Pattern formation in non-Newtonian Hele–Shaw ﬂowkondic/heleshaw/PFl_nonnewt.pdf · 2014. 10....

PHYSICS OF FLUIDS VOLUME 13, NUMBER 5 MAY 2001

Pattern formation in non-Newtonian Hele–Shaw flowPetri Fasta)Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

L. KondicDepartment of Mathematical Sciences and Center for Applied Mathematics and Statistics,New Jersey Institute of Technology, Newark, New Jersey 07102

Michael J. ShelleyCourant Institute of Mathematical Sciences, New York University, New York, New York 10012

Peter Palffy-MuhorayLiquid Crystal Institute, Kent State University, Kent, Ohio 44242

~Received 29 December 1998; accepted 7 February 2001!

We study theoretically the Saffman–Taylor instability of an air bubble expanding into anon-Newtonian fluid in a Hele–Shaw cell, with the motivation of understanding suppression oftip-splitting and the formation of dendritic structures observed in the flow of complex fluids, suchas polymeric liquids or liquid crystals. A standard visco-elastic flow model is simplified in the caseof flow in a thin gap, and it is found that there is a distinguished limit where shear thinning andnormal stress differences are apparent, but elastic response is negligible. This observation allowsformulation of a generalized Darcy’s law, where the pressure satisfies a nonlinear elliptic boundaryvalue problem. Numerical simulation shows that shear-thinning alone modifies considerably thepattern formation and can produce fingers whose tip-splitting is suppressed, in agreement withexperimental results. These fingers grow in an oscillating fashion, shedding ‘‘side-branches’’ fromtheir tips, closely resembling solidification patterns. A careful analysis of the parametricdependencies of the system provides an understanding of the conditions required to suppresstip-splitting, and an interpretation of experimental observations, such as emerging length-scales.© 2001 American Institute of Physics.@DOI: 10.1063/1.1359417#

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I. INTRODUCTION

While flows of non-Newtonian fluids are of considerabtechnological importance, their understanding is oftenscured by their complexity. For this reason, we concenton a rather simple situation: fluid flow in the essentially twdimensional setting of a Hele–Shaw cell, where the flowdescribed by a balance between pressure and viscous foand which for a Newtonian fluid is governed by Darcy’s laSuch thin-gap flows of non-Newtonian fluids are relevantindustrial processes such as injection molding1 or displaydevice design.2 In particular, a two-phase flow in this settinis a scientifically important one, given the close analogytween the Saffman–Taylor instability of driven Newtoniafluid with quasistatic solidification~and the Mullins–Sekerkainstability3!, and many other physical problems, such as etrochemical deposition.4

To make contact with a large body of experimental atheoretical work on pattern formation in such systems,concentrate here on the interfacial dynamics of a gas buexpanding into fluid in a radial Hele–Shaw cell. When tfluid is Newtonian, a dense branching pattern morphologcommonly observed~see McCloud and Maher5 and the ref-

a!Present address: Center for Applied Scientific Computing, Lawrenceermore National Laboratory, P.O. Box 808, L-661, Livermore, Califnia 94551.

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erences therein! as the outcome of the nonlinear develoment of the Saffman–Taylor instability. Such patterns acharacterized by successive tip-splitting of the interface,formation of branched structures, and the competitiontween them. This morphology has been observed in carnumerical simulations,6 and described in some of its aspectheoretically.7 It is also well known that flow structures remniscent of solidification—dendritic fingers, side-branchinsuppressed tip-splitting—can be produced in such Newtian flows by imposing an anisotropy on the system, for eample, by scoring lines on the plates,8 or by introducing aperturbation~bubble! in the fluid itself.9 Again, some de-tailed understanding of these systems has been achithrough analysis and simulation~see, for example AlmgrenDai, and Hakim10!.

However, experiments performed with complex liquisuch as liquid crystals,11,12 polymer solutions and melts,13,14

clays,15 and foams,16 have shown that similar structures cabe induced by the bulk properties of the fluid itself. Thatthe response of the fluid, which may itself be isotropic, cproduce an effect akin to anisotropy. One property sharedthese different liquids is that they are shear-thinning~theshear viscosity decreases with the local shear rate!, and wewill concentrate on this property.

As a motivating example, Figs. 1~a! and 1~b! show anexperiment of the pattern formation that results from push

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1192 Phys. Fluids, Vol. 13, No. 5, May 2001 Fast et al.

air into a dilute solution of PEO, a standard shear-thinnpolymer ~figures courtesy of R. Ennis and P. PalffMuhoray, LCI, Kent State!. A relative lack of tip-splitting isapparent, and one sees the appearance of isolated finHolding the driving pressure fixed, Figs. 1~c! and 1~d! showthe effect of decreasing the gap width by 2.5 times. Tdecreases the non-Newtonian effect by lowering the Wsenberg number~defined as the ratio of the material relaation time to a fluid flow time!, and one observes the emegence of tip-splitting and of a more densely branchpattern. In either case, the similarity to dendritic structureclear.

