our simulations used the equations of Eggers and Dupont (4), which areapproximate versions of the Navier-Stoke>equations (for a Newtonian fluid) that neglect non-axisymmetric effects (4, 15, 16):

' d v d v \ 3 t \ d I d v \ d pP | - + « 7 h - , .■ - . - ( * » - ] - _ + „d< dz h2 dz

dh2 d(h2v)+ = 0

dt dz

(2)

(3)

>mkafcv 21-1/2

_ y Bh\2Yi/2d2hI -dz T T ( 4 )dz'

where h(z) is the radius of the fluid neck adistance z from the nozzle, and v(z) is thefluid velocity. Equation 2 expresses forcebalance: the acceleration is due to viscousstresses, a pressure gradient, and gravity;Eq. 3 expresses local mass conservation;and Eq. 4 specifies that the pressure jumpacross the interface is determined by it-mean curvature. These equations weresolved with a modified version of a finite-difference scheme supplied by Eggers anJDupont (4). An adaptive grid, similar tothat of Bertozzi et al. (17), allowed highresolution of the singularity. At prescribed

220 SCIENCE • VOL.265 • 8 JULY 1994

rime intervals, the grid was adjusted.In Fig. 3, we present the results of

numerical simulations for a liquid with aviscosity of 1 P falling from a nozzle with adiameter D = 4.5 mm. Figure 3 A shows thedrop profile approximately 10-8 s beforerupture. The shape of the interface near thedrop is similar to that shown in Fig. 2A(18). In a magnification of the region of thesecondary neck (Fig. 3B), the interfacethins near the main drop. This thinningcontinues smoothly to arbitrarily smallscales. Near the singularity, the interfacial>hape is well approximated by the similaritysolution proposed by Eggers and first seen byEggers and Dupont (4). However, thesesimulations show no evidence of the repeated formation of necks, as shown in Fig. 2.

To induce the repeated formation ofsuch necks in the simulations, it was necessary to introduce a noise source explicitly.We did this by placing fluctuations, on theorder of 10-4 the minimum thickness of theneck, on the interface during the simulation. Both the temporal and spatial separation between subsequent fluctuations were-mall compared to the characteristic scalesof the Eggers similarity solution. Our results(10~8 s before break-off) show a magnification of the secondary neck (Fig. 4A) andthe highest resolution attainable with ourpresent code (Fig. 4B). The smallest neckshown in Fig. 4 is only 2 A wide. At eachlevel, before a new neck starts to form, theshape of the interface was approximated bythe similarity solution of Eggers. This resultdemonstrates that, at the point of breaking,fission proceeds by means of the formation

hID0

B / i / D f x I O - 2 )1 - 7 . 5 0 5

Hg. 3. Simulation of the profile for a drop of aglycerol in water mixture (85 weight %; viscosity= 1 P) falling from a nozzle of diameter D = 4.5mm. (A) A view of the entire drop, at approximately 10-8 s before snap-off, shows that nearIhe bottom of the long neck there is a region■''here the thickness decreases and forms asecondary neck. (B) An enlargement of theregion in (A) enclosed by brackets. Note thatthe scales are different along the two axes, so;hat the shape can be easily seen. The insetshows another, greater magnification.

of necks that themselves form necks downto the smallest level detectable.

Noise fluctuations in the shape of theinterface become more effective at creatingadditional structures as the thickness of theinterface decreases (19). The amount ofnoise introduced in the simulations abovecorresponds to angstrom-size fluctuations ofthe interface when the thickness of a neckis still on the order of 1 u.m. Thus, eventhermal fluctuations may be sufficient toproduce these effects. The only caveat isthat because this is a strongly nonequilib-rium situation, we do not understand thespectrum of the noise in the experiment.However, there are many sources of noise,ranging from surfactants on the interface,to pressure fluctuations in the nozzle, to aircurrents in the room, shear heating in theliquid, and thermal fluctuations (for example, excitation of capillary waves). In ourexperiment, we might also expect noisegenerated from fluctuations in the concentration of the two liquids as well as fromevaporation from the surface. On repeatingthe experiment, we found that the details ofthe interfacial shape can vary: the preciseposition of the necks can change from dropto drop and blobs can form on the interfaceat irregular intervals. Nevertheless, the repeated formation of necks appears to berobust. (Our simulations with a noisesource also produce such variations whendifferent realizations of the noise are used.)

