Fuid mechanics and the environment

Point Vortex Models and the Dynamics of Strong Vortices in the Atmosphere and Oceans Bubble Disconnection: Self-Similarity and Cascading Physics Implicit Multigrid Computation of Unstead

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John L Lumley (Ed.)

Fluid Mechanics

and the Environment:

Dynamical Approaches

A Collection of Research Papers

Written in Commemoration of the 60th Birthday

of Sidney Leibovich

Springer

Fluid mechanics and the environment: dynamical approaches; a

collection of research papers written in commemoration of the 60th

birthday of Sidney LeibovichIJohn L Lumley (ed.) - Berlin;

Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris

; Singapore; Tokyo: Springer, 2001

(Lecture notes in physics; 566)

(Physics and astronomy online library)

ISBN 3-540-41475-4

ISSN 0075-8450

ISBN 3-540-41475-4 Springer-Verlag Berlin Heidelberg New York

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The papers in this volume were written by his students and colleagues to honorSidney Leibovich, Samuel B Eckert Professor in the Sibley School of Mechanicaland Aerospace Engineering at Cornell University, in commemoration of his 60thbirthday, 2 April 1999 They were presented at a symposium held at Cornell, 23and 24 August 1999

Sid obtained his Bachelor of Science degree with honors from The CaliforniaInstitute of Technology in 1961, graduating first in his class He came to Cornell

to work with Geoffrey Ludford on Magnetohydrodynamics, and obtained hisPh.D in 1965 in the Department of Theoretical and Applied Mechanics Hespent a year at University College, London as a NATO Postdoctoral Fellow, andreturned to Cornell as an Assistant Professor He has been here ever since, and

is currently Director of the Sibley School

Since returning to Cornell, Sid has concentrated on rotating fluids and linear waves, in various combinations and applications, producing some 3.2 pa-pers a year with an applied-mathematical bent In particular this interest led toboth Langmuir circulation and vortex breakdown, two areas in which Sid has hadenormous influence, and both, of course, examples of rotating fluids interactingwith waves It was impossible to work in this area without being distracted bythe study of the nonlinear dispersive and dissipative waves themselves, and Sidhas made substantial contributions in this area

non-Interest in the ocean (presumably aroused by the study of Langmuir cells, aswell as by a sabbatical partly spent at Exxon) led to a study of oil-spill dispersal,which was eventually combined with the study of Langmuir cells

Although the general areas of Sid's interest have been fairly constant, thatdoes not imply that his work has been in stasis He has been delving deeperand deeper into these areas, and the nature of his interest has been evolvingalong with the field Dynamical systems theory has made its appearance (leading

to studies of 0(2) symmetry and Hopf bifurcations), as well as thermosolutalconvection and secondary instabilities The mathematical nature of the equationsthemselves has been examined The instabilities investigated have been stronglynon-linear I think it is fair to say that Sid probably knows more about non-linearevolution of disturbed rotating flows than any person alive

Sid has supervised 24 research students, who are now scattered at variousuniversities and national laboratories At least one preferred the turbulence ofthe financial markets to fluid turbulence Sid maintains extraordinarily warm

of Cambridge Monographs on Applied Mechanics and Applied Mathematics.Sid has been very active in what we may call scientific politics He wasChairman of the US National Committee on Theoretical and Applied Mechan-ics, Chairman of the Applied Mechanics Division of the American Society ofMechanical Engineers, Chairman of the Division of Fluid Dynamics of the Amer-ican Physical Society, Chairman of the National Academy of Sciences - NationalResearch Council delegation to the General Assembly of the International Union

of Theoretical and Applied Mechanics, and Chairman of the Timoshenko MedalCommittee of the American Society of Mechanical Engineers, as well as lesseroffices too numerous to mention Sid is extraordinarily smooth in committee:warm, friendly and generous, while at the same time being firm and effective

He manages to get things done without offending, a very rare talent which healso puts to good use as School Director As a result, he has been extremelyinfluential

It is hardly surprising that Sid's work has been recognized by his colleagues

In 1992 he was elected a Fellow of the American Academy of Arts and Sciences,and in 1993, a Member of the National Academy of Engineering

Sid Leibovich has been my friend since 1977, when he recruited me fromPenn State We have exercised together three time a week since then, or some