As an illustrative case, our analysis uses the JohnsSegalman–Oldroyd~JSO! model17 for a viscoelastic fluid,though our results apply to more general differential mod~Sec. II!. This model considerably simplifies in the thin-galimit e5b/L!1, where b is the separation between thplates andL is some typical lateral dimension~Sec. II A!. Tothe leading order ine, we find that there is a distinguishelimit—where the natural Weissenberg number of the flowO(1)—where shear-thinning is retained. In this limit, thviscoelastic fluid is reduced to a generalized Newtonfluid, where elastic effects enter only through the definitiof a Weissenberg number. Following our previous work,18,19

we obtain then a generalized Darcy’s law governing the bfluid flow,

u52“2p

m̄~We2u“2pu2!, “2•u50, ~1!

whereu is the gap averaged longitudinal velocity,p is thefluid pressure, We is a Weissenberg number, andm̄ is aderived effective viscosity depending upon the squared psure gradient. This yields a nonlinear, elliptic boundary vaproblem ~BVP! for the pressure in the driven fluid~Sec.II B !. Issues related to boundary conditions for this BVPdiscussed in Sec. II B 1. As it appears appropriate for

FIG. 1. Graphs~a! and~b! show the temporal development of a pattern thresults from pushing air into a dilute, shear-thinning PEO solution inHele–Shaw cell. Graphs~c! and~d! show the resulting pattern in a cell withgap width 2.5 times smaller. The driving pressures are the same in ecase. Figures courtesy of Ennis and Palffy-Muhoray, LCI, Kent State.


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parametric regime which we study, in our simulations we uthe Laplace–Young condition, as is typically used for Netonian Hele–Shaw flows.

In recent theoretical work, Poire´ and Ben Amar20,21haveused our prescription to study the formation of ‘‘fracturesor ‘‘cracks’’ in clays and associating polymer solutions20

Using a shear-thinning power-law fluid, they examined t‘‘width selection’’ problem for a gas finger propagatinsteadily down a channel. They consider the displaced fluibe slightly shear-thinning~in the shear-thinning exponent!,and show within this asymptotic limit, the selected fingwidth decreases to zero~i.e., a crack! as surface tension goeto zero. Lindner, Bonn, and Meunier22 recently studied ex-perimentally the propagation of a finger into a shear-thinnliquid. Also using Eq.~1! for a power-law fluid, they findexcellent agreement with their experimental data.

In Sec. III, we examine the linear stability of a circulaexpanding gas bubble, where the driven fluid is governedthe generalized Darcy’s law. We consider first a weakly noNewtonian model where, in the limit of a small Weissenbenumber, the nonlinear boundary value problem for the prsure is simplified to a linear one, and the linear stabilproblem can be solved exactly. This suggests that shthinning can modify the Saffman–Taylor instability to givincreased length-scale selection. We expand on this furby solving numerically the linear stability problem forstrongly shear-thinning fluid. In Sec. IV we perform fullnonlinear, time dependent simulations of a bubble growinto a strongly shear-thinning fluid. These simulations shthat shear-thinning influences considerably the evolutionthe interface, and in agreement with experiments with coplex fluids, can lead to the formation of fingers which do nsplit, and that grow in an oscillating fashion. They cansemble closely the dendritic structures observed in solidcation. We also analyze the dependence of the interface mphology on nondimensional parameters, which allowscomparison with and interpretation of available experimenresults~Sec. V!. Finally, we also explore some different viscosity models and discuss computed and experimentallyserved length-scales. In the Appendix, we discuss the mematical aspects of solving for the effective viscositym̄, andthe relation to the solvability of the nonlinear BVP~1!.

II. EQUATIONS OF MOTION

First, for thin gap~Hele–Shaw! flow we show how a‘‘typical’’ visco-elastic flow model reduces asymptotically tthe non-Newtonian Darcy’s law~1!. We then discuss boundary conditions at the gas/fluid interface. At the end of tsection, we formulate the flow problem as the dynamicsthe gas/fluid interface, whose velocity is found by solvingnonlinear BVP over the fluid domain encompassed byinterface.

A model for the motion of an incompressible, isothemal, viscoelastic fluid is given by

rDv

Dt5“•t, ~2!

“•v50, ~3!

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1193Phys. Fluids, Vol. 13, No. 5, May 2001 Pattern formation in non-Newtonian Hele–Shaw flow

together with a constitutive relation for the stress tensot.Herev5(u,v,w) is the velocity field,D/Dt5] t1v•“ is thematerial derivative, andr is the ~constant! density. The ve-locity gradient is (“v) i j 5] jv i , andD5(“v1“vT)/2 is therate-of-strain tensor.

When studying a particular flow problem, it is importato pick a constitutive relation that reproduces the experimtal observations for the relevant rheometric flows. In the cof Hele–Shaw flow, we show how to derive the generalizDarcy’s law~1! for a broad class of differential models. Willustrate this within the context of the Johnson–SegalmaOldroyd ~JSO! model17,23 with a single relaxation time. Thisis perhaps the simplest viscoelastic model that capturesmal stress differences and shear-thinning of the viscosThe JSO equations are

t52pI12msD1s, ~4!

s1lDas

Dt52mpD, ~5!

wherep is the pressure,s is the extra stress tensor,l is therelaxation time, andms andmp are the solvent and polymeviscosities.