Taken together, our results show thatthe singularity in the fission of a drop ischaracterized by the repeated formation ofnecks, which may proceed ad infinitum inthe presence of a small noise source. The

A / 7 / D ( x i o - 3 ) B h / o r x - i o - 4 )- 4 0 4 - 2 . 5 0 . 0

2.501-

Fig. 4. Simulation of the same drop as in Fig. 3in the presence of noise. A view of the entiredrop is indistinguishable from that in Fig. 3A.(A) An enlargement of the first neck (as in Fig.3B) shows a third, fourth, and (very thin) fifthneck sprouting from the second neck. (B) Anenlargement of the bracketed region of (A)shows the fourth and fifth necks of the previousfigure, as well as the sixth neck. In the inset, amagnification of the sixth neck shows a seventhneck.

shape of the interface near this singularity istherefore rough, with a time-dependentstructure in contrast to the scaling hypothesis of Eq. 1 (20). The description of thissingularity is more complicated, with newstructures evolving progressively more frequently in time (21). In a real fluid, suchrepeated formation of necks can occur onlya finite number of times. Once the neckbecomes thin enough, a hydrodynamic description breaks down as long-range van derWaals forces and molecular effects becomeimportant. In this regime, simulators haveused molecular dynamics techniques (22,23) to study such a breakup.

In our experiments, after the drop detached from the bottom of the neck a satellite droplet formed from the long neck itselfand separated from the fluid near the nozzle.The singularity in this case had a structuresimilar to the one described above (wherethe drop separated from the bottom of theneck), and in both cases the thinnest neckswere attached to a large, almost sphericalmass of fluid. However, for the upper singularity, the diameters of the necks increasedwith distance from the nozzle (in the direction of gravity), which is opposite to whathappens in the situation at the bottom of theneck where the diameters decreased (againstthe direction of gravity). The form of thesingularity thus appears to be invariant tothe direction of the external forcing.

We have shown that the singularityfound in the fission of a viscous drop ofliquid is different from what had been expected from earlier observations of less viscous fluids. Instead of a smooth interface,we found (to the limit of our resolution) arough interface with necks "growing" othernecks. The existence of this phenomenonappears to depend on the presence ofexperimental noise. However, it should bepointed out that because our simulationsused only an approximation of the Navier-Stokes equations, it is possible that such acascade of necks could form by a deterministic mechanism in the full equations. Ourresults raise many questions: To what extent does the regular repetition of theformation of these necks depend on thecharacteristics of the noise source? Underwhat circumstances can the interfacialshapes develop even more structure? Andwhat is the spectrum of noise in a typicalexperiment?

REFERENCES AND NOTES1. A. Pumir and E. D. Siggia, Phys. Rev. Lett. 68.

1511 (1992).2. W. S. Rayleigh, Proc. London Math. Soc. 4, 10

(1878).3. J. Plateau, Statique Experimental et Theoretique

des Liquides Soumis aux Seules Forces Molecu-laires (Gauthier-Villars. Paris, 1873).

4. J. Eggers and T. F. Dupont, J. Fluid Mech. 262.205 (1994).

SCIENCE • VOL.265 • 8 JULY 1994 221

5. D. H. Peregrine, G. Shoker, A. Symon, ibid. 212,25 (1990).

6. M. Tjahjadi. H. A. Stone, J. M. Ottino, ibid. 243,297 (1992).

7. We note that, nevertheless, the solution near thesingularity must still be measured in terms of alength scale relevant to the dynamics in theproblem. This scale is the viscous length definedbelow.