3500 times We tell each other stories, we shout at each other, we have Talmudicarguments about obscure points of science, we gossip, and we discuss politics,both university and national And we are still friends

Acknowledgment

The editor would like to take this opportunity to thank the various sources whoprovided generous support for the symposium which produced these papers: TheOffice of Naval Research (Physical Oceanography Program) and the National Sci-ence Foundation, Programs in Fluid Dynamics and Hydraulics, ComputationalMathematics and Physical Meteorology, as well as the College of Engineering ofCornell University

Point Vortex Models and the Dynamics of Strong Vortices

in the Atmosphere and Oceans

Bubble Disconnection: Self-Similarity and Cascading Physics

Implicit Multigrid Computation of Unsteady Flows

with Applications to Aeroelasticity

Second-Harmonic Resonance

with Parametric Excitation and Damping

Bubble and Temperature Fields in Langmuir Circulation

Computing Periodic Orbits

Dynamics of Layers in Geophysical Flows

Radiative Transport in Anisotropic Media

Vortex-Wake Pollution: A Problem in Fluid Mechanics

Turbulent Bursts in Couette-Taylor Flow

P.S Marcus • 183

Surface-Wave Effects on Winds and Currents

in Marine Boundary Layers

Point Vortex Models and the Dynamics of

Strong Vortices in the Atmosphere and Oceans

Hassan Aref and Mark A Stremler

Department of Theoretical and Applied Mechanics

University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Dedicated to Sidney Leibovich on his 60th birthday

The models to be considered are the simplest imaginable: The flow is assumedtwo-dimensional and inviscid, and each strong vortex is represented as a 8-function singularity of the vorticity field This simple Ansatz leads to what

is called the point vortex model It was introduced already by Helmholtz

in his seminal 1858 paper on vortex dynamics [9] (English translation byTait [21]) and has been a mainstay of fluid mechanics modeling ever since

J. G Charney extolled the virtues of this approximation in the context ofnumerical simulations of atmospheric flows He wrote [8]:

the continuous vorticity distribution in two-dimensional flow may

be approximated by a finite set of parallel rectilinear vortex filaments

of infinitesimal cross-section and finite strength, whose motion is erned by a set of ordinary differential equations This is analogous

gov-to replacing a continuous mass distribution by a set of gravitatingmass points It has the virtue that mass, energy, linear and angularmomentum continue to be conserved, and that the motions repre-sented are those of conceivable, though idealized, physical systems

It is, in a sense, the dual or complement of [the] functional sentation, the Green's function being the dual of the eigenfunction,

repre-or the 'particle' the dual of the 'wave' Which representation is themore suitable depends on the nature of the field of motion to beapproximated Fields with wave-like properties are more amenable

to functional representation, whereas those with discontinuities orvortex-like properties are more naturally represented by discrete vor-tices

In the context of oceanographic flows H M Stommel returned to the pointvortex model many times, notably in a couple oflate papers [10,11] in whichbaroclinic point vortices, called 'hetons', were considered In the early 1980's

Point Vortex Models and the Dynamics of Strong Vortices 3

What have modulated vortices to do with Christmas? This is a tion that has answers at many levels First, they are toys for grown-ups Then there is the question of belief in approximate models andSanta Claus Even chimera can instruct us

ques-At about the same time Stommel and Fredkin were exploring lated vortices, Morikawa introduced a modified form for point vortices onthe I-plane [15] The 1/1' interaction is here replaced by a Bessel function.Morikawa and Swenson later studied steady configurations of both these vor-tices and classical point vortices [16] They developed extensive catalogs ofstationary configurations and explored their stability Such steady patternshad been observed to resemble large-scale atmospheric circulations associatedwith blocking events

modu-In summary, the point vortex model has been used repeatedly to exploredynamical features of various geophysical flows, just as it has a long history

of use in modeling engineering flows, in particular the wakes behind bluffbodies The vortex wake configurations associated with the names of Foppland von Karman are particularly well known Indeed, Karman vortex streetshave been observed in satellite images to occur behind certain islands Themain reason these simple models have had such staying power is precisely, asCharney stressed [8], that point vortices yield solutions to the equations ofmotion that conserve the basic integrals while using only a small number ofdegrees of freedom Thus, point vortex motions can be seen as idealized yetsensible approximations to what real flows do