The Gordon–Schowalter~GS! convected derivative,24

Das

Dt5

Ds

Dt2$“v s1s“vT%1~12a!$Ds1sD%,

~6!

in JSO models the nonaffine motion of polymer chains. Thare not locked into a rubber network, which deforms with tflow, but rather the chains are allowed to slip past the ctinuum. Fora51 the motion is affine, and JSO reducesthe Oldroyd-B model, which forms50 is the same as thupper-convected Maxwell model. Decreasinga increases theslippage, and softens the response of the material by incring shear-thinning in shear flows, and reducing strain haening in extensional flows.@The slip-parametera should berestricted to 0.2,a,0.89 for consistency with experiments25

using dilute solutions of a variety of commercial polymeThe ratio of the second normal stress difference to the fiequal to 2(12a)/2 for JSO, was found to lie betwee20.40 and20.055, and to be independent of the shear-ra#The GS-derivative reduces to the corotational derivativea50, and to the lower-convected derivative fora521.

A. A Hele–Shaw scaling of JSO

In this section, we use the small aspect ratio«5b/L ofthe gap widthb to the lateral length-scaleL to derive thegeneralized Darcy’s law~1!. This is done by choosing thpressurep}«21, which makes shear-thinning a dominaeffect and leaves the elastic response a higher order cotion.

1. One-dimensional steady shear flow

We expect a non-Newtonian Hele–Shaw flow to behalocally like a one-dimensional steady shear-flow in the dirtion of the pressure gradient. To understand the scaling ofull equations ~2!–~4!, we consider a steady, onedimensional shear flowv(z)5„u(z),0,0…. Partial derivatives


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are denoted by a subscript and the components of a tensoa superscript, e.g., thexx-component ofs is sxx.

The extra stress tensors can be solved explicitly from~5! without approximation.26 The extra stress components agiven by

sxz5mp

11~12a2!l2uz2 uz , s

xx5~11a!luzsxz, ~7!

syy50, szz52~12a!luzsxz. ~8!

The steady momentum equations~2! are then

px5]

]z„m~uz

2!uz…, ~9!

py50, ~10!

pz52]

]z„~12a!lm~uz!uz

2…, ~11!

where the shear-rate dependent viscositym is defined by

m~uz2!5m0

11a~12a2!l2uz2

11~12a2!l2uz2 , ~12!

andm05ms1mp is the total, or zero shear-rate, viscosity.The ‘‘shear-thinning parameter’’a5ms /m0 determines

the behavior of the viscosity function:a51 yields a New-tonian fluid with a constant viscosity, anda,1 yields aviscosity that increases with decreasing shear-rate, i.e.,viscosity is shear-thinning. The constrainta.1/9 is neces-sary for the stress–strain relation to be invertible~see Sec.II B !.

We now nondimensionalize Eq.~9!, and study the effectof different scalings of the pressure. For a given fluid, texperimentally adjustable quantities are the driving~gauge!pressuredP and the plate separationb. The lateral lengthscaleL is given by some typical dimension in the horizontdirection, such as the size of the cell or an initial bubble siwhich is large in comparison tob, so that«5b/L!1 ~Fig.2!. The characteristic lateral velocityU will also depend onthe driving pressure. We scalez;b, x;L, p;dP and u;U, and write Eq.~9! nondimensionally as

dP

L

]p

]x5

m0U

b2]

]z S 11a ~12a2!We82 uz211~12a2!We82 uz2 uzD . ~13!Here We85lU/b is a Weissenberg number.26

FIG. 2. Hele–Shaw cell.


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We now choose the pressure scaling. Balancing presand viscous forces givesU5«2 dPL/m0 . If dP is indepen-dent of«, then We8}«, and ~13! reduces to the Newtoniacase with viscositym05ms1mp . If dP}«

22, then We8}«21, the shear-viscosity will be constant, and elasticfects would become important in an unsteady flow.

The choice that makes shear-thinning apparent isdP}«21. Then U}«, We8}1, and ~13! retains its shear-thinning character in the leading order as«→0. This is adistinguished limit, as a specific scaling of the independand dependent variables is required to retain some desquality, in our case the shear-rate dependency of the visity.

2. Nondimensional form of JSO for Hele –Shaw flow

In this section, the full equations of motion~2!–~5! arenondimensionalized. The one-dimensional shear flow stion suggests the following scaling~the nondimensionaquantities are primed!:

t5m0

« P0t8, x5Lx8, u5«

P0 L

m0u8, ~14!

p512P0

«p8, y5Ly8, v5«

P0 L

m0v8, ~15!

s5P0s8, z5«Lz8, w5«2

P0 L

m0w8. ~16!

Here, the pressurep, cross-gap directionz and lateral direc-tions x, y are scaled as in the previous subsection, as ischaracteristic velocityU5« P0 L/m0 . Time is scaled asL/Uand the cross-gap velocityw is scaled as«U. The typicalsize of viscous and viscoelastic stresses ism0U/b5P0 . Thevelocity gradient and the rate-of-strain tensor are then

“v5P0m0

“v8, where “v85S «ux «uy uz«vx «vy vz«2wx «

2wy «wzD ,

and

D5 P0m0

D8,

where D85 12 S 2«ux «~uy1vx! uz1«2wx«~uy1vx! 2«vy vz1«2wy

uz1«2wx vz1«

2wy 2«wzD .