8. C. Domb and M. S. Green, Eds.. Phase Transitions and Critical Phenomena (Academic Press.London, 1972).

9. P. Constantin et al.. Phys. Rev. £47, 4169 (1993).10. R. E. Goldstein, A. I. Pesci. M. J. Shelley. Phys.

Rev. Lett. 70, 3043 (1993).11. J. Eggers, ibid. 71. 3458 (1993).12. A. Pumir, B. I. Shraiman, E. D. Siggia, Phys. Rev.

A 45, R5351 (1992); T. Dombre, A. Pumir, E. D.Siggia, Physica D57, 311 (1992).

13. G. W. C. Kaye and T. H. Laby, Tables of Physicaland Chemical Constants (Longman, New York,1986); J. B. Segur and H. E. Oberstar, Ind. Eng.Chem. 43. 2117 (1951).

14. E. A. Hauser, H. E. Edgerton, W. B. Tucker, J.Phys. Chem. 40, 973 (1936); H. E. Edgerton, E. A.Hauser, W. B. Tucker, ibid. 41, 1017 (1937).

15. Similar equations have been derived independently by S. E. Bechtel, J. Z. Cao, M. G. Forest [J.Non-Newtonian Fluid Mech. 41.201 (1992)] in thecontext of a general mathematical study of thedynamics of viscoelastic fluids. Their derivationincludes Eqs. 2 and 3 as a special case, with aslightly different form for Eq. 4.

16. A similar equation (without the correct coefficientin front of the term with viscosity) was proposedby R. W. Sellens [Atomization Sprays 2, 236(1992)].

17. A. L. Bertozzi, M. P. Brenner, T, F. Dupont, L. P.Kadanoff, in Centennial Edition, Applied Mathematics Series, L. Sirovich, Ed. (Springer-Verlag

Applied Mathematics Series, New York, 1993).18. The boundary conditions used at the top of the

drop in the simulation are different from those thatactually apply in the experiment. We believe thatthis accounts for the differences in Ihe two figuresnear the nozzle.

19. M. P. Brenner, X. D. Shi, S. R. Nagel, unpublishedresults.

20. From the Eggers similarity solution, one can showthat the Reynolds number is =1. This situation isdistinct from other models considered (1. 12).

21. Tjahjadi et al. (6) observed a different hydrodynamic phenomenon that also had a fractal spatialstructure. By allowing a stretched drop to relax,they found a sequence of droplets with sizes thatdecreased in a geometric progression. Our situation differs from theirs in that the repeated formation of necks is a property of the singularityitself and not the result of multiple breakup events.

22. J. Koplik and J. R. Banavar, Phys. Fluids A 5, 521(1993).

23. U. Landman, W. D. Luedtke, N. A. Burnham, R. J.Colton, Science 248, 454 (1990).

24. We thank J. Eggers and T. Dupont for many helpfuldiscussions and for generously sharing their numerical code and for showing us how to use it. We thankR. Leheny for initiating the experiments on the droplet breakup. R. Jeshion for help in the photographicdevelopment, O. Kapp for help in obtaining a fastcamera, and E. Siggia for useful comments. D. Grierand D. Mueth helped us analyze the photographsdigitally, and N. Lawrence made measurements ofdrop mass. We also thank A. Bertozzi for advice andL Kadanoff for, among many other things, stimulating our interest in this subject. Supported by NSFgrants DMR-MRL 88-19860 and DMR 91-11733,and Department of Energy grant DE-FG02-92ER25119.