There is another area of fluid mechanics, with important repercussions forgeophysical fluid dynamics, where the point vortex approximation has played

an important role This is the subject of two-dimensional turbulence, bywhich one means the study of solutions of the Navier-Stokes equation for 2Dflow with many degrees of freedom excited Strictly speaking 2D turbulenceexists only as a computer simulation [13] Experimental realizations have beenattempted by constraining the fluid motion to be approximately 2D usingmagnetic fields, high rates of rotation or soap films Flow in the atmosphereand oceans is similarly constrained and so many features of 2D turbulentflows are seen in large-scale geophysical flows Point vortices enter this subjectbecause the flow field appears after a relatively short time to be dominated bystrong vortices that move approximately as if they were a system of mutuallyinteracting point vortices There is one important difference: Two distributedvortices of the same sense, arising in a 2D turbulence simulation, can mergeand form a single vortex, and such events take place repeatedly and w~thnon-negligible frequency Indeed, it has been suggested that these merger orrecombination events are the key element governing the self-similarity of 2Dturbulence

The overall tendency in a 2D turbulence simulation is for the vorticityfield to coarsen over time, in the sense that the number of vortices continuallydecreases via merger events and due to viscous dissipation Half a century ago

4 Hassan Aref, Mark A Stremler

Onsager, in a profound paper [17], proposed a very different explanation forthe appearance of large coherent vortices in 2D flows He based this explana-tion on Gibbsian statistical mechanics applied to the Hamiltonian dynamics

of interacting point vortices

Point vortex dynamics has a special 'flavor' since it combines fluid ics, which usually entails the solution of partial differential equations, withthe theory of dynamical systems, which is usually concerned with a smallnumber of ordinary differential equations Modern developments have shownthat very complicated things can happen already for a small number of ODEs

mechan-In the case of classical point vortices on the unbounded plane, the two-vortexproblem is integrable and very simple Furthermore, one has the interestingresult that the three-vortex problem is integrable Generally, dynamical three-body problems are non-integrable unless the system in question is integrablefor any number of particles Four point vortices, on the other hand, can dis-play chaotic motion Indeed, the transition from integrability to chaos occursalready for the problem of passive advection of a particle in the flow fieldproduced by the integrable three-vortex motion The motion of the advectedparticle can be chaotic This insight led to the concept of chaotic advection,which has since emerged as a general paradigm of flow kinematics and fluidmixing, and has been actively pursued in the context of very viscous flows.Here again there are important connections to geophysical fluid dynamics.Thus, Zimmerman [24] and collaborators have emphasized that the mixing

in shallow tidal basins, such as the Wadden sea, is probably more ately described by chaotic advection than by turbulent transport models

The problem of three interacting point vortices on the unbounded plane plays

a pivotal role, not just as the largest integrable system but as the smallestsystem one must consider if new scales of motion are to be generated dynam-

ically The rate of change of the distance between any two vortices in an

N-vortex system is a sum of contributions over all 'third' vortices, where eachsuch vortex contributes a term proportional to its circulation, and dependent

on the distances between the three vortices in question In particular, if oneattempts to write down the kinetics of a point vortex 'gas', as did Marmanis

in his MS thesis, later published as [14], the basic interaction to be treated

is the three-vortex interaction In the usual kinetics of an ideal gas it is, ofcourse, the two-body interaction that enters and plays the main role.Much of the early work of one of us (HA) on the three-vortex problem onthe unbounded plane was completed while a student at Cornell Sid Leibovichplayed an important role as a mentor, among other things guiding the paper[1J to its final publication venue In view of this and in view of the generalimportance of the problem, it seems appropriate to describe progress withthe three-vortex problem since 1980 In particular, during the past couple

Point Vortex Models and the Dynamics of Strong Vortices 5

of years the present authors have achieved considerable progress with theintegrable three-vortex problem in a periodic domain

