Iterated stretching of viscoelastic jets

PHYSICS OF FLUIDS VOLUME 11, NUMBER 7 JULY 1999

Iterated stretching of viscoelastic jetsHsueh-Chia Chang,a) Evgeny A. Demekhin, and Evgeny KalaidinDepartment of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556

~Received 4 February 1998; accepted 23 March 1999!

We examine, with asymptotic analysis and numerical simulation, the iterated stretching dynamics ofFENE and Oldroyd-B jets of initial radiusr 0 , shear viscosityn, Weissenberg numberWe,retardation numberS, and capillary numberCa. The usual Rayleigh instability stretches the localuniaxial extensional flow region near a minimum in jet radius into a primary filament of radius@Ca(12S)/We#1/2r 0 between two beads. The strain-rate within the filament remains constant whileits radius~elastic stress! decreases~increases! exponentially in time with a long elastic relaxationtime 3We(r 0

2/n). Instabilities convected from the bead relieve the tension at the necks during thisslow elastic drainage and trigger a filament recoil. Secondary filaments then form at the necks fromthe resulting stretching. This iterated stretching is predicted to occur successively to generatehigh-generation filaments of radiusr n , (r n /r 0)5&(r n21 /r 0)3/2 until finite-extensibility effects setin. © 1999 American Institute of Physics.@S1070-6631~99!01307-0#

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I. INTRODUCTION

There has been considerable recent progress in ourderstanding of Newtonian jet dynamics. Numerical simution can now significantly extend the classical linear Raleigh theory for the initial small-amplitude evolution1

However, singular stresses that occur as the jet radiusproaches zero have prevented accurate numerical resolof the final breakup dynamics. Instead, recent mathemaanalysis of the self-similar, finite-time singularity formationear breakup has provided significant insight,2–7 including aninteresting study of observed iterated jet pinching leadingbreakup.5 Universal scalings of the near-breakup evolutiare now well understood, eventhough the longwave apprmation invoked in the theory may prevent it from resolvithe dynamics at or beyond breakup when drops beginform. The hope is that one can ‘‘patch’’ the breakup analyfor the numerically inaccessible interval to numerical simlation of the evolution prior and beyond breakup. Since thare only a few parameters in the governing equations, deeation by numerical simulation can be readily carriedaway from the breakup stage.

Such a luxury is lost in another classical jet breakproblem—evolution of non-Newtonian jets. In additionthe usual capillary forces that drive the breakup, viscoeticity effects introduced by polymers are known to signicantly alter the breakup dynamics. However, viscoelastinot only introduces additional rheological parametersalso renders the equations hyperbolic. Both factors exclexhaustive numerical analysis even with modern-day coputers. In any case, the myriad of physical effects introduby the polymers can probably be best elucidated withanalysis that can isolate each effect.

Linear stability analysis that amounts to an extensionthe classical Rayleigh theory can be readily carried out

a!Electronic mail: [email protected]

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viscoelastic jets. However, since viscoelastic effects can obe triggered when the polymers are significantly stretchedthe flow, viscoelasticity is not expected to be of significaninitially when the flow within the unperturbed jet of radiusr 0

is either zero or a uniform axial flow. Prior linear theories8,9

indeed confirm that viscoelasticity does not alter the classRayleigh wavelength 2&pr 0 significantly and only slightlyincreases the growth rate.

However, as uniaxial extensional disturbance flowscreated by the initial disturbance, the polymers are stretcconsiderably at the stagnation points and the late-stagenamics are profoundly affected by viscoelasticitExperiments8 show that the breakup is delayed by ordersmagnitude. In some cases, the viscoelastic jet may not ebreak up over the entire duration of the experiment. Instof pinching asymmetrically about the pinch point likeNewtonian jet to form satellites, a unique filament-bead cfiguration is observed. This configuration is extremely roband the drainage from the stretched filament to the copressed beads is extremely slow. If the viscoelastic jet dbreak, it breaks at the necks joining the filament to the beaThis bead-filament configuration has also been observenumerical simulation by Bousfield etc.10 for an Oldroyd-Bfluid. Due to the slow drainage from the filament, the simlation is unable to proceed beyond the bead-filament cfiguration and determine the final fate of the jet.

Instead, a number of theoretical analyses have focuon the breakup dynamics of slender filaments.11–13 Thesetheories11,12 have uncovered the exponential drainage dnamics of an elastic filament. This drainage is driven bycapillary pressure difference between the bead and filamA more detailed force and mass balance across this neckbe offered here but the scalings of earlier elastic draintheories remain valid. Because the radii of both beadfilament vary very slowly, the constant capillary drivinforce approximation is valid quasi-steadily. The reason bradii vary slowly, on the other hand, is because the ela

7 © 1999 American Institute of Physics

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1718 Phys. Fluids, Vol. 11, No. 7, July 1999 Chang, Demekhin, and Kalaidin

axial stress, created by the stretched polymers duringment drainage, exactly cancels the slowly varying capillpressure. As a result, a linear uniaxial extensional flow exwithin the filament with a constant strain-rate. Due to tdrainage, the filament radius decreases and the axial sincreases but the strain-rate remains constant. This undrainage mechanism yields a distinctive exponentialcrease in time for the filament radius with a large elastic tiscale. The exponential thinning implies that an Oldroydjet, in contrast to the Newtonian case, does not breakufinite time. It is only when finite extensibility in a FENEmodel is introduced that finite-time breakup is predicted.

However, these analyses omit inertial effects and foonly on slender filaments. Since the Newtonian self-simbreakup solution of Eggers3 involves inertia, it is not clearthat its omission is valid in late-stage filament dynamics wfast axial flow. More importantly, experimental data fNewtonian jets5 and non-Newtonian jets8 clearly show thatmuch of the late-stage jet dynamics, including breakup,cur at the neck joining the filament to the bead. For examiterated pinching has been observed in Newtonian jets5 at thenecks. Such dynamics escape the analyses of Renardy11 andEntov and Hinch12 for slender filaments without inertia. Important dynamics at the neck of the jets have hence escour understanding thus far. In this report, we endeavodelineate both the formation mechanism for the befilament configuration and the dynamics at the necks.shall examine both an Oldroyd-B jet and a FENE jet areveal an interesting recoil and iterated stretching dynam

II. LONGWAVE SIMPLIFICATION AND SIMULATION

We use the FENE-CR model of Chilcott and Rallison14

a simplification of the classical FENE dumbbell model,15 todetermine the stress tensor

t5msg1G f~R!~A2I !, ~1!

whereR25traceA. The spring force law with

f ~R!51

12R2/L2 , ~2!

represents finite extensibility withL as the ratio of the lengtha fully extended dumbell to its equilibriu length andA beingthe ensemble average of the dyadic product of the end-tovector of the dumbbell, normalized by the equilibrium sepration. The matrixA is taken to evolve by

]A

]t1u•¹A5A•¹u1¹uT

•A2f ~R!

D~A2I !. ~3!

The parametersms , G, and D represent solvent viscosityelastic modulus, and relaxation timeD, respectively. Themagnitude of non-Newtonian stresses is measured bc5GD/ms such that the steady shear viscosityn5(11c)ms /r. The tensorg5¹u1(¹u)T is the rate-of-straintensor.

The appropriate boundary conditions are the normaltangential balances at the jet interface defined byr 5h(z).There is also the kinematic condition for mass conservat


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in

sr

c-e,

edo-eds.

nd-

d

n

]h2

]t1

]

]z E0

h

2rudr50. ~4!

In the longwave limit whenh(z) varies slowly with re-spect toz, the axial velocity, pressure, and the stress comnentstzz andt rr are almost uniform with respect tor whilethe radial velocityy and the off-diagonal stress component rz and tzr are nearly zero. Hence, the proper ansatzslender jets is a Taylor expansion inr

u;u01u2r 2¯ , ~5a!

v;21

2

]u0

]zr 2

1

4

]u2

]zr 3¯ , ~5b!

p;p01p2r 21¯ , ~5c!

tzz;tzz0 1¯ , ~5d!

t rz5tzr;Tr1¯ , ~5e!

t rr ;t rr0 1¯ , ~5f!

Azz;Azz0 , ~5g!

Arz5Azr;Arz0 r , ~5h!

Arr ;Arr0 , ~5i!

where all the coefficients of expansion are only function otandz.

Upon substituting this ansatz into the equations of mtion and boundary conditions, nondimensionalizing with tinitial undisturbed radiusr 0 as the characteristic lengthr 0

2/nas the characteristic time, wheren5ms(11c)/r is the shearviscosity due to both solvent and polymer, andn/r 0 as thecharacteristic velocity, one gets to leading order inr, withuniform pressure and axial flow and negligible off-diagonstresses, the following dimensionless longwave equation

]u

]t1u

]u

]z5

1

Ca

]k

]z1

1

h2

]

]z@h2~tzz2t rr !#, ~6a!

]h2

]t1

]

]z~h2u!50, ~6b!

]A

]t1

]

]z~uA!23A

]u

]z1

f ~R!

We~A21!50, ~6c!

]B

]t1

]

]z~uB!1

f ~R!

We~B21!50, ~6d!

tzz52S]u

]z1

12S

Wef ~R!~A21!, ~6e!

t rr 52S]u

]z1

12S

Wef ~R!~B21!, ~6f!

whereu denotesu0 , k the jet curvature, the radially uniformaxial velocity, A and B represent the polymer stretchingthe axial and radial directions,Azz

0 andArr0 , respectively, and

tzz and t rr the dimensionless versions of their counterpa


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1719Phys. Fluids, Vol. 11, No. 7, July 1999 Iterated stretching of viscolelastic jets

in Eq. ~5! with a superscript 0. All these quantities are funtions of the dimensionlessz and t only. The spring law Eq.~2! now becomes

f ~R!5L2

L22~A1B!, ~7!

and the other parameters are the usual capillary, Weissenand retardation numbersCa5rn2/sr 0 , We5Dn/r 0

2 and S51/(11c).