1 February 1994; accepted 4 April 1994

Thermophysiology of Tyrannosaurus rex:Evidence from Oxygen Isotopes

Reese E. Barrick* and William J. ShowersThe oxygen isotopic composition of vertebrate bone phosphate (8p) is related to ingestedwater and to the body temperature at which the bone forms. The 5p is in equilibrium withthe individual's body water, which is at a physiological steady state throughout the body.Therefore, intrabone temperature variation and the mean interbone temperature differences of well-preserved fossil vertebrates can be determined from the 5p variation. Valuesof 5 from a well-preserved Tyrannosaurus rex suggest that this species maintainedhomeothermy with less than 4°C of variability in body temperature. Maintenance of ho-meothermy implies a relatively high metabolic rate that is similar to that of endotherms.

Dinosaurs dominated the terrestrial landscape for 163 million years, from theirorigin in the mid-Triassic to their extinction at the end of the Cretaceous (1).Whether they were warm-blooded (ta-chymetabolic endotherms), cold-blooded(bradymetabolic ectotherms), or something in between (2) has been uncertain.A group as successful and diverse as theDinosauria may have employed a widerange of thermal physiological strategies,

Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University, Raleigh, NC27695-8208, USA.'To whom correspondence should be addressed.

222

some of which are not represented inextant terrestrial vertebrates.

Endothermy is a pattern of thermoregulation in which body temperature dependson a high and controlled rate of metabolicheat production (3). In ectothermy, bodytemperature is dependent on behaviorallyand autonomically regulated uptake of heatfrom the environment. These patterns ofthermoregulation are end members along acontinuum of physiological strategies. Inhomeothermy, cyclic body temperaturevariation, either nyctohemerally or seasonally, is maintained within ±2°C despitemuch larger variations in ambient temperature (3). Large dinosaurs must have grown

SCIENCE • VOL. 265 • 8 JULY 1994

quickly to reach sexual maturity within anecologically feasible length of time. Thehigh metabolic rates of endotherms and thehigh efficiency of energy production of ectotherms are contrasting physiological strategies for solving this problem (2). Arguments for particular strategies of thermoregulation have been based on posture, bonehistology, predator-prey relations, feedingmechanics, brain size, postulated behaviors, and paleobiogeography. Though innovative, these lines of evidence have ultimately proven to be inconclusive.

In this study, we used the oxygen isotopic composition of bone phosphate (8 ) tocalculate the body temperature variabilityof Tyrannosaurus rex. The phosphate-waterisotopic temperature scale was developedfor marine invertebrates (4) and later confirmed to be accurate for fish (5) and mammals (6). Vertebrate 8p is a function ofthe body temperature at which bone formsand of the isotopic composition of bodywater (7, 8). The isotopic composition olbody water (8bw) depends on the 8180 olwater ingested during feeding and drinkingas well as on the metabolic rate relative towater turnover rates (8, 9). The bodywater of air-breathing vertebrates is at aphysiological steady state but is not inequilibrium with environmental water.Several studies have demonstrated that 5ris linearly offset from the 8180 values ofthe local meteoric water, from the relativehumidity, or from the ambient water (inthe case of marine mammals) (10). Without knowledge of the 8180 values (11) ofbody water for each fossil species, it is notpossible to calculate actual body temperatures. However, all body water compartments in individual mammals have essentially identical 8180 values (12). Thevariations in 8 among skeletal element>in an individual should therefore reflectrelative differences in body temperature.Temperature differences between the corebody (the trunk) and the extremities canbe calculated by use of the bone isotopicdifferences (A8p) and the slope of thephosphate-water paleotemperature equation (4, 5).

An assessment of the isotopic integrity ofthe fossil bone material being examined iscrucial for paleophysiological interpretations.We analyzed a T. rex specimen from theMaastrichtian Hell Creek Formation for 5rand for the isotopic composition of both thestructural bone carbonate (8C) and the coexisting calcite cements (8CC) (13). This specimen was chosen because of its excellent preservation (14)- Two lines of evidence demonstrated that the bone phosphate analyzed waslikely to reflect the original isotopic signal 01growth. One test compared 8 and the isotopic composition of structural carbonate (8C)with the isotopic composition of the diage-