To separate the orders of«, we write these tensors withouapproximation as

“v85L01«L11«2L2 , D85D01«D11«2D2 .There are two nondimensional parameters associ

with the scaling given by Eqs.~14!–~16!,

Re5«2rUL

m05«3

rP0L2

m02 , Reynolds number; ~17!

We85lU

«L5

lP0m0

, Weissenberg number. ~18!


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A natural Deborah number would be defined as De5lU/L5« We8. We will find it convenient to use the modifieWeissenberg number,

We5~12a2!1/212We85~12a2!1/212lP0

m0. ~19!

Introduce the lateral velocityu5(u,v), lateral gradient“25(]x ,]y) and lateral LaplacianD25]xx1]yy . Then,dropping the primes, the nondimensional momentum convation equations~2! are

ReDu

Dt5212“2p1a~uzz1«

2D2u!1FszxzszyzG1«Fsxxx1syxysxxy1syyyG , ~20!

«2 ReDw

Dt5212pz1« sz

zz

1«2„sxxz1sy

yz1a~wzz1«2D2w!…, ~21!

the incompressibility condition~3! is

“2•u1wz50, ~22!

and the constitutive relation~5! with the convected derivative ~6! is

s2We8~L0 s1s L 0T2~12a!~D0s1sD0!!

52~12a!D02« H We8S DsDt 2L1 s2s L 1T1~12a!3~D1s1sD1! D22~12a!D1J 1«2 $We8„L2 s1s L 2T2~12a!~D2s1sD2!…12~12a!D2%. ~23!

3. Leading order equations

Assume the horizontal velocityu, the pressurep and theextra stress tensors have asymptotic expansions of the for

u~x,t !5u(0)1O~«!,

p~x,t !5p(0)1O~«!, ~24!

s~x,t !5s(0)1O~«!,

and substitute these expansions into Eqs.~20!–~23!.The leading orderO(1) contribution to the momentum

equations~20! and incompressibility condition~22! are~afterdropping the superscripted 0 from the notation! a set of re-duced Stokes equations:

12“2p5auzz1FszxzszyzG , ~25!pz50, ~26!

“2•u1wz50. ~27!


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The constitutive relations~23! yield the set of linear equations for the stresses:

sxx2We8 ~11a!sxzuz

5szz1We8 ~12a!~sxzuz1syzvz!50, ~28!

syy2We8 ~11a!syzvz

5sxy2We811a

2~sxzvz1s

yzuz!50, ~29!

sxz2We8 szzuz1We812a

2„~sxx1szz!uz1s

xyvz…

5~12a!uz , ~30!

syz2We8 szzvz1We812a

2„~syy1szz!vz1s

xyuz…

5~12a!vz . ~31!

Equations~28!–~31! can be solved by first findingsxx,syy, szz andsxy in terms of the shear stressessxz andsyz,and then substituting these into Eqs.~30! and ~31!. After amoderate amount of algebra, the shear-stressessxz andsyz

are found to satisfy the equations

AFsxzsyzG5~12a!FuzvzG , ~32!where

A5S 114Cuz21Cvz2 3Cuz vz3Cuz vz 114Cvz

21Cuz2D , ~33!

andC5(12a2) We82/4. Note thatC.0 as long as the assumptionuau,1 holds. The matrixA is nonsingular, with apositive determinant,

detA5~11C uuzu2!~114C uuzu2!.The shear stresses, along with the rest of the componenthe extra stress tensor, can now be solved from~32! and~28!–~29!. They are given by

sxz512a

duz , s

yz512a

dvz , ~34!

sxx5aWe8~11a!

duz

2 , syy5aWe8~11a!

dvz

2, ~35!

szz52aWe8~12a!

duuzu2, sxy5a

We8~11a!

duzvz ,

~36!

whered511(12a2)We82uuzu2. Notable is the presence onormal stress differences. This is in contrast to the HeShaw flow of a Newtonian, or a generalized Newtonian fluin which case the normal stress differences are zero. Hever, these normal stress differences only enter atO(e), andare not present in the leading order reduced Stokes’ etions ~25!–~26!.

Substituting the shear stresses into Eqs.~25!–~26! yields


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]z~ma„~We/12!

2uuzu2…uz!, ~37!

pz50, ~38!

“2•u1wz50, ~39!

as the leading order equations of motion for a JSO fluid iHele–Shaw cell. The nondimensional shear viscosity, wboth the polymer and the solvent contribution, is

ma„~We/12!2uuzu2…5

11a~We/12!2 uuzu2

11~We/12!2 uuzu2, ~40!

where a5ms /m0 , and the modified Weissenberg numbeWe, is given by~19!. This agrees with the one-dimensionsteady shear flow result.

B. Generalized Darcy’s law

The reduced Stokes equations~37!–~39! can be used toderive a generalized Darcy’s law, as in Kondic, PalffMuhoray and Shelley.18 The discussion applies to a genershear-rate dependent viscosity function, but we specializeresults to the case of JSO.