The parameterWe measures the elasticity of the polymers related to the relaxation timeD. We are interested inthe strongly elastic limit withWe@1. The retardation parameter S, on the other hand, is associated with the ratioretardation time scale due to non-Newtonian stress torelaxation time scaleD. It is bounded between zero~New-tonian limit! and unity. The capillary number is also a unorder parameter relative toWe. We shall be exploiting thesmallness ofWe21 in subsequent asymptotic analyses. Textensibility parameterL, on the other hand, can range frounit order toO(We), depending on the molecular weight,16

with L→` being the Oldroyd-B limit.To render the hyperbolicity of the stress constituti

equations more apparent, it is convenient to separatepolymer elastic stress from the quasi-viscous retardastress by defining the excess stresses

tzz5tzz22S]u

]zand t rr 5t rr 1S

]u

]z, ~8!

to remove the velocity derivative in time in the stress eqtions that result when Eqs.~6c!–~6f! are combined. The resulting equations are

]u

]t1u

]u

]z5

1

Ca

]k

]z1

1

h2

]

]z@h2~ tzz2 t rr !#

13S

h2

]

]z S h2]u

]z D , ~9a!

]h2

]t1u

]h2

]z1

]u

]zh250, ~9b!

]A

]t1u

]A

]z22A

]u

]z1

f ~R!

We~A21!50, ~9c!

]B

]t1u

]B

]z1B

]u

]z1

f ~R!

We~B21!50, ~9d!

tzz512S

Wef ~R!~A21!, ~9e!

t rr 512S

Wef ~R!~B21!. ~9f!

The inertial terms lie to the left of the equation of motio@Eq. ~9a!# and they are balanced by the capillary pressgradient, the gradient of the normal stress difference andpolymer retardation stress terms on the right. The constive equations@Eqs.~9c!–~9e!#, capture the convection of th


-

erg

fe

e

hen

-

eheu-

stresses along the streamline, the stretching due to the veity gradient (]u/]z), finite extensibility in f (R) and the re-laxation of the stretched polymers.

Several limits of Eq.~9! can be readily derived. TheextensibilityL is practically infinite when (A1B)!L2 in Eq.~7!. In this limit, Eqs.~9e! and~9f! yield the Hookean springlaws

A511 tzz

We

~12S!and B511 t rr

We

~12S!, ~10!

and, upon substitution into Eqs.~9c! and~9d!, the stress evo-lution of an Oldroyd- B fluid results

]

]ttzz1u

]

]ztzz22tzz

]u

]z1

1

WeH t rr 12~S21!]u

]zJ 50,

~11a!

]

]tt rr 1u

]

]zt rr 1 t rr

]u

]z1

1

WeH t rr 2~S21!]u

]zJ 50.

~11b!

The Oldroyd-B limit is hence not a singular limit.If one further neglects elastic and retardation effec

We50 and S50, a Newtonian limit is obtained withtzz

52(2]u/]z) and t rr 52(]u/]z);22(]v/]r ).It is far simpler to integrate the longwave equation E

~9! or Eq. ~11! than the full equations of motion. Howevestrictly speaking, the longwave equation is only valid ffilaments whose radii vary gradually. This is not true at tobserved beads which are spherical. Nevertheless, the spcal beads should obey the axisymmetric Laplace–Youequation with constant curvature to leading order. Hencewe retain the full curvature in Eq.~9a!

k5hzz

~11hz2!3/22

1

h~11hz2!1/2, ~12!

the spherical beads would also be captured to leading oby Eq. ~9!. We have successfully applied this composite aproach to capture both the bead and annular film during dformation when a vertical fiber is coated17 and to captureboth the finger tip and the thin wetting films in the Bretheton problem of air fingers replacing liquid in capillaries anchannels.18 It is nevertheless anad hocapproach that is onlyvalid to leading order. It must be verified against numerisimulation of the full equations to examine if there is adiscrepancy due to higher order effects.

To this end, we compare in Fig. 1 our computed profifrom Eq. ~11! for the Oldroyd-B fluid (L→`) at Ca510,We5300, andS50.25 in a domain of sizel 520 to thecomputation of the full equations by Bousfieldet al.10 Due toa different scaling, their dimensionless timeu corresponds tot/Ca and their length corresponds toz/ l of the present nota-tion. The results are presented inu andz/ l . As is evident, theevolution is faithfully captured by the longwave equatioeven after the bead-filament configuration is established.simulation of the Newtonian jet (We50) is also in agree-ment with earlier simulations by Eggers,2,3 Papageorgiou,4

and Brenneret al.6

As is consistent with the experiments, the longer simlations allowed by the longwave simplification reveal impo


d

d


FIG. 1. Simulation of the jet radiush(z,t) of the Oldroyd-B jet from thelong-wave equation on the right anfrom the full equations of motion onthe left by Bousfieldet al. ~Ref. 10!.The parameters areCa510, S50.25,We5300, and a domain size ofl520. The graphs are plotted in anaxial scale ofz/ l and a time scale oft/Ca.

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tant jet dynamics at the necks joining the beads to thement. Such late-stage dynamics develop long afterformation of the bead-filament configuration and is missby earlier numerical studies. An extreme case ofWe510 000 is shown in Fig. 2. When the retardation numbeSis not near its two limits of zero and unity, a distinctive recof the filament develops at the necks. The simulated evtion begins with the formation of a minimum in the jet radidue to the usual Rayleigh capillary instability. This createstagnation point at the minimum and an uniaxial extensioflow near it. The extensional flow stretches the polymersgenerates elastic stresses of positivetzz and negativet rr .The profiles oftzz during the evolution are seen in Fig.This axial elastic stress develops a symmetric maximumthe first stagnation point. As the jet profile near this pointstretched into a filament bounded by two beads att56.5, thestress profile evolves into a constant value within the fi

FIG. 2. Evolution of a highly elastic Oldroyd-B jet from the Rayleigh istability, to the formation of a filament by stretching and to the beginningrecoil at the necks of the draining filament. The nodes during the Raylinstability, which bound the jet interval that is stretched into a filament,marked. (We510,000,S50.25,Ca510, andl 54p).


-ed

lu-

aald

ats

-

ment. As pinching begins symmetrically at the two necnear t57.0, two additional uniaxial extensional flows acreated locally at the necks and the stress again exhibitssharp maxima. The excess axial stress plays an imporrole in the recoil process.

The recoil that follows the pinching is shown in Figs.and 5 for a different Oldroyd-B jet. It is evident that seconary filaments are created at the necks by the stretchingfollows the recoil of the primary filament. The bead is unafected during the recoil and the secondary filament joins ia neck that is quite similar to the neck of the primary filment. However, the secondary filament is much thinner tthe primary one and, as shown in Fig. 4~b!, has a much largerelastic stress. The simulated elastic stress evolution shthat the stress actually drops at the primary neck befforming a sharp maximum due to the stretching that creathe secondary filament. This suggests the recoil of the

fheFIG. 3. The built-up of the axial elastic stresstzz in the stretched filament ofFig. 2. The elastic stress is constant within the straight filament until reat the necks triggers two sharp maxima.


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mary filament is triggered by a relief of the tension at tneck. The fully formed secondary filament, in the preseof the bead and the primary filament, is shown in Fig. 5.

We are unable to numerically track the jet dynamicster the formation of the secondary filament. However, sinits neck with the bead is quite similar to that of the primaone, we expect another recoil to initiate there. Iterated reand stretching dynamics can then proceed indefinitely atnecks of Oldroyd-B jets. In our subsequent analysis, we sdevelop a theory for Oldroyd-B filaments and show that thsimilarity allows us to relate their radii and elastic stress.a result, with proper scalings ofWe, Ca, andS, the evolutionand recoil of the primary filament can be used to dedthose of higher-generation filaments. We shall also demstrate preliminary experimental evidence of this self-simiterated stretching dynamics.

The evolution of the Oldroyd-B jet radius, the axistresstzz and the velocityumax at the neck of the first fila-ment are shown in Fig. 6. There are two distinct slow sta

FIG. 4. Blow ups of the Oldroyd-B interface recoil and elastic stress elution at one of the necks. For clarity, snapshots at different time, measfrom the onset of pinching at the bottom figure, are taken in the two plNote that the elastic stress is first relieved at the neck before the smaximum develops due to secondary stretching. (We5300, S50.25, Ca510, andl 54p).


e

-e

ilell

irs

en-r

s

prior to the first pinch recoil that define the lifetime of thprimary filament. The jet radius drops and the axial strrises precipitously during the stretching stage neart5100 toform the constant-radius filament. The profiles shown in F7 indicate the transformation to an axisymmetric filamewith a constant stress and a linear axial velocity profile treaches6umax at the necks. However, this stretching staends abruptly asumax approaches zero and both filament rdius and its stress reach constant values. An even sloelastic drainage then takes over after a short transient ft.100. The radius continues to decrease and the stresstinues to increase within the filament after this short hestion, but at distinctly slower rates than the stretching intervThe maximum axial velocity at the necks, however, remaconstant during this long interval. Due to the linear uniaxflow, this implies the strain rate in the filament remains costant during this interval.

In Fig. 8, the evolution of jet radius at the first neckshown for a large range ofWe and Ca for an Oldroyd-Bfluid. The stretching, drainage and recoil stages show apciable sensitivity to these values.