2.1 Vortices on the Unbounded Plane

Around 1985 Nicholas Rott contacted HA inquiring about the fate of WalterGrobli, the young Swiss applied mathematician who in his thesis of 1878 hadsolved the three-vortex problem on the unbounded plane Rott recalled histeacher Ackeret speaking about this work in lectures many years before Hewas intrigued by two things: First, the historical question of what became

of Grobli and, second, the case of three vortices with sum of strengths zero,which seemed to be a bit of a 'stepchild' in Grobli's treatment and in the laterpapers on this problem by Synge [20] and Aref [1] Rott's historical quest led

to the review paper [3] published as the historical lead-in to volume 24 of

Annual Reviews of Fluid Mechanics. The technical quest led to a paper byRott [18] with an appendix by Aref [2], both dedicated to Hans Thomann onthe occasion of his 60th birthday

We briefly recall the Rott-Aref solution for three vortices on the

unbound-ed plane with net circulation zero One first shows, by subtraction of theequation of motion for one of the vortices from the equation of motion foranother vortex, that the separation vector between two vortices is the solution

of an advection problem for a fictitious particle in the field of three stationaryvortices, the circulation and locations of which follow from the original prob-lem This simple advection problem is integrable Once it is solved, individualtrajectories of the three original vortices can be found from the solution byperforming one additional quadrature The nature of the solution space ofthe original three-vortex problem is clearly displayed by the various 'regimes

of motion' found in the advection problem

Figure la illustrates the four different regimes that arise in the tion problem for the case when the circulations in the original three-vortexproblem are in the ratios rl : r2 : r3 = 2 : 1 : (-3). Only the relativemagnitudes of the circulations really matter - the absolute values simply setthe time-scale of the motion Given the ratio of circulations in the originalthree-vortex problem, rl :r2 :r3, the circulations of the three fixed vortices

advec-in the advection problem are 1/r3 : 1/r2 : 1/rl.Thus, while the original culations sum to zero, the circulations of the three advecting vortices do not.The first advecting vortex is at the origin The second is at -(Q + iP)/ r2,

cir-where Q +iP is the linear impulse of the original three-vortex system (Here

we are using the common convention that the flow plane is the complex planeand the cartesian coordinates of the vortices are concatenated into complexpositions with the abscissa taken as the real part and the ordinate as theimaginary part.) The third advecting vortex is at (Q +iP) /rl.The advectedparticle path in each of the four regimes of the streamline pattern is, clearly,

a rather simple closed curve This implies that the relative motion of thethree vortices is periodic Figure Ib illustrates a sample motion from each

Fig.3 Sample vortex trajectories corresponcling to regimes I, IV, VI, VIII, and X

in Fig 2 'Base' vortices are shown by solid circles and their trajectories by heavylines; perioclic images are open circles and their trajectories are light lines Initialpositions are labelled 1,2,3 and final positions 1',2',3' Final positions ofperioclicimages are labelled 2",3"

problem in Fig 2 using one additional quadrature Note the complexity ofsome of the motions, in particular how individual vortices propagate overmany strips widths before re-assembling into the original triangle In regimes

I, II, IX, X and XI the re-assembly of the original triangle occurs via periodicimages of the original vortices from neighboring strips

The linear impulse plays the role of a bifurcation parameter in this lem As it changes, the nature of the connections between saddle points inthe advection problem changes and there may be fewer regimes of motion.However, there always exist regimes of motion resembling I and II Theseregimes have the very interesting property that one of the original trio ofvortices heads off in a different direction than the other two

prob-This problem is not just of mathematical interest A situation that callsfor a model with three vortices in a periodic strip, and where it is believedthat the three vortices have total circulation zero, is the wake of an oscil-lating cylinder This problem was studied extensively by Williamson andRoshko [23], among others The wakes of interest are the ones that they labelP+S (for pair plus singlet) The feature of regimes I and II in Fig 2 just men-tioned above is readily observed in their experiments (d. [4]) The richness

14 Hassan Aref, Mark A Stremler

3 Braids and Topology of Stirring

One further development is worth mentioning The qualitative tion of 'regimes of motion' used above, where a 'regime' was taken to be anapparent area in the advection problem streamline plot, can actually be mademuch more precise [7] If one augments the plane of motion with a time axisperpendicular to it, thereby creating a (2+I)-dimensional 'space-time', the'world lines' of the three vortices in this space-time will trace out what iscalled a physical braid consisting of a 'strand' for each vortex This physicalbraid can be formalized into a mathematical braid by adopting certain con-ventions for treating the apparent crossings of strands in the physical braid.Figure 9 illustrates the procedure starting from a case of three-vortex motion

characteriza-on the unbounded plane, ccharacteriza-onstructing the physical braid and then formalizingthis into the mathematical braid