Integration of the reduced Stokes equation~37! yields

12z“2p5ma„~We/12!2 uuzu2…uz , ~41!

where we seek flows symmetric aboutz50, and use theindependence ofp from z. We would like to expressuz as afunction of“2p, as in the usual Darcy’s law. Squaring~41!gives an implicit equation foruuzu2 in terms ofz2u“2pu2. Theinvertibility of this equation, or lack thereof, is a central isue. A sufficient condition for finding a well-behaved inverof ~41! is that

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m̄a~We2 u“pu2!

512E21/2

1/2

dzz2

m̃a~We2 z2u“pu2!

. ~46!

Equations~45!–~46! are a generalized Darcy’s law for a noNewtonian fluid, with the viscosity expressed as a functof u“pu2. The subscript on“ andm, and the bar onu will beomitted hereafter.

Figure 3 showsm̄a(We2 u“pu2) for various values ofa,

with the Weissenberg number simply rescaling the absciWe summarize the relations between the four different ‘‘vcosities:’’ m is the shear viscosity given by~40!, m̃ is theinverse ofm, m̄ is the gap-average~46! of m̃, and later wewill need m̂, which is the inverse of the gap-averaged vcosity. For a Newtonian fluid, these would all be equal toconstantm0 in dimensional terms, and simply 1 nondimesionally. The ‘‘inverse’’ is always to be taken in the sendiscussed above. The viscositym̄a(We

2u“pu2) inherits theinvertibility and monotonicity ofma(We

2uuzu2). In Appen-dix A we give a detailed discussion of the properties ofviscosity functions, and anexplicit expression form̄ in thecase of JSO.

Remarks. ~1! The analysis of this section generalizes immediately to JSO models with multiple relaxation timeConsider a model where the extra stresss is the sum ofNmodessk , each satisfying a constitutive relation of the forof Eq. ~5!. That is, the equations of motion are~2!, conser-vation of mass“•v50 and the constitutive relation

sk1lkDask

Dt52hkD ~k51,...,N!, s5 (

k51

N

sk .

~47!

Herelk andhk are the relaxation time and viscosity of thk-th mode, respectively.

FIG. 3. The effective viscositym̄ for some typical values ofa with We51 ~changes in We rescale the abscissa!.


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The analysis of Sec. II A can now be applied to Eq.~47!for each mode separately. The reduced Stokes equationthis case are~37!–~39! with the viscosity functionma5mgiven by

m„~We/12!2 uuzu2…5a01 (k51

Nak

11bk2~We/12!2uuzu2

,

~48!

where a05ms /m0 , ak5hk /m0 , bk5lk /l1 , and m05ms1(hk . The modified Weissenberg number should befined as in Eq.~19! using the longest relaxation timel1 . Thegeneralized Darcy’s law~45! holds now with the effectiveviscosity m̄ defined using the inverse viscosity of Eq.~48!.

~2! More general constitutive models can be treated. Jis a special case of the Oldroyd 8 constant model,26

t52pI1s,

s1l1D1sDt 1l3~Ds1sD!1l5tr~s!D1l6~s:D!I

52m0S D1l2 D1DDt 12l4DD1l7~D:D!ID ,which is the most general differential model linear in textra stress tensor. HereA:B5tr(AB T), and the upper con-vected derivativeD1 /Dt is given by Eq. ~6! with a51.There are certain restrictions on the constantsm0 , l1 ...l7for the model to be physically reasonable.26

Applying the scaling~14!–~16! to the Oldroyd 8 con-stant model yields as the leading order the reduced Stoequations~37!–~39!, but with a different viscosity. Here, theshear-rate dependent viscosityma5m is given by

m„~We/12!2uuzu2…511b2~We/12!

2 uuzu2

11b1~We/12!2 uuzu2

, ~49!

where

b15a31a51a3~12a32a5!1a6~12a3232 a5!,

b25a2~a31a5!1a4~12a32a5!1a7~12a3232 a5!,

and a j5l j /l1 for j 52,...,7. The modified Weissenbernumber should be defined as in Eq.~19! usingl1 instead ofl. Note that no generality is gained by using the Oldroydconstant model instead of JSO in the present context;though the Oldroyd 8 constant model yields more geneexpressions for the extra stresses, the resulting reduStokes equations are the same as with JSO, with the shrate dependent viscosity of the same form.

~3! The model used by Bonnet al.,27,28 similar to Eq.~45! but with the viscosity depending onuuu, now follows byusing the viscositym̂ instead of the viscositym̄a(We

2u“pu2)in Eq. ~45!, thereby expressing“p explicitly in terms ofu.In this case, the functional form is such that this inversican always be accomplished. We consider Eq.~45! to be theÿ.more natural form of the flow equations; it leads toboundary value problem for pressure, as in the Newtoncase.

~4! Darcy’s law could be formulated in the same mannwith a ‘‘power-law’’ viscosity m(uuzu2)5(uuzu2)a. The pa-


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is-ri-Ca

he

asa

onalest-

theDe

e in


rameteruau,1/2 is negative for a shear-thinning fluid. Dacy’s law is still given by Eq.~45!, with the effective viscositym̄ now given by

m̄a~ u“pu2!5314a

3~112a! S 14 u“pu2Da/~112a!