We examine the dynamics of the FENE jet in Fig. 9 a

-eds.rp

FIG. 5. The entire jet profiles before and after the recoil of the Oldroydjet in Fig. 4. A secondary filament is clearly visible.


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a

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, ititea

here-

re-enen

tsta-sbenga-d.

aryare

fila-s ofh-ro-

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jetbed, itsitialn-anim-or-

e


function of extensibilityL. The formation dynamics of theprimary filament and the subsequent elastic drainage dynics are insensitive toL for L in excess of 10. This suggesthat the stretchingA1B, is much smaller thenL2 in both theinitial jet and the primary filament under such conditionHowever, the recoil dynamics in Figs. 3 and 4 suggestthe secondary filament formed after the recoil will havemuch higher axial elastic stress and hence highA1B is ex-

FIG. 6. Evolution of the jet~filament! radius, elastic stress~measured at themiddle of the filament!, and strain-rate~maximum axial velocity at the neck!of an Oldroyd-B jet~filament! prior to recoil. Theoretical predictions aralso shown. (We5300,S50.25,Ca510, andl 520).


m-

.at

pected. Correspondingly, the low-L evolution in Fig. 9 maywell represent the dynamics of higher-generation filameThis will be further verified by an analysis that relates trecoil dynamics of filaments of different generation.

As seen in Fig. 9, the low-L primary filament drainsmuch faster than the highly extensible filaments. In factdoes not recoil at the neck and seems to pinch off in fintime. Entov and Hinch12 have predicted this outcome forconstant-radius filament. An insert of the low-L filament-bead profile immediately before pinch off is shown in tinsert of Fig. 9. Instead of a recoil, the straight filamentmains during the final precipitous drop inh(t) of Fig. 9. Amuch thinner filament drains rapidly at this stage andmains stable to the instabilities that trigger recoil. This thsuggests that iterated stretching will eventually stop whA1B, the stretching, is the same order asL for high-generation filaments.

Our analysis to establish the self-similarity of filamenof different generation begins with the linear Rayleigh insbility and the ‘‘hyperbolic’’ stretching it creates that formthe primary filament. This formation dynamics can thenused to fully specify the slow exponential elastic drainidynamics for the Oldryod-B jet shown in Fig. 6. The instbility that triggers the recoil at the neck is then scrutinizeIn contrast to the Rayleigh instability that creates the primfilament, the resulting recoil begins with Egger’s self-similpinching with negligible elastic effect and followed by thsame stretching and drainage dynamics of the primaryment. We are then able to estimate the radius and stresthe secondary filament and, by induction, relate all higgeneration filaments to the previous generation. In the pcess, we delineate the self-similarity of all high-generatfilaments until finite extensibility becomes important. Whextensibility comes into play, the drainage is too rapid for trecoil instability to take effect and Fig. 9 indicates that pinoff will occur instead.

III. LINEAR STABILITY THEORY AND ONSET OFSTRETCHING DYNAMICS

We shall examine jets with largeWeandL. As seen inFig. 9, the initial instability, the filament formation dynamicand the drainage dynamics are insensitive toL as long as it isin excess of 10. We hence focus only on the Oldroyd-Bhere. The stretching dynamics will be shown to be describy a coupled set of hyperbolic equations and, as suchevolution has a strong memory that remembers the incondition and evolution. Fortunately, the initial evolution ivolves small-amplitude deviations from the initial jet and cbe captured by a standard linear analysis that is further splified by our longwave expansion. Consider a standard nmal mode perturbation of the straight jet basic state

S hu

tzz

t rr

D ;S 1000D 1S h8

u8tzz8

t rr8D eiaz1lt. ~13!



FIG. 7. Radius, velocity, and stress profiles of Fig. 6 at various times during the filament formation and drainage stages.

-fro

n-

at

sort

-at

lityx-

ity,lyqs.

ehas

blehis

a

In the limit of largeWe, one obtains the following relationships between the stresses and the deviation radiusthe linearized versions of Eqs.~9! and ~11!

tzz8 522ia~S21!

Weu852

4~12S!

Weh8, ~14!

t rr8 5ia~S21!

lWeu85

2~12S!

Weh8. ~15!

The growth ratel is determined from the dispersion relatioship

2Wel31~216a2SWe!l21a2F62We

Ca~12a2!Gl

2a2~12a2!

Ca50. ~16!

The simplest limit is that of a Newtonian jet (We50)and it yields the classical longwave quadratic growth rwhich vanishes ata50 and at the neutral wave numbera0

51. Its maximum growth rate and wave number are

lmaxNewt5

1

2A2Ca~113ACa/2!

and ~17!

amaxNewt5

1

&~113ACa/2!1/2.

In the limit of large We, the elastic effect becomenegligible—the relaxation time approaches infinity. This cresponds to a zero eigenvalue which can be factored ou


m

e

-of

the cubic polynomial@Eq. ~16!#. The resulting quadratic corresponds to a longwave growth rate with a neutral modea051 and a maximum-growing mode with

lmax51

2A2Ca~113SACa/2!

and ~18!

amax51

&@113SACa/2#1/2.

The extra mode whose growth rate vanishes atWe→` canbe determined by standard expansion to be stable

l3;21

WeS 116Ca

We~12a2!1¯ D . ~19!

These results are consistent with earlier linear stabianalysis of the full equations for the Oldroyd-B jet, the Mawell jet (S50) at largeWe, and the Newtonian jet atWe50.8,9 Since the retardation number must be less than unhighly elastic jets yield slightly longer waves and slightlarger growth rates than Newtonian jets, as seen from E~17! and~18!. The limiting Maxwell jet is the most unstablwith the longest disturbances. Nevertheless, elasticitylittle effect in the initial evolution.

Despite the negligibly small elastic stresses, we are ato decipher its creation mechanism at inception from tlinear theory. The phase difference betweenh8 andu8 in Eq.~14!, h8;2( ia/2l)u8, implies that a node inu with a posi-tive slope appears at the minimum inh511h8. This corre-sponds locally to an axisymmetric extensional flow with


ip

en-ers

iss thetics

arthent

wojettheds.m-nlyntly

n

l-6.


FIG. 8. Evolution of the neck radius of an Oldroyd-B jet forS50.25 andl 520 but for the indicated ranges ofWeandCa. All exhibit the stretching,drainage, and recoil stages.


stagnation point at the jet minimum. The next relationsh~15! indicatestzz8 and t rr8 are in phase with2h8 and h8,respectively. This reflects the result that the uniaxial extsional flow at the stagnation point has stretched the polymand induces a maximum intzz and a minimum int rr at thestagnation point in the middle of the computation domain

tzz112S

We;

~12S!

We~124h8!;

~12S!

We~11h8!4 ;~12S!

Weh4 ,

~20a!

t rr 112S

We;

~12S!

Weh2, ~20b!

during the initial evolution with small-amplitude waves. Thset of invariance between the stresses and the jet radius irelationship that will be propagated along the characterisduring the hyperbolic stretching stage.

IV. FILAMENT FORMATION BY STRETCHING

The axisymmetric extensional flow revealed in the lineanalysis will trigger a stretching evolution that enlargessmall region near the jet minimum, with a locally constaradius, a linear axial velocity and a constant positivetzz,until a straight filament is formed. There are, of course, tadditional converging stagnation points at the twomaxima bounding the extensional stagnation point atminimum. These regions will be compressed into beaHence, the stretching of the filament at the minima is accopanied by compression at the maxima. We shall focus oon the extensional flow near the minima and consequeonly on filament stretching.

The scalings from the linear theory in Eqs.~14!, ~15!,and ~20! suggest thattzz and t rr at the above stagnatiopoint are a factor ofWe21 smaller thanh and (]u/]z),which are of unit order, in the stretching evolution that folows. This is consistent with our numerical results in Fig.

FIG. 9. The effect of extensibilityL on the jet evolutionfor a FENE jet (We5300, S50.25, Ca510, and l520). There is little sensitivity toL until L.10. Theinsert is the filament-bead profile att5243 for L52.


ovjto

re

u

oes

o

owl

-i

tio

tb

hde he

t

nrain-


Also, anticipating the length of stretching stage to be gerned mostly by the slow extension flow near the slenderminimum at the stagnation point, we expect fluid inertiabe negligible in the stretching filament and the curvaturek inEq. ~12! to be well approximated by the azimuthal curvatuonly, k521/h.

Hence, the dominant terms in Eqs.~9! and ~11! duringthe filament stretching stage are

1

h2

]

]z S h

Ca1h2~ tzz2 t rr !13Sh2

]u

]z D50, ~21a!

S ]

]t1u

]

]zDh252]u

]zh2, ~21b!

S ]

]t1u

]

]zD tzz52]u

]z S tzz112S

We D , ~21c!

S ]

]t1u

]

]zD t rr 52]u

]z S tzz112S

We D . ~21d!

The hyperbolic nature of the kinematic and stress eqtions in Eqs.~21b!–~21d! is quite apparent. It originates fromthe fact that both the liquid mass and the polymers are cvected by the nearlyr-independent axial velocity. Hence, thevolution of h2, tzz, and t rr are along characteristic linedefined by

dx

dt5u, ~22!

on which they behave as

dh2

dt52

]u

]zh2, ~23a!

d

dttzz52

]u

]z S tzz112S

We D , ~23b!

d

dtt rr 52

]u

]z S t rr 112S

We D . ~23c!

Since the equation of motion@Eq. ~21a!# becomes asteady force balance among capillary, elastic and viscforces, a simple integration yields az-independent forcef (t)that can only be a function of time

h

Ca1h2~ tzz2 t rr !13Sh2

]u

]z5 f ~ t !. ~24!