For periodic boundary conditions the construction is somewhat more plicated The periodic strip case is handled by mapping the strip conform allyonto a disk with the origin removed Then, in the construction of the physicalbraid, one needs to also track an additional point corresponding to the origin.Figure 10 illustrates how this is done for a simple case

com-The point of all this is that a precise characterization of regimes of motion

is possible in terms of braids and their symbolic representation However,there are deeper insights It is possible to characterize the nature of theadvection by the original three vortices on the basis of the nature of the braidproduced by their trajectories The Thurston-Nielsen classification theorem[22] tells us that there are three possibilities, called the 'finite order', the'pseudo-Anosov' (or pA), and the 'reducible' cases The most intriguing isthe pA case because it assures us that within the flow domain there will be

Fig 10 Construction of a braid for vortex motion in a periodic strip (a) ries of three vortices from regime II in Fig.2 (b) Conformal mapping of the vortexmotion to the singly-periodic plane 'Vortex' 4 shows the location of the origin (c)Physical braid of the transformed motion (d) The resulting mathematical braidfor this motion

Trajecto-a region in which the Trajecto-advection of Trajecto-a pTrajecto-assive scTrajecto-alTrajecto-ar will be like thTrajecto-at of the'cat map' (except for some isolated singularities) This kind of advection mustlead to the most efficient mixing one can imagine, since the passive scalar will

be exponentially stretched with the same stretching rate everywhere withinthat region This is a topological result, so the extent of the region where thispA-type stirring takes place cannot be predicted However, the stretching ratecan be calculated from the braid

The nature of the entities orbiting one another and thus stirring the fluid

is of little consequence The same topological considerations apply if we ually move about three stirring sticks in a vat of viscous liquid as if we letour three mutually interacting point vortices stir up a domain of ideal fluid.The former option is much more readily realized in the laboratory (!) and

man-we have performed such experiments with the expected results The rathermathematical developments and the simple but visually convincing outcomeare discussed in a forthcoming paper [6] The main lesson of importance here

is that three vortices in a periodic domain can produce stirring that is asefficient as one can imagine Adding further vortices will not, in this sense,produce more intense stirring (although the region over which pA stirringtakes place could well be larger) The conclusion is that, in certain parameterregimes, integrable three-vortex motion in a periodic strip or parallelogram

16 Hassan Aref, Mark A Stremler

produces stirring of a passive scalar that is as intense (in a quantifiable sense)

as any 2D turbulent flow

We thank P L Boyland for his crucial involvement in the developmentdiscussed in §3 and for constructing the mathematical braids shown in Figs 9and 10 This work was supported by NSF grant CTS-9311545 and an ONRFellowship

References

1 Aref, H (1979) Motion of three vortices Phys Fluids 22, 393-400

2 Aref, H (1989) Three-vortex motion with zero total circulation: Addendum

J Appl Math Phys (ZAMP) 40, 495-500

3 Aref, H., Rott, N & Thomann, H (1992) Grobli's solution of the three-vortexproblem Ann Rev Fluid Mech 24, 1-20

4 Aref, H & Stremler, M A (1996) On the motion of three point vortices in aperiodic strip J Fluid Mech 314, 1-25

5 Aref, H & Stremler, M A (1999) Dynamics of vortex lattices Bull Am Phys.SoC 44, 174

6 Boyland, P L., Aref, H., & Stremler, M A (2000) Topological fluid mechanics

of stirring J Fluid Mech 403, 277-304

7 Boyland, P L., Stremler, M A., & Aref, H (1999) Topological fluid mechanics

of point vortex motions Physica D (submitted)

8 Charney, J G (1963) Numerical experiments in atmospheric hydrodynamics.In: Experimental Arithmetic, High Speed Computing and Mathematics, Pro-ceedings of Symposia in Applied Mathematics 15 Am Math Soc., Providence,R.I., 289-310

9 Helmholtz, H von (1858) Uber Integrale der hydrodynamischen Gleichungenwelche den wirbelbewegungen entsprechen Journal fiir die reine und ange-wandte Mathematik 55, 25-55