. ~50!

This model was used by Poire´ and Ben Amar,20,21who stud-ied the Saffman–Taylor instability of a weakly sheathinning power-law fluid (a!1) in a channel geometryLindner, Bonn and Meunier22 have also used this expressioto describe their experimental results.

Boundary conditions on the pressure.Applying thedivergence-free condition to the generalized Darcy’s lyields

“•S 1m̄a~We2 u“pu2! “pD50, ~51!which is a nonlinear BVP for the pressure. As a boundcondition we will use in this work the Laplace–Young codition:

@p#5Ca21 k, ~52!

for the pressure jump@p#, wherek is the interfacial curva-ture, Ca512«22m0U/g is a ~modified! capillary number,and g is a surface tension parameter. This boundary contion is typically used, and has been justified,29,30for Newton-ian Hele–Shaw flows. For the parametric regime whichstudy—moderate Weissenberg and capillary numbers—current state of theoretical and experimental evidence sgests that this boundary condition remains appropriate. Fther, as our simulations do capture important qualitative ftures observed in experiment, our results mightinterpreted as ana posteriori justification for this assumption. Nonetheless, as the question of boundary conditionnon-Newtonian flows is a complicated one, we give a brreview of what seem to be the relevant issues.

The derivation of the bulk fluid equation~1! assumes aseparation of length-scales into a large lateral length scaLand a small gap-thicknessb so that «5b/L!1. No suchseparation of length-scales is available near the meniswhere in fact the flow is fully three dimensional~see Smith,Wu, Libchaber, Moses and Witten,31 and Tabeling, Zocchiand Libchaber32!, and at which the fluid satisfies a strejump condition. An analysis of the flow near the meniscustherefore required to derive a consistent approximation toboundary conditions on the gap-averaged pressure.

It is known that an air bubble displacing a fluid inHele–Shaw cell leaves behind a thin residual film. The thiness of this film varies with bubble velocity and so gives rto variations in the pressure jump across the meniscus.analogous problem is that of a long air bubble displacinfluid in a capillary tube,33,29,34,30which for the Newtoniancase was analyzed by Bretherton33 by expanding in a verysmall capillary number Ca˜5mU/g!1. This analysis wasgeneralized to Newtonian Hele–Shaw flow by Park aHomsy,29 again for small Ca˜, who showed that the LaplaceYoung condition gave the leading order contribution. Reinand Saffman30,34 removed the small Ca˜ restriction through


y

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e

inf

s,

se

-en

a

d

lt

direct numerical solution of the flow, and determined tcorrect pressure boundary conditions up to Ca˜5O(1).Reinelt35 studied viscous fingering in a channel Hele–Shcell geometry, and extended the work of McLean aSaffman36 by using the improved boundary conditions: hanalysis recovered the experimentally observed finwidths. ~These improved boundary conditions have not beapplied in numerical simulations of radial Hele–Shaw floand, in fact, are not generally used.! In this work, we con-centrate on the case Ca˜!1, and do not consider these corections.

In the case of a non-Newtonian fluid in a Hele–Shacell, only partial theoretical results are available. Ro aHomsy37 generalized the analysis of Park and Homsy,29 andfound corrections to the pressure boundary condition forOldroyd-B fluid model.26 However, they do not consideterms depending on the lateral curvature, and only accofor changes in the thickness of the residual film.

We remark that Wilson38 has usedad hocboundary con-ditions in an attempt to account for normal-stress effectshis study of the non-Newtonian Saffman-Taylor instabilitHe derived a normal-stress jump condition at the interfaceassuming the stress distribution in the bulk fluid can betended up to the interface; however Ro and Homsy,37 work-ing in the limit Cã5m0U/g!1, show that this assumption iincorrect.

Several experimental studies have identified normstress differences as being important in some instancenon-Newtonian Hele–Shaw flow~see Smith, Wu, Lib-schaber, Moses and Witten,31 Gauri and Koelling39 andHuzyak and Koelling40!. However, this is highly dependenon the parametric regime considered. Indeed, the aforemtioned experiments reveal that the Newtonian and nNewtonian case yield an almost identical response if the cillary number and Weissenberg number are moderate, athe present paper.

Gauri and Koelling39 study the flow dynamics at the tipof the meniscus of a long air bubble that displaces a vcoelastic fluid with a constant shear-viscosity. Their expements are characterized in terms of a capillary number˜and a Deborah number De,

De5l U

b5We85

We

12A12a2'

1

4We,

whereU/b is the wall shear-rate in Hele–Shaw flow, and tlast approximative equality applies foruau'0.9. They findtwo distinct flow patterns at the tip of the meniscus,sketched already by Taylor:41 a complete bypass flow, andrecirculation flow. When De>1 the flow completely by-passes the tip of the bubble, creating a strong extensiflow field. The response of the non-Newtonian fluid changdramatically at De'1. This transition could be perhaps atributed to a ‘‘coil–stretch’’ transition~De Gennes42!, whichoccurs due to the sudden uncoiling of polymer strands instrong extensional flow near the tip of the meniscus for>1.