This quasi-steady balance then yields how the local fl~actually flow gradient! is determined by the local azimuthacapillary pressure and elastic stress difference.

The forcef (t) evolves in time during the stretching interval between the small-amplitude evolution describedthe last section and the slow elastic drainage of next secDuring this interval~aroundt;100 in Figs. 6 and 8!, tzz

increases dramatically andu drops precipitously while the jeevolves into a filament with constant radius. We are unato obtain the force evolutionf (t) explicitly but our numeri-cal results in Fig. 10~a! indicate that it does not vary mucduring this stretching interval. During the small-amplituevolution initially, the elastic stresses of Eq.~20! are small at


-et

a-

n-

us

nn.

leO(We21) and the axial velocityu and its gradient (]u/]z)are both small at the order of the perturbation radiush8 fromthe original straight filament of unit radius. As a result, tinitial value of the force during the stretching interval is

FIG. 10. ~a! Evolution of the forcef (t) at the jet minimum in Eq.~24!showing a decrease from 1/Ca to 0.6/Ca during the stretching interval a0,t,70 for the same Oldroyd-B jet of Fig. 6.~b! Simulated value ofCahtzz for the Oldroyd-B jet of Fig. 6, showing convergence to aasymptotic value 1.6 close to the estimated value 2 during the elastic dage stage fort.100. ~c! Comparison of the simulated jet radiush* at theend of the stretching interval to estimate Eq.~30!.


tho

ot

qsi

henc

ne

axe

la

tio-e

s

thatutic

si-

hetheichts isnidelas-e ate isnd

ehens.ostly

stsan

slowrre-essssurein

us

of

w-ab-ent

si-

hefol-

tog’’

ow-thedy

ing


f ~ t !;1

Ca. ~25!

We shall use this value throughout the entire interval—force is assumed to be also time independent. A more rigous treatment would be to expand the evolution in bWe21 andt to discern the full evolutionf (t). This would bea tedious endeavor and we will numerically verify that E~25! is a valid approximation. With this estimate, the quasteady force balance becomes

3S]u

]z5

1

Ca

~12h!

h2 2~ tzz2 t rr !, ~26!

that is valid at every point within the stretched filament. Tcapillary pressure and normal stress difference are balaby polymer viscous dissipation.

We now focus on the jet minimum with a stagnatiopoint (u50). Its vanishing axial velocity implies that thcorresponding characteristics line is vertical in thet-x planewhile neighboring characteristic lines diverge from it. Asresult, the linear axisymmetric extensional flow region is epanded throughout the region bounded by the jet nodMoreover, a simple analysis of Eqs.~23a! and ~23c! revealsthe following invariance along the characteristics during fiment stretching:

S tzz112S

We Dh45const1

and ~27!

S t rr 112S

We Dh225const2 .

Hence, applying this to the characteristic at the stagnapoint of the jet minimum where the initial condition is provided by Eq.~20! after the small-amplitude evolution, onobtains

tzz2 t rr 5S 12S

We D S 1

h42h2D , ~28!

at the stagnation point.Hence, at this minimum where (]u/]z).0, h and t rr

decrease monotonically whiletzz increases monotonically athe filament is stretched. Combining Eqs.~23a!, ~26!, and~28!, we acquire the thinning rate at the minimum

6Sdh

dt52

1

Ca

12h

h1

~12S!

We S 1

h32h3D , ~29!

where the right side represents the flow gradient betweenjet minimum and the jet node as driven by the azimutcurvature difference and retarded by the elastic stresses aminimum. This stretching ceases when the capillary pressincreases sufficiently ash decreases to balance the elasstress in Eq.~29!. This occurs when the thinning jet radiushapproaches

h* 5FCa~12S!

We G1/2

, ~30!


er-h

.-

ed

-s.

-

n

helthere

when ]u/]z and u approach zero as seen in Fig. 6. Phycally, the elastic stress, which scales as@(12S)/We#h24, asseen in Eq.~20!, has reached such a high value within tfilament that the liquid cannot continue to drain towardsnode due to the gradient in the azimuthal curvature, whscales only as (1/Ca)h22. That the stretched filamenreached an intermediate asymptote with constant radiuevident in all simulations seen in Fig. 8. An ‘‘inflectiopoint’’ when the evolution ‘‘hesitates’’ is seen after the rapdecrease during the stretching stage and before the slowtic drainage stage. The precipitous drop in the strain ratthis intermediate stage between stretching and drainagalso evident in the velocity evolution depicted in Figs. 6 a7.

Despite the approximation made onf (t) in Eq. ~25!, Eq.~30! is seen in Fig. 10~b! to be in good agreement with thsimulated filament radius at the inflection point, prior to tslow elastic drainage stage, for a wide range of conditioSome scatter is observed but the measured values are mbounded betweenh* and 2h* .

Note that this intermediate stretching interval only exiwhen S is not zero or unity. Since we have carried outexpansion inWe21, S, and 12S must actually be larger thanWe21. When polymer retardation is absent (S50), thequasi-steady force balance cannot be assumed anddrainage described by Eq.~29! breaks down. The singulalimit of S51 corresponds to the coincidence of polymerlaxation and retardation times. At this condition, the excelastic stresses are never triggered and the azimuthal pregradient drives the jet to pinch off without stretching, asPapageorgiou’s viscous jet breakup.

V. ELASTIC DRAINAGE

At the end of the stretching interval, the filament radihas reached a small constant valueh* ;O(We21/2), the ra-dial stresst rr remains small but the axial stresstzz is large atO(h

*24/We);O(We) by Eq. ~27! and the strain rate

(]u/]z) has dropped from unit order at the beginningstretching to negligibly small values by Eq.~26! such thatthere is no flow out of the filament due to stretching. Hoever, at this point, the bead-filament configuration is estlished and a new capillary driving force between the filamand the bead replaces that of the initial jet in Eq.~25! duringstretching. This different driving force changes the quasteady force balance and the magnitude oftzz. It also per-mits a small but finite drainage from the filament to tbounding beads. This is the elastic drainage stage thatlows the stretching stage.

Instead of carrying out detailed matched asymptoticsmatch these two stages, we adopt a lead-order ‘‘patchinscheme to resolve the relaxation dynamics during this slest intermediate drainage stage. We begin by determiningproper scalings. During this interval, the filament has alreabeen stretched to a jet with a uniform radiush(t) of orderWe21/2 as stipulated by Eq.~28!. For relaxation to be in-cluded in the stress dynamics as the polymer is bestretched,We21tzz, (]tzz/]t), and (]u/]z) tzz must all bal-ance in Eq.~11a!. Since the filament length is unit order,z


n-ngthoocai-

aththesnd

ngi

hara

aisegek

h

hethc

e

en

anba

aennc

ie

ne

llyip

la

the

re-notr-e to

la-ady.

itys

the

thand

m-ess

ce-

sthe

bead

stic

asson-

len-’s


;O(1) and this yields the scalingu;O(We21) and t;O(We) for the draining filament. Relaxation must be icluded to effect the pull of the stretched polymers acounter the capillary driving force such that a slow drainainto the beads can now proceed. This slow drainage islongest process in the jet dynamics and its duration is a gestimate of the drainage time. The remaining unknown sing is for tzz which will be determined through a quassteady force and mass balance at the neck.

Once the straight filament is formed, the hyperbolic nture of the evolution is lost. During the stretching stage,dynamics on each characteristic line are not affected byevolution on the other characteristics. However, with thetablishment of a straight filament, the linear uniaxial extesion flow permeates the entire filament and the drainagenamics over the entire filament is in unison. The driviforce has also changed. During the stretching stage, therequasi-steady force balance governed by Eq.~24! which stipu-lates that the stretching flow is driven by the local azimutpressure gradient and countered by local elastic stress gent and viscous dissipation. When a straight filamentformed, Eq.~25! becomes invalid as there is zero local grdient within the filament. The driving force for drainagenow provided by the azimuthal pressure drop across the njoining the filament to the bead. To quantify this drivincapillary force, a more detailed order assignment at the nregion is required. This analysis of the quasi-steady necabsent in earlier slender filament theories.

In lieu of the quasi-steady, slender jet force balance tyields Eq.~24!, we return to Eq.~9a! with the full curvaturek. Within the neck,k varies fromh21;We1/2 at the filamentto O(1) at the bead. We shall hence assign it the higWe1/2 order in our dominant balance as it corresponds tocapillary pressure difference across the neck. Using the sing u;O(We21), t;O(We), k;O(We1/2) and still an un-known scaling forz, the width of the small neck region, wcan already conclude from Eq.~9a! that the inertial terms onthe left are negligible compared to the curvature gradi(1/Ca)(]k/]z), regardless of the scaling ofz. This leavesthe stress gradient with mostlytzz contribution and the poly-mer viscous dissipation on the right for possible dominbalance with the curvature gradient. Each or both canance the curvature gradient since both the elastic stressviscous dissipation serve to reduce the flow from the filamto the bead while the curvature gradient drives it. We hefirst seek the scalings fortzz and z when all three termsbalance. This is possible whentzz;h21;O(We1/2) to bal-ance capillary pressure and elastic stress andz;O(We23/2) to balance viscous dissipation to the earltwo.

Since the neck width decreases in time as fluid is draiinto the bead, the scalingz;O(We23/2) to match viscousdissipation to capillary pressure is not established initiaAs the neck width decreases with drainage, viscous disstion increases. However, the initial width is larger at sayz;O(We21/4), to ensure the longwave approximationO(h)!O(z) remains valid, and elastic stress balances capilpressure at the neck.