10 Hogg, N G & Stommel, H M (1985) The heton, an elementary interactionbetween discrete baroclinic geostrophic vortices and its implications concerningeddy heat-flow Proc Roy Soc (London) A 397, 1-20

11 Hogg, N G & Stommel, H M (1985) Hetonic explosions: The breakup andspread of warm pools as explained by baroclinic point vortices J Atmos Sci

(Lon-15 Morikawa, G K (1960) Geostrophic vortex motion J Meteor 17,148-158

16 Morikawa, G K & Swenson, E V (1971) Interacting motion of rectilineargeostrophic vortices Phys Fluids 14, 1058-1073

17 Onsager, L (1949) Statistical hydrodynamics Nuovo Cimento 6 (Suppl.), 287

279-Point Vortex Models and the Dynamics of Strong Vortices 17

18 Rott, N (1989) Three-vortex motion with zero total circulation J.Appl Math

Phys (ZAMP) 40, 473-494

19 Stremler, M A & Aref, H (1999) Motion of three point vortices in a periodicparallelogram J Fluid Mech 392, 101-128

20 Synge, J 1.(1949) On the motion of three vortices Can J.Math 1,257-270

21 Tait, P G (1867) On integrals of the hydrodynamical equations, which expressvortex-motion Phil Mag 33, 485-512

22 Thurston, W (1988) On the geometry and dynamics of diffeomorphisms ofsurfaces Bull Amer Math Soc 19,417-431

23 Williamson, C H K & Roshko, A (1988) Vortex formation in the wake of anoscillating cylinder J Fluids Struct 2, 355-381

24 Zimmerman, J T F (1986) The tidal whirlpool: A review of horizontal persion by tidal and residual currents Netherlands J.Sea Res 20, 133-154

dis-Bubble Disconnection:

Self-Similarity and Cascading Physics

Olu§ N Boratav1, Yi-Ju Chen2, and Paul H Steen1,3

1 School of Chemical Engineering, Cornell University, Ithaca, NY 14853

2 Department of Engineering Sciences and Applied Mathematics

Northwestern University, Evanston, IL 60208

3 Center for Applied Mathematics, Cornell University, Ithaca, NY 14853

Abstract Surface tension disconnects a capillary bubble from a capillary bridge whiledriving a surrounding inviscid flow Spatial and temporal behavior is studied just prior

to and just after disconnection, via computation and experiment They are in ment both before and after the event but anticipated self-similarity is observed onlybeforehand Computation coincides with observation due to modeling that identifiesthe cascade of physics probed as decades of length scales are traversed

Physics The nonequilibrium trajectory that carries the system from state A to

state B, by its very nature, probes many decades oflength (and time) scales.The cylinder may start on a centimeter scale (diameter) while the ultimatedistribution of material must occur on a molecular scale Molecular bondsare broken in the disconnection event As length scales cascade toward themolecular, there is a corresponding cascade of relevant physics More thanfour decades in time and three decades in lengths are probed below

Mechanics In the continuum description, which we shall restrict to out, topological change necessarily involves a double-valued mapping fromthe material to spatial configuration That is, at the disconnection instant,there is at least one material point that gets mll-pped into two spatial points

through-By this pathology, the bounding surface does" not remain a material face and Lagrangian trajectories (material points) are not unique (Dussan1976) Disconnection is a limiting case of fracture If one cannot understand'breaking' in the simplest of contexts, there is little hope for solving moretechnically relevant problems In bubble disconnection, the interface is shown

sur-to turn over just before disconnection and velocities blow-up at the singularinstant; the disconnected pieces possess different accelerations afterwards

20 Boratav et al.

Mathematics. A partial differential system provides a model for the dynamics

of topological change The capillary surface is modeled as a mathematicalsurface The surrounding flow (on either side of the surface), taken to beinviscid and irrotational, is governed by Laplace's equation for the velocitypotential while the pressure difference across the bounding interface is re-lated to the mean curvature by the Young-Laplace equation Disconnectionoccurs as finite-time blow up of this system Various questions are raised Is

there a graceful passage from pre- to the post-singular behavior, for ple? We report asymmetric behavior about the singularity; beforehand there

exam-is self-similarity while, for comparable times afterwards, each disconnectedcomponent follows a different scaling law The issue of jump conditions orgraceful passage remains open