However, such behavior is an example of the responsa particular parametric regime. FormoderateDe,1 and


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Ca,103, Gauri and Koelling39 show that there is no stronextensional flow near the tip. Further, for De,0.35, the flowfield near the meniscus tip for non-Newtonian fluids wsimilar to that for Newtonian fluids, and the thicknessestheir residual films were nearly identical. In our numericstudies~Sec. V! we concentrate on flows where Ca,103,and the modified Weissenberg number is moderate,,0.5 (De,0.125). This parametric regime should be belothe transition to bypass flow and a possible ‘‘coil–stretcresponse.

In a channel Hele–Shaw cell, Smithet al.31 studied theproperties of Saffman–Taylor fingers in very dilute solutioof polystyrene dissolved in a Newtonian solvent. They foua transition to narrow fingers when the modified capillanumber Ca;103, and the shear rateU/b was comparable tothe inverse of the polymeric relaxation timel ~estimatedusing Zimm theory!. Their analysis31 yields again the criticalDeborah number De'1, at which the abrupt change in response is attributed to a ‘‘coil–stretch transition.’’ Anagain, for the capillary numbers considered herein, Smet al.31 found an essentially Newtonian response.

Although it is reasonable to assume that in this paramric regime normal-stress effects are negligible near the tipthe meniscus, it is possible that viscoelastic effects in thefilm region become important.37 However, for moderateWeissenberg numbers (We8,1), Gauri and Koelling39 andHuzyak and Koelling40 show that in a purely elastic polymeric ~Boger! fluid the residual film thickness scales almoidentically in Cãfor Newtonian and non-Newtonian fluidsthe Ro and Homsy analysis37 suggests in this case that thnon-Newtonian effects at the meniscus are negligible,Newtonian boundary conditions are applicable. Henceseems reasonable to neglect elastic effects at the interfathe parametric regime in which we are interested.

The effect of shear-thinning near the meniscus has bless researched: in light of theoretical studies~Ro andHomsy37 and Fast43! it is possible that shear-thinning giverise to corrections to the Laplace–Young condition. A modetailed analysis is clearly warranted: It is likely that a fnumerical simulation, as performed by Reinelt aSaffman30 for Newtonian fluids, is required to settle thquestion of shear-thinning and viscoelastic non-Newtoncontributions to the pressure boundary conditions.

C. Dynamics of an expanding bubble

We now consider the evolution of a gas bubble expaing under an applied pressure into a non-Newtonian fluida radial Hele–Shaw cell. The fluid domain is taken to beannular regionV bounded by an inner boundaryG i and anexternal boundaryGe . Let the inner and outer boundary bgiven by the curvexi ,e(b,t), respectively, where we assumb to be the Lagrangian parametrization of the curve.

We will study initial data, for an expanding interfacwhich is a small perturbation from a circle. Accordingly,defining the nondimensional parameters, Ca and We, weas characteristic length and velocity scales the initial bub

radiusR0 and initial velocityṘ0.


sfl

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d

h

t-f

in

t

ditin

en

e

n

-nn

sele

The full evolution problem forV is the nonlinear BVPfor the pressure,

“•S 1m̄a~We2 u“pu2! “pD50,~53!

puG i512Ca21k i , puGe5Ca

21 ke ,

and the kinematic condition,

]xi ,e]t

~b,t !5u„xi ,e~b,t !,t…, ~54!

which states that the boundaries are material curves.Remarks:~1! Nonlinear BVPs similar to Eq.~53! arise as

the steady states of nonlinear conservation laws in mother physical contexts, such as gas dynamics44 andmagnetostatics.45 The solvability of Eq.~53! is established inAppendix A3 using classical results; inequality~42! is also asufficient condition for Eq.~53! to have a unique solution.

~2! Consider a finite patch of fluid, denoted byV withboundaryG, surrounded by gas at uniform pressure~set tozero!. Then the length of the boundary curve decreases wtime, so the dynamics is curve shortening. To show this,nonlinear BVP~53! with the Laplace–Young boundary condition must be augmented with the kinematic condition~54!where the boundary curvex is parametrized with the Lagrangian parameterb. The velocity is obtained from thepressure through Darcy’s law~1!. This free-boundary prob-lem describes the relaxation of the bubble under capillforces. A direct calculation shows that the lengthL of theboundary curve decreases in time since

dLdt

52CaEV

u“pu2

m̄~We2u“pu2!dA,0.

~3! The Weissenberg number could be removed fromproblem by rescaling Eqs.~53!, ~54! and ~46! asL→WeL,t→We2 t, and Ca→Ca/We. However, we retain a We dependence in what follows to keep a fixed physical lengscale for our initial data.

III. LINEAR STABILITY ANALYSIS

We study the linear stability of a circular bubble of rdius R(t), which is perturbed by a small azimuthal distubance, and expands into a non-Newtonian fluid in anbounded Hele–Shaw cell. For simplicity, we impose in thsection a constant mass flux as the driving force, so thatareaS(t) of the bubble satisfiesSt/2p5RRt51. The nondi-mensionalization is chosen so thatR(0)51, Rt(0)51.