Using the scalingsh;O(We21/2), tzz;O(We1/2), z


deedl-

-ee--y-

s a

ldi-is-

ck

ckis

at

real-

t

tl-ndte

r

d

.a-

ry

;O(We21/4), u;O(We21), andt;O(We) from the abovescaling arguments, we obtain the following equations forneck region from Eqs.~9! and ~11!

1

Ca

]k

]z1

1

h2

]

]z~h2tzz!50, ~31a!

]

]z~uh2!50, ~31b!

u]

]ztzz22

]u

]ztzz50, ~31c!

where we have omitted the negligiblet rr at the end of thestretching stage. As long as the longwave approximationmains valid in the neck such that the neck width doesexceedO(We21/2), the radial stress remains negligible duing the drainage stage and all subsequent dynamics. Duthe relatively small width of the neck compared to the fiment, the force and mass balances are both quasi-steSimple integration of Eqs.~31b! and ~31c! from the end ofthe filament, where the linear extensional axial velocreaches its maximum valueu0 and where the filament radiuand elastic stress retain the same values throughoutstraight filament ath0 and t0 , to any location within theneck, yields

tzz5t0~u/u0!2 and u5u0~h0 /h!2. ~32a!

The filament quantities with subscript 0 actually vary witime as the drainage proceeds. However, the neck stressvelocity are slaved to them according to Eq.~32! due to thenarrow width of the neck. These invariances can be cobined to yield a simple relationship between the neck strand the neck radius any where within the neck

tzz5t0h04h24, ~32b!

wheret0h04 is a slowly varying function of time only.

This invariance allows us to simplify the force balanacross the neck@Eq. ~31a!#. It can be converted into an integrable form by Eq.~32b!

1

Ca

]k

]z1

1

2

]

]ztzz50. ~33!

We now integrate Eq.~33! again but now completely acrosthe neck from the filament to the bead. The curvature andstress at the filament are large compared to those at theand neglecting the subscript 0 in Eq.~32!, we obtain animportant invariance between the curvature and the elastress of the straight filament

Cahtzz52. ~34!

The unique factor of 2 again arises from the force and mbalance across the neck. This predicted invariance is cfirmed by our numerical simulation shown in Fig. 10~c!where an asymptotic limit ofCahtzz51.6 is reached soonafter the stretching interval att5100. The invariance~34! isdistinct from earlier drainage theories which assume a sder filament without necks. A reanalysis of Renardyresult,11 for example, yields a constant of1

2 instead of 2 inEq. ~34!.


n

ct

io

ena

an-vanao-ai

-

eue

-o

ing

rei

r-

sthE

y-

the

. 4.h-

erce,ageandon,oilq.ni-theion

ads.

si-

er-e


Respecting the constant radius and linear extensioflow, the proper dimensionless variables are

u~z,t !5zU~ t !Wi21, Q5tWi21, ~35!

wherez is now O(1) as we return to the filament. The fathat tzz(Q) and h(Q) are only functions ofQ and notzallows us to construct the leading-order filament equatfrom Eqs.~9b! and ~11a!

dh

dQ52

1

2hU and tzz1

d

dQtzz22U tzz50. ~36!

We have neglected to scaletzz;O(We1/2) and h;O(We21/2) explicitly for simplicity.

The kinematic equation@Eq. ~36!# is simply a mass bal-ance for a straight filament while the stress equation@Eq.~36!# represents stress relaxation within the filament. Thboth evolve exponentially for a constant uniaxial extensioflow, as shown by earlier straight-filament theories,11,12

h(t);exp(2Ut/2) and tzz(t);exp@(2U21)t#. However, thenew force balance across the neck@Eq. ~34!# stipulate theyare correlated in time such that their product is a constThis is only possible ifU5 2

3, the maximum filament velocity at the neck remains constant during this drainage interThis implies that the strain-rate of the uniaxial extensioflow within the draining filament remains constant at twthirds the rate at which the stress would relax at fixed strAlthough Entov and Hinch’s straight-filament theory usesdifferent correlation constant betweenh and tzz, it still cap-tures this unique constant strain rate.12 After inserting theinitial conditionh* of Eq. ~30! for matching with the stretching stage, we obtain the large-time asymptotic behavior

u~z,t !52

3Wez, ~37a!

h~ t !5h* exp~2t/3We!, ~37b!

tzz~ t !52

h* Caexp~ t/3We!, ~37c!

for the draining filament in the original variables. Thuniaxial extensional flow is clearly evident but the uniqfeature is the correlated exponential decay ofh(t) and expo-nential growth oftzz(t) due to matching of the filament solution to the bead. These asymptotic predictions are favably compared to the simulated evolution in Fig. 6. Matchwith the final radius of the stretching stage in Eq.~30! andthe proper capture of thetzz and h correlation in Eq.~35!from the neck analysis are essential to obtain the cordescription of the draining filament. They are not availableearlier slender or straight filament theories.11,12

In both the neck analysis of Eq.~31!, which yields cor-relations~32! for the neck, and the derivation of the impotant stress-radius correlation of Eq.~34! for the filament, thebeads are never explicitly included. So long as their radiumuch larger than the filament, the beads do not affectleading-order filament drainage dynamics captured in~37!. This also suggests that Eq.~37! is a universal drainagedynamics valid for all Oldroyd-B filaments bounded bbeads. We had used the initial jet radiusr 0 as the character


al

n

yl

t.

l!l

n.a

r-

ctn

ise

q.

istic length scale but one can use the actual value ofinitial jet radiush* . We also expect Eq.~37! to describe thedrainage of the secondary filament after the recoil in FigHowever, the recoil dynamics are different from the stretcing dynamics of the filament in Sec. IV since the formbegins at the neck while the latter at a jet minimum. Henthe radius of the secondary filament at the onset of draincannot be described by the stretching analysis of Sec. IVwill, instead, be addressed in Sec. VII. In the next sectiwe shall determine the instability that triggers the neck recby analyzing the stability of the drainage dynamics in E~37!. Since it describes all draining filaments except the itial jet, which is not bounded by beads, we also expectsame recoil mechanism to apply for all higher generatfilaments.

VI. STABILITY OF A FILAMENT DRAINING INTOBEADS

Due to the slow elastic drainage with time scale 3We,one can analyze the stability of the draining filament-beconfiguration at any given instant in time by linearizing Eq~9! and ~11! about the draining state as if the jet is quastationary

AS u8h8t8D 5lS u8

h8t8D , ~38!

where the differential operatorA is

A1152d

dz~u• !1

3S

h2

d

dz S h2d

dz• D ,

A1251

Ca S d3

dz3 •11

h2

d

dz•2

2

h3

dh

dz• D2

2

h3

d

dz~h2tzz!•

26S

h3

d

dz S h2du

dz D •11

h2

d

dz~2htzz• !

13S

h2

d

dz S 2hdu

dz• D ,

A1351

h2

d

dz~h2

• !,

A21521

2h

d

dz~h2

• !,

A22521

h

d

dz~hu• !,

A235A3250,

A3152d

dz~ tzz• !13tzzS d

dz• D2

2~S21!

We S d

dz• D ,

A3352d

dz~u• !13

du

dz•2

1

We•.

The dynamics oft rr have been neglected due to the slendness of the neck andt8 represents the disturbance to thexcess axial elastic stress.


nt

onres


FIG. 11. The spectrum of the Oldroyd-B bead-filameconfiguration with a draining filament in Fig. 6 att5500. The dominant structures are the stable bandsthe ellipse and the negative real axis. The fine structunear the origin are shown in Fig. 12.

om

aisotiv

tn-

3ndshe

tur-of

bothnen-ver,

theAt

t allds.con-ttere theni-rob-ent-

thethe

hehesns,neng

q.

.a

the

o1rer

sethbe

Assuming the same periodicity, over the domainlength l, for the disturbance and the evolving jet, the coputed spectrum with a spectral numerical method att5500for the draining filament in Fig. 6 is shown in Fig. 11 withblow-up of the origin in Fig. 12. Most of the spectrumstable and the stable eigenvalues form a nearly continuband of ellipse and a nearly continuous line on the negareal axis, as is evident in Fig. 11. Continuous spectrumobviously impossible with a finite domain sizel but bothstable bands do approach continuum asl becomes large. Thestable ellipse terminates on the negative real axis nearorigin, as seen in Fig. 12.~Discrete eigenvalue 1 is an exte

FIG. 12. Blow-up of the fine structures of the spectrum near the originthe complex plane. The vertical branch to the right of the ellipse in Fig.is branch 3 here. Mode 1 is part of the ellipse and branch 2 is a stablebranch. The most unstable branch 4 is opposite branch 2 on the positiveaxis. The radius eigenfunctions of branches 1, 3, and 4 are shown in inThe former two are confined to the filament in the middle and decay intobeads at the two ends. The eigenfunction of branch 4 is confined to thein the middle of that insert.


f-

use

is

he

sion of the ellipse.! Several additional branches~2 and 3!bifurcate from the ellipse near the origin. Complex branchtends to approach the imaginary axis while branch 2 exteslightly into the stable region on the negative real axis. Tmember of the eigenfunction corresponding to radius disbance is also shown in Fig. 12 for some typical membersthe more unstable branches 1 and 3. The disturbances ofbranches are confined to the filament and decay expotially into the beads. The disturbances of branch 3, howealso decay towards the middle of the filament.