We will focus on a simple analog of the cylindrical interface prototype Inthe laboratory, an axisymmetric soap-film bridge can be made by pinning itstwo ends at coaxial circular contact rings Since the rings are open, there iscommunication between 'inside' and 'outside' Indeed, in the absence of motion,the pressure drop across the surface is everywhere zero Such a soap-film isknown to be well-modeled as a minimal surface, called the catenoid and firststudied by Euler (1744) This equilibrium surface is generated by revolving acatenary curve around its axis It is the only minimal surface that is a surface ofrevolution (Kreyzig 1959) State A corresponds to the neutrally stable catenoidthat occurs at a critical separation of the contact rings, beyond which thereexists no connected solution State B corresponds to a disconnected state Itconsists of a bubble and of flat planar films spanning the circles of contact.The details of droplet formation are important in applications such as com-bustion and ink-jet printing while bubble disconnection can be important in theentrainment of gases into liquids at breaking interfaces (e.g foams and froths).Below, after some estimates about scaling in the space-time neighborhood ofthe singularity, the physics of the soap-film experiment is outlined, the govern-ing equations are then summarized, simulations before and after pinch-off arereported (in synopsis), after which we return briefly to the physics with a com-parison of simulation and experiment We close with some concluding remarks

2 Disconnection: spatial & temporal scalings

3 Soap-film physics

Various means have been employed to break a fluid mass in the laboratory,including the gravity-driven droplet and a droplet placed in an extensional flow.For the dripping tap, gravity and surface tension act on a pendant drop to pull itfrom its mother The motion is resisted by liquid inertia and viscosity For water,inertia dominates viscosity on lengths greater than 1O-5mm On the other hand,surface tension dominates gravity for lengths smaller than about a millimeter.Hence, if the goal is a balance between surface tension and inertia, there is awindow of about 3 or 4 decades of lengthscale available For the extensional flow,gravity is unimportant for two liquids of nearly the same density, but viscousshear stress pulls the mass apart In both cases, an external force drives thebreak up process

In contrast, for the soap-film bridge, the break up is initiated at a neutrallystable equilibrium by disturbances in the ambient Surface tension drives theprocess from its beginning Gravity is never important for thin enough films andair viscosity enters first on lengths on order of lO-6mm (The film cannot be

so thin to be susceptible to 'black-film' rupture, though.) Hence, the soap-filmbridge offers advantages of 'self-initiated' break up and the unimportance ofgravity and viscosity

Oleic acid is the surfactant that stabilizes water in the soap-film Glycerin isadded to enhance the viscosity Film thickness IS has been independently mea-sured to be on the scale of microns The bridge is initially established by bringingthe circular contacts (equal radius) close together The bridge length is then in-creased slowly with a quasistatic approach to the neutrally stable state (state

A) The critical length occurs at 1.325 times the contact radius Ambient

distur-bances destabilize and capillarity drives the surrounding air as the film collapses

on itself leading eventually to disconnection Surface tension and air inertia vide the primary balance However, as length scales decrease other effects com-pete with air inertia to influence the dynamics

pro-5 Simulations

Chen and Steen (1997) have identified different scaling regimes in the solution

of equations 13 and 15 up to disconnection Since, we have completed new ulations with greater overall resolution and better~resolved small scales Table

sim-2 summarizes selected work on inviscid collapse problems which use boundaryintegral techniques We start with a low resolution (N = 81) and double it assharp gradients in curvature and velocity are formed during the evolution Thesurface elements are redistributed at every step in such a way that the spacingbetween two elements is inversely related to a local quantity such as velocity or

Bubble disconnection 27

on (6) above but can, nevertheless, be easily understood The balance (6) doesnot recognize gradients of curvatures as scaling differently from ratios of length

scales It turns out that the gradient of the radial curvature can dominate that

of the axial curvature even though a cusp-like scaling holds Indeed, this occursand accounts for the observed scaling (Boratav & Steen 2000) Earlier regimeshave also been accounted for by simple balances (Chen &Steen, 1997) In sum-mary, figure 3 shows that as the interface approaches turnover, first the radialcurvature gradient and then the axial curvature gradient dominates Turnoveritself allows the axial curvature to grow unbounded in unison with the radialcurvature

30 Boratav et al.

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