The most unstable branch 4, however, protrudes intoleft half of the complex plane on the positive real axis.t5500, the most unstable mode of this branch is atl r

.0.019 and this value is nearly constant up tot51000. Itseigenfunction is also shown in Fig. 12 which suggests thathe disturbances of this branch 4 are confined to the bea

There are hence two classes of disturbances, onefined to the filament and one to the beads. Although the laseems more unstable, both decay towards the neck wherrecoil initiates. We can better understand why the recoil itiates at the neck with a deeper analysis of the spectral plem ~38!. If we omit the beads and use the estimated filamsolution of Eq.~37! over an unbounded domain, the computed spectrum shown in Figs. 13 and 14 yields bothstable ellipse in Fig. 13 and branches 1, 2 and 3 nearorigin in Fig. 14 att5500. Other than some details near torigin, due to mode interaction with the beads, the brancare all quantitatively reproduced. Even the eigenfunctioincluding the odd branch 3, are captured correctly if oallows for the fact that they do not decay into the missibeads.

In fact, analytical expressions can be derived from E~38! in the limit of infinite We. In this limit, thez-dependentuniaxial velocity profileu(z,t) in Eq. ~37a! vanishes andhence the coefficients of the operatorA become constantThis allows the usual normal mode expansion to yieldsimple dispersion relationship. Due to the omission ofuniaxial extensional velocity field,l50 is always a solution

f2al

ealrts.ead


-m


FIG. 13. Spectrum of an infinitely long draining filament. The computed values are indistinguishable frothe analytical result withh5h* exp(2t/3We) at t5500.

si

a

trura

ticheone

with no velocity fluctuationu850. Omitting this neutralmode, the other two modes are determined by the disperrelationship

l213Sa2l2a2S 1

2Cah2 tzzD1

a4h

2Ca50. ~39!

Comparing this to the Rayleigh dispersion relationship forunstretched filament of unit radius (h21) in Eq. ~18!, onecan easily see that the positive elastic stresstzz has a stabi-lizing effect.

If one further introduces the derived correlation~34! of astretched filament under drainage into beads, the speccan be estimated explicitly as a function of the filamentdius h


on

n

m-

l523

2Sa26A9

4S2a41

h

2Caa2~12a2!2

2a2

Cah.

~40!

This analytical dispersion relationship for an highly elas(We→`) stretched filament quantitatively captures tstable ellipse and stable real branch of Figs. 11 and 13, ifuses the estimate~37b! for the filament radiush(t) at t5500.

The ellipse corresponds to smalla and is well approxi-mated by

l523

2Sa26 iaA 2

Cah. ~41!

gc-

FIG. 14. The spectrum of the infinitely long draininfilament near the origin and representative eigenfuntions.


r

ine

omngaernlv

of

ineinofhe

hdlaRa

ls aing

nce

dlin-

vem

ingan.entthe

ofting

ra

t

ithc-atg

therr inm.ck

the

tely

u-


The stable real mode is at largea, corresponding to23Sa2.Obviously, the actual filament lengthl imposes an uppebound ofa5(2p/ l ).

The bead branch 4 can likewise be estimated by linkit with the Rayleigh instability of a cylinder. We insert in thmiddle of the bead a cylinder of lengthL and a radius iden-tical to that of the bead. We than remove the filaments frthe elongated bead such that the structure has a leroughly equal to the sum of the diameter of the original beandL. The results are insensitive to the exact location whthe filaments are removed. We then impose periodic bouary condition for the disturbance over this structure and sothe full eigenvalue problem with the prescribedh and withnegligible elastic stresstzz. At infinite L, we obtain the con-tinuous Rayleigh spectrum of an infinitely long cylinderEq. ~18!, after correcting for the new cylinder radius:

lRayleigh523

2Sa26A9

4S2a41

a2

2Cah2

a4

2Ca. ~42!

As L becomes finite, this continuous spectrum breaks updiscrete modes. But as is evident in Fig. 15, the discrmodes lie approximately at the same location as the contous modes of Eq.~42! and approach those on branch 4Fig. 12 at L50. Branch 4 can hence be attributed to tRayleigh instability of a bead with openings to filaments.

The relative dominance of the unstable bead brancand the unstable filament branch 1 at the neck must betermined by a different spectral theory. The continuous fiment spectrum Figs. 13 and 14 and the continuous bead

FIG. 15. The spectra of the bead with a cylindrical insert of lengthL. TheL5` limit is the Rayleigh instability and theL50 limit is the bead mode ofbranch 4 in Fig. 12.


g

thded-e

toteu-

4e--y-

leigh spectrum of Fig. 15 atL5` correspond to normamodes in unbounded domains. However, if one introducegeneric localized disturbance, the local effect of the resultwave packet is not determined by the spectruml~a! of thenormal modes. Specifically, consider a generic disturbaof the form

h8~z,t !5E2`

`

A~a!eiaz1l~a!tda, ~43!

whereA(a) is the Fourier coefficient of the initial localizedisturbance and the evolution in time is specified by theearized equations of Eq.~38!.

At a specific location,z50 say, the dynamics~43! aredominated by a single complex modea* derivable byWentzel–Kramers–Brillouin~WKB! theory19

dl

da~a* !50, ~44!

wherea5a r1 ia i is complex and so isl5l r1 il i . Hence,the contribution of all modes at the neck~and any otherlocation! is determined by the growth rate at the abosaddle pointa* in the complex plane. An unstable spectruin an unbounded domain is ‘‘absolutely’’ unstable ifl r(a* )is positive. Otherwise, it is convectively unstable—disturbances will connect pass the neck without triggerany local instability. The growth would then occur only inmoving frame and would not be felt at any specific locatioThis classification of an unstable spectrum is most pertinto instabilities which possess a specific sensitive spot—neck in the present example.

We determine the absolute and convective stabilityboth the filament and bead continuous spectra by exploithe Cauchy–Riemann condition. Since bothl and a arecomplex in Eq. ~44!, a* can be determined from(]l r)/(]a r)5(]l r)/(]a i)50 only. We optimize with re-spect toa r anda i sequentially to locate the saddle pointa* .In Fig. 16, we fixa i for both the filament and bead spectand plot the spectral(a r1 ia i) as parameterized bya r fromzero to infinity. We then seek the maximuml r with respectto a r on these spectra,lmax(ar

max,ai). For all a i in bothcases, this optimum is located on the real axis,a r

max50. Wethen varya i to optimizelmax with respect toa i along thereal axis. As seen in Fig. 16, the filament spectrum at5500 is convectively unstable witha* 52.2i and l(a* )520.6 while the bead spectrum is absolutely unstable wa* 51.15i and l(a* )50.466. The unstable filament spetrum of Fig. 14 hence does not contribute to local growthany location, including the neck. Its growth is in a movinframe. The unstable bead spectrum of Fig. 15, on the ohand, contributes to a local growth rate of 0.466 that is faexcess of its maximum value 0.019 on the original spectruIt is the ability of Rayleigh modes to accumulate at the nethat accounts for the absolute instability which triggersrecoil.

The same conclusion that the bead mode is absoluunstable has also been verified at largert during the drainageinterval. Although our analysis is carried out for the contin


ndd ar-rved. Ifandacehthethe

nin-

po-

on-ith

tesilitytoathees.

ther a

inedck.re-

e

tru


FIG. 16. The spectral(a r1 ia i) parameterized bya i . The solid spectracorrespond to those whosel r

max decrease witha i and the dashed lines arthose that increase.~a! The filament spectrum witha* 52.2i and l(a* )520.6. ~b! The bead spectrum, as approximated by the Rayleigh specwith a* 51.15i andl(a* )50.466.


ous band atL5`, we expect the discrete modes atL50 tobehave likewise due to their similarity in Fig. 15.

To verify that it is the disturbances from the bead anot the filament that trigger neck recoil, we have performelarge number of numerical simulations. If localized distubances are placed on the draining filament, they are obseto convect pass the necks and vanish within the beadsthey are placed on the beads, they expand into the neckquickly trigger a recoil. To show that the recoil is indeedresult of this instability, we place the same disturbanwithin the bead att5400, 600, and 800 in Fig. 17. Eacdisturbance triggers a recoil as seen in the figure. Withoutdisturbance, the drainage would continue undisturbed bypredicted dynamics of Eq.~37!. The simulation in Fig. 17 iscarried out with a FENE model with extensibilityL5100. Itis clear that, forL in excess of 10, the above recoil initiatiodue to absolutely unstable disturbances from the bead isdependent ofL. For smallerL, however, Fig. 9 indicates thefilament drainage is much faster than the Oldroyd-B exnential drainage of Eq.~37!. Earlier theory12 suggestsh(t)decreases linearly until breakup. The linear thinning is csistent with Fig. 9 but the predicted rate does not agree wour simulation. Nevertheless, this fast thinning invalidathe quasi-steady assumption in the current filament stabanalysis. From the simulations, a low-L filament seems‘‘outrun’’ the convective instability from the bead such thatrecoil is never triggered. We are unable to determinecritical L that separates unstable filaments from stable on

VII. FORMATION OF HIGH-GENERATION FILAMENTS

The absolutely unstable Rayleigh disturbances frombeads relieve the tension at the necks and quickly triggerecoil of the primary high-extensibility filament. As seenFig. 4, the relieved tension is almost immediately replacby a sharp maximum in the axial elastic stress at the neThis elastic stress grows very rapidly as the stretching c

m

r-

el

FIG. 17. Recoil triggered by small localized distubances placed in the beads att5400, 600, and 800.This simulation was actually done with a FENE modwith all conditions identical to those of Fig. 6 but withextonsibility L5100.


th

an

me

ly-

ice to

coilinEq.-theuchheta-

ur-ndasticnt

iscsike

gain.am-outrsin

e-

fila-

nduiteingtednd

oil

on


ates a much finer secondary filament near the neck wimuch larger azimuthal curvature. The small spike intzz ofFig. 4 rapidly grows into a large maximum, much larger thtzz of the primary filament, as seen in Fig. 18~a!.

FIG. 18. Rescaling of the stress evolution near the neck beyond the recFig. 4 by~a! Newtonian pinching scaling and~b! viscoelastic pinching scal-ing. (We510,000,S50.25,Ca510, andl 512.5.)

FIG. 19. Normalization of the interface evolution near the neck beyrecoil. Conditions same as those in Fig. 18.


a

The secondary filament formed must still obey the saquasi-steady force balance~26! and the kinematic condition~23a! of the primary filament before its drainage due to pomer relaxation. Combining these equations, we get

6Sdh

dt5 tzz2

1

Cah, ~45!

where we have neglected 1/h relative to 1/h2 in the capillaryterm and have omittedt rr . We hence expect the elaststress to again balance the azimuthal capillary pressurform a secondary filament of radius

h* 5S 1

tzzCaD 1/2

. ~46!

However, since the stretching here arises from the reand not the elastic stretching of the Rayleigh instabilitySec. III, we cannot use the stress-radius correlation of~28! in Eq. ~46! to obtain an explicit prediction for the secondary filament radius. Instead, we need to resolvepinching dynamics during the recoil. We shall associate sdynamics with a self-similar solution that evolves from tprimary filament, after being triggered by the Rayleigh insbility from the beads.

The pinching dynamics triggered by the bead distbances push fluid rapidly from the neck into the filament athe bead. This relieves the stretching and reduces the elstresstzz at the neck. As a result, inertia terms are importafor the first time in the jet evolution while elastic effectnegligible at the beginning of a recoil. The recoil dynamiare hence similar to that of a Newtonian jet. However, unlinertia pinching of a Newtonian jet,2,3,5 the extensional flowabout a newly created stagnation point near the neck acreates a large local maximum intzz seen in Figs. 4 and 18This elastic stress mediates the subsequent pinching dynics considerably. Since the neck profile is asymmetric abthe minimum during pinching, asymmetric stretching occuinitially that evolves later into a straight filament, as seenFig. 4.

We first attempt to simplify the force balance and kinmatic conditions with the Oldroyd-B model of Eq.~11! bythe self-similar transform of a Newtonian jet7

j5x2x0

S1/2At02t, t5

t02t

S2 ,

h5SH~j!t/Ca, u5S21/2t21/2V, ~47!

tzz5X~j,t!

St, t rr 5

Y

St,

where t0 is a nominal ‘‘pinchoff’’ time when the filamentradiush vanishes atx5x0 . Although this pinchoff is nevercompleted due to elastic effects that form the secondaryment in Eq.~45!, scaling Eq.~47! is still appropriate in anintermediate interval beyond the initiation of the recoil abefore the secondary filament is established. This is qanalogous to the termination of the hyperbolic stretchstage in Eq.~23! when the jet nodes have been convecinto the beads. After that, a straight filament is formed a

of

d


-

ilald

he

ndethit

thals-

n

n

ec

ch

gth

ching-

g,ri-

ated

as-

nce-ch-f a

of

ofoth

by

is

-ian-

rtu-


begins to drain by the elastic time scale 3We, as describedby Eq. ~37!. Unlike the self-similar pinching solutions studied earlier, the current one at the neck in Eq.~47! will even-tually evolve into the beads and terminate the self-simbehavior. Nevertheless, as the hyperbolic stretching yiethe initial filament radiush* in Eq. ~30! prior to the drainageof the primary filament, transformation~47! yields an impor-tant intermediate pinching solution that links the recoil to tstraight secondary filament.

Under transformation~47!, the Oldroyd-B jet of Eqs.~9!and ~11! becomes, in the limit oft→0

~V1j/2!dV

dj1

V

25

1

H2 H dH

dj1

d

djH2~X2Y!

13d

djH2VjJ , ~48a!

~V1j/2!dH

dj52~Vj/221!H, ~48b!

2t]X

]t1

1

2~2V1j!

]X

]j5~2Vj21!X, ~48c!

2t]Y

]t1

1

2~2V1j!

]Y

]j52~Vj11!Y. ~48d!

Without the normal stress difference, Eqs.~48a! and~48b! are just the inertial self-similar equations of motioand kinematic operation of a Newtonian jet. There arescribed by an ode. However, the hyperbolic nature ofstress equations must be retained to propagate the instress profiles. Hence, one cannot omit thet~]/]t! terms inEqs. ~48c! and ~48d!. Otherwise,X andY vanish exactly asthe system has no memory of the past—it collapses intoNewtonian self-similar pinchoff. Hence, the Newtonian scing Eq. ~47! is not the self-similar transform for a viscoelatic jet.

Nevertheless, the hyperbolic nature of Eqs.~48c! and~48d! renders them amenable to another self-similar traform by the methods of characteristics. DefiningT52 ln t,one obtains along each characteristic defined by

dj

dT5

1

2~2V1j!, ~49!

the stress evolution

dX

dT5~2Vj21!X and

dY

dT52~Vj11!Y. ~50!

We shall carry out Taylor expansion inj about thepinch-off stagnation pointj5j0 to facilitate numerical solu-tion of Eqs.~48a! and ~48b!. However, as in the Newtoniacase, the coefficients in the kinematic equation@Eq. ~48b!#stipulate that the expansion is only possible~a smooth self-similar solution only exists! if

Vj~j0!52, V1j0/250. ~51!

Hence, the leading-order expansion of the velocity is spfied

V;2j0/212~j2j0!. ~52!


rs

-eial

e-

s-

i-

The axial velocity is again a uniaxial extensional flow whiflows to the right and left from the stagnation pointj0 . How-ever, unlike a straight-filament extensional flow, its strenincreases in time,u;2t23/2(x2x0).

Substituting Eq.~52! into Eq. ~50!, one again concludesthat the radial excess stress approaches zero as the pinprogresses,T→` and t→0. The axial excess stress, however, increases monotonically as described by

dX

dT53X. ~53!

As for the characteristic lines during initial stretchinthe characteristic lines on the plane of the self-similar vaables j and T also fan out from the stagnation pointj0 .Hence, the elastic stress near the pinching point is dominby the evolution on the characteristic lines nearj5j0 . Sub-stituting the expansion of the velocity nearj08 in Eq. ~52!into Eq. ~49!, we get

dj

dT5

5

2~j2j0!. ~54!

Combining Eqs.~54! and~53!, it is clear that any initialstress profileF( j) nearj0 at T50 would be propagated bythe characteristics to produce a stress fieldX(j,T)5F( je25T/2)e3T where j5j2j0 is the distance from thestagnation point. ExpandingX(j,t) in powers ofj, we ob-tain

X~j,t!5F~ je25T/2!e3T;F0~t!1F2~t!j2, ~55!

where F0(t)5F(0)t23 and F2(t)5 12F9(0)t2. Hence,

knowing the initial profileF(j) for X, we can derive thetime-dependent coefficientsFi(t).

The dominant stress behavior nearj0 from Eq.~55! sug-gests the invariant scalingtzz(t02t)4 is the true self-similartransform for the pinching dynamics of the current viscoeltic jet. This is distinctly different from thetzz(t02t) scalingof a Newtonian jet from Eq.~47!. The deviation originatesfrom thee3T factor of Eq.~55! which, in turn, arises from thehyperbolic stress convection and elastic stretching enhament. The universal stress scaling during self-similar pining of a viscoelastic jet is hence quite distinct from that oNewtonian jet eventhough the radius and velocity scalingsEq. ~47! are identical.

In Figs. 18 and 19, we verify these universal scalingsthe pinching dynamics in a viscoelastic jet by collapsing bthe stress evolution and the interface evolution at neckdemonstrating that, at the neck

tzz5tzz~0!t0

4

~ t02t !4 , h5Hmin

SCa~ t02t !, ~56!

whereHmin is a universal constant. The neck radius scalingidentical to that of a Newtonian jet for which Brenneret al.6

have determinedHmin to be 0.0304. The stress scaling, however, is unique to viscoelastic jets. In fact, the Newtonscalingtzz5 tzz(0)t0 /(t02t) fails to collapse the stress evolution in Fig. 18~a!.

It would be difficult to solve Eq.~48! with expansion~55! to obtain the actual radius and stress profiles. Fo


e.

r-

og

el

,

teh

nt

i

ntaie

y

by

ch,inrethe

redre-areeri-ini-fila-

. 6d isoise

tion

y

achjustpoly-e

, ala-up

tricrfor

isllyaf--fornt

pa-

fila-he

-

ch-mu-ings atr-

r-


nately, scalings~56! are already sufficient for our purposConsider a primary filament in drainage, with stresstzz(0)and radiush(0), when its neck recoils due to bead distubances. We do not know the values oftzz(0) andh(0) pre-cisely, as they are determined by the disturbances as shin Fig. 17. However, we do know they are related throuthe stretched filament correlation~34! from the neck analysis~that specifies the driving force during drainag!Cah(0)tzz(0)52. We have sett50 to be the onset of recoiand hence to be consistent, Eq.~56! must yield t0

5@h(0)SCa#/Hmin . The scalings~56! can now be insertedwith these matching conditions into the force balance~45! todetermine the radius of the secondary filament. Consiswith our earlier leading-order matching in time, we use tself-similar recoil stress and radius of Eq.~56! in the subse-quent quasi-steady force balance of the secondary filameEq. ~46! during its stretching interval. AlthoughHmin is notknown exactly, the power-law expressions allow us to elimnateHmin , t0 , SCa, tzz and tzz(0), to yield an explicit rela-tionship betweenh* , the radius of the secondary filameafter the recoil and stretching stages but prior to elastic drage, andh(0), theradius of the primary filament when threcoil initiates

h* 5&~h~0!!3/2. ~57!

This simple correlation is verified numerically in Fig. 20 btriggering the recoil in Fig. 17 at different radiush(0) of theprimary filament during drainage. The recoil is initiatedplacing localized disturbances at the bead.

VIII. DISCUSSION

The beads do not participate actively during the streting and drainage stages. They act as sources of noiseaccumulators and transmitters of noise from the surroundfluid, that trigger the recoil dynamics. However, they aunaffected by the recoil and the subsequent formation ofsecondary filament. We hence expect this secondary filamto drain like the primary one as described by Eq.~37!, but

FIG. 20. Comparison of the predicted filament radius iterationh*5&@h(0)#3/2 from various experiments with varying conditions by triggeing the recoil in Fig. 17 at different radius of the primary filament.


wnh

nte

in

-

n-

-org

ent

with h* replaced by Eq.~57!. We also expect it to suffer thesame instability at the neck as the primary one, as captuin Sec. VI. An iterated stretching sequence is hence pdicted, creating finer and finer filaments, even though weunable to capture the higher-generation filaments numcally. If the disturbances are large, such that the recoiltiates before significant drainage has taken place, thement radius is a constant andh(0) in Eq.~57! corresponds tothe undrained filament. From the drainage history of Figsand 17, this requires very little disturbance at the bead anquite reasonable for a jet in the presence of constant nfrom the environment. In this realistic limit, Eq.~57! yields arecursive relationship for filaments of successive genera

~r n /r 0!5&~r n21 /r 0!3/2, ~58!

in dimensional filament radiusr and the original jet radiusr 0 . Equivalently, if we allow the first iterate to be given bEq. ~30! and all subsequent ones by Eq.~58!, one obtains

~r n /r 0!5FCa~12S!

We G3~n21!/4

, ~59!

and the axial elastic stress of thenth filament, from Eq.~34!,

tzz~n!52

Ca FCa~12S!

We G23~n21!/4

. ~60!

The elastic stress hence increases very rapidly with esuccessive filament, as we have observed in Fig. 18 forone iteration. As the elastic stress increases, so does themer stretchingA is the axial direction, as described by thOldroyd-B model. Eventually,A;L2 in the spring law~2!and extensibility becomes important. As seen in Fig. 9low-L primary filament, analogous to a high-generation fiment, will no longer recoil and the iteration ceases. Breakis expected at that point.

The above universal scalings arise from the asymmeself-similar pinching after recoil. This particular self-similasolution necessarily involves inertia, as does Egger’sNewtonian jets.2 In fact, it is the ony stage where inertiaimportant. We have carried out simulations by artificiaremoving the inertial terms. Only the pinching dynamicster recoil differ from those with inertia. The recoil still initiates at the neck as the Rayleigh instability responsibletriggering it is independent of inertia. A secondary filamestill forms but it does not obey correlation~57!. The pinchingis also asymmetric, unlike the inertialess pinching of Pageorgiou’s solution4 for Newtonian jets without inertia. Weare hence unable to predict the radius of the secondaryment if inertia is omitted. However, as is evident from tnaturally scaled governing equations@Eq. ~9!#, inertia is neg-ligible for a jet surrounded by an inviscid fluid only for excessively smallCa and S in the limit of largeWe. This isimpractical and we expect inertia to enter during the pining and recoil at the neck, as we have observed in our silations. In fact, to obtain Papageorgiou’s symmetric pinchsolution, one must introduce large-amplitude disturbancethe middle of the filament that will transform the lineauniaxial extensional flow into a uniform axial flow with vanishing (]u)/(]z). Only then would the inertial term


-fo

nfilaonl

fanhe

di-the

d-ice

ad-no

nts.ntgi-

outvelike21.enter-ote

uc-his

tionoresal

ed

up-

et

,’’

ys.

ng

ties

ak-ids

a

rts,’’

up

onn

r

isthT

on


u(]u)/(]z) in Eq. ~9a! be negligible. Hence, we expect inertia to be important for recoil and pinching at the neckmost practical values ofCa and S of a highly elastic jet(We@1) in an inviscid surrounding. Since we have showthe neck is the most unstable portion of the stretchedment, we expect inertia to play an important role in the recand stretching iterations. In our simulations, we have oseen symmetric pinching for the singular limit ofS51. Aconsequence of this argument is that recoil may not occurjets in a viscous fluid where inertia can be independentlyartificially suppressed. However, such jets can trigger ot

FIG. 21. Photographs taken from a primitive experiment in which a vcoelastic bead-filament configuration is created in a Newtonian fluid ofsame density. The photographs are taken with about 20 s intervals.primary filament clearly stretches, recoils, and restretches to form a secary filament.


r

-ily

ordr

viscoelastic effects as their axial velocity has a radial graent even in a straight filament. This scenario is beyondcurrent theory.

All prior experimental studies of the viscoelastic beafilament configuration involve flying jets instead of statones confined within finite domains. In a long flying jet, thbeads can slide along the filaments8 and disrupt the recoildynamics. The jet also bends and twists when the befilament configuration appears. Not surprisingly, there isreported observation of recoil and high-generation filameTo remedy this, we have performed a primitive experimeby squeezing a viscoelastic fluid of uncharacterized rheolocal properties through a tube into a Newtonian fluid of abthe same density but much lower viscosity. We hasqueezed an excess of fluid initially to create a bead-head followed by a narrower filament, as seen in Fig.Despite the crudeness of the experiment, the filamstretches immediately by draining into the bead and undgoes a distinct recoil to generate a secondary filament. Nthe recoiled primary filament forms another bead-like strture that bounds the other end of the secondary filament. Tsuggests that recoils and stretchings to form high-generafilaments are to be expected from viscoelastic jets. Mcareful experiments are underway to verify the universcalings of Eq.~58!.

ACKNOWLEDGMENTS

We are grateful to Michael Renardy for pointing out thpossibility of recoil. The experiment in Fig. 21 was carrieout by an undergraduate A. Rastaturin. This work is sported by NASA.

1N. N. Mansour and T. S. Lundgren, ‘‘Satellite formation in capillary jbreakup,’’ Phys. Fluids A2, 114 ~1990!.

2J. Eggers, ‘‘Universal pinching of 3D axisymmetric free-surface flowPhys. Rev. Lett.71, 3458~1993!.

3J. Eggers, ‘‘Theory of drop formation,’’ Phys. Fluids7, 941 ~1995!.4D. T. Papageorgiou, ‘‘On the breakup of viscous liquid threads,’’ PhFluids 7, 1529~1995!.

5M. P. Brenner, X. D. Shi, and S. R. Nagel, ‘‘Iterated instabilities duridroplet fission,’’ Phys. Rev. Lett.73, 3391~1994!.

6M. P. Brenner, J. Lister, and H. A. Stone, ‘‘Pinching threads, singulariand the number 0.030...4,’’ Phys. Fluids8, 2827~1996!.

7M. P. Brenner, J. Eggers, K. Joseph, S. R. Nagel, and X. D. Shi, ‘‘Bredown of scaling in droplet fission at high Reynolds number,’’ Phys. Flu9, 1573~1997!.

8M. Goldin, J. Yerushalmi, R. Pfeffer, and R. Shinnar, ‘‘Breakup oflaminar capillary jet of a viscoelastic fluid,’’ J. Fluid Mech.38, 689~1969!.

9R. Keunings, J. Comput. Phys.62, 199 ~1986!.10D. W. Bousfield, R. Keunings, G. Marrucci, and M. M. Denn, ‘‘Nonlinea

analysis of the surface tension driven breakup of viscoelastic filamenJ. Non-Newtonian Fluid Mech.21, 79 ~1986!.

11M. Renardy, ‘‘A numerical study of the asymptotic evolution and breakof Newtonian and viscoelastic jets,’’ J. Non-Newtonian Fluid Mech.59,267 ~1995!.

12V. M. Entov and E. J. Hinch, ‘‘Effect of a specturm of relaxation timesthe capillary thinning of a filament of elastic liquid,’’ J. Non-NewtoniaFluid Mech.72, 31 ~1997!.

13M. Renardy, ‘‘Some comments on the surface-tension driven break-up~orlack of it! of viscoelastic jets,’’ J. Non-Newtonian Fluid Mech.51, 97~1994!.

14M. D. Chilcott and J. M. Rallison, ‘‘Creeping flow of dilute polymesolutions past cylinders and spheres,’’ J. Non-Newtonian Fluid Mech.29,381 ~1988!.

-ehed-


seol-id

r-

s,’’

a-


15R. B. Bird, R. A. Armstrong, and O. Hassager,Dynamics of PolymericFluids ~Wiley, New York, 1977!, Chap. 10, pp. 471–510.

16M. J. Soloman and S. J. Muller, ‘‘Flow past a sphere in polystyreme-baBoger fluids: the effect on the drag coefficient of finite extensibility, svent quality and polymer molecular weight,’’ J. Non-Newtonian FluMech.62, 81 ~1996!.


d

17S. Kalliadasis and H.-C. Chang, ‘‘Drop formation during coating of vetical fibers,’’ J. Fluid Mech.261, 135 ~1994!.

18J. Ratulowski and H.-C. Chang, ‘‘Transport of gas bubbles in capillariePhys. Fluids A1, 1642~1989!.

19P. Huerre and P. A. Monkewitz, ‘‘Local and global instabilities in sptially developing flows,’’ Annu. Rev. Fluid Mech.22, 473 ~1990!.


Iterated stretching of viscoelastic jets

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Iterated stretching of viscoelastic jets