The bubble is centered at the origin and the positionR ofthe interfaceG is given by

R~u,t !5R~ t !@11eh~u,t !# r̂ , ~55!

where e!1, andh is the perturbation. Assuming a pureradial flow far from the expanding bubble, the far-fieboundary condition simplifies to

p~r !;2 ln r , as r→`,similarly to the Newtonian case.


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Since h(u,t) can be written as a Fourier series in tazimuthal angleu, and the linearized equations are separawe consider without loss of generality a perturbation ofform h(u,t)5N(t)cosmu, wherem is a wave number, andN(t) is the amplitude of the perturbation. We derive an epression for the growth ratesm5Nt /N for a weakly non-Newtonian fluid, as well as for a general shear-thinning flu

A. Weakly non-Newtonian limit

We start by considering the weakly non-Newtonian limWe2!1, where we can obtain an explicit expression for tgrowth rate. This limit can be attained experimentallychoosing a fluid with a short relaxation timel, or by choos-ing P0 to be small, as is suggested by~19!.

Expanding the viscosity function~40! for JSO in We2

!1 yields

ma~We2uuzu2!512We2 ~12a!uuzu21O~We4!.

All dependency on the specific viscosity function is cotained in the parametera. By introducing in Eq.~41! a smallWe expansion forp5p01We

2p1 andu5u01We2 u1 , inte-

grating and gap-averaging as in Sec. II B, we obtain theterm hierarchy,

u052“p0 , “•u050, ~56!

u152“p123~12a!

20u“p0u2“p0 , “•u150, ~57!

with the boundary conditions

p0uG52Ca21k, p1uG50, ~58!

p0→2 ln r , p1→C, as r→`. ~59!The constantC is determined as a part of the solution.

By solving the perturbation pressurep1 from Eqs.~56!–~59! and using the kinematic boundary condition, we find tinstantaneous growth rate

sm5211mS 11B m21m11D1Ca21 m~12m2!S 11B 2mm11D . ~60!

In this weakly non-Newtonian limit, the non-Newtoniacharacter of the fluid is contained in the single small positparameterB5(3/20)(12a)We2.

In the absence of surface tension (Ca2150), thegrowth-ratesm is always positive and grows essentially liearly with the wave numberm, making the system ill-posedIntroducing surface tension (Ca21.0) stabilizes the largewave numbers, and yields a band of unstable modes at inmediate wave numbers. This is similar to the case of a Ntonian fluid.

The shear-thinning of the fluid has several effects ongrowth rate. The wave number of maximum growth forNewtonian fluid is given by46

mmaxNewt5A11Ca

3. ~61!


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-

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e

From Eq. ~60!, one can obtain an explicit solution for thwave numbermmax with a maximal growth rate in theweakly non-Newtonian case. This expression is rather coplex, but in the limit Ca21!1 it simplifies to

mmax'mmaxNewtS 12 B2 D . ~62!

The maximal growth rate is increased by shear-thinningCa.17. Typical experimental values of the capillary numbare much larger than this.12 Similarly, the critical wave num-ber mc , the maximum wave number for which the growrate is still positive, is shifted towards lower wave numbeas shear-thinning is increased. We find that the relatmc

Newt5)mmaxNewt holds approximately for a shear-thinnin

fluid as well.To summarize, in the weakly non-Newtonian lim

shear-thinning decreases the wave number of maximgrowth, increases the maximum growth rate and tightensband of unstable modes. This suggests an increased seleity of wavelengths in the pattern formation problem. Fshear-thickening fluids, the results are reversed: the grorate for the wave number of maximum growth is decreafor all reasonable values of the capillary number, andwave numbers of maximum and critical growth are icreased.

B. Linear stability: General case

Let us now return to the general case@We5O(1)# of anon-Newtonian fluid whose viscosity is given by Eq.~40!.We do not obtain an explicit expression for the growth rain this case, but can find the growth rates numerically.

In the absence of perturbations, the radiusR(t) of a cir-cular bubble evolves asRt51/R, since we impose a constanmass-flux at infinity. The corresponding velocity fieldgiven by ū(r ,t)5 r̂ /r . We define the pressurep̄ through

p̄r~r !52m̂~We2/r 2!

1

r, p̄„R~ t !…52

1

CaR~ t !,

by expressing the viscosity as a function ofu. The connec-tion of m̂ to the previously defined viscosities is discussedSec. II B.

The perturbation of the interface induces perturbationsthe pressure and velocity fields, which we expand as

u~r ,u,t !5ū~r !1«ũ~r ,u,t !,

p~r ,u,t !5 p̄~r ,t !1« p̃~r ,u,t !.

By expanding Darcy’s law~45! and the boundary condition~53! in «, we obtain for the pressure perturbationp̃ the linearboundary value problem,

“•F m̄22 We2 m̄8“ p̄“ p̄Tm̄2 “ p̃G50, for r .R~ t !, ~63!p̃~R,u!52 p̄r~R! Rh1

1

R Ca~h1huu!, ~64!

p̃→0, as r→`. ~65!Here m̄5m̄(We2 p̄r

2), andm̄8 is defined analogously.


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The separation of variables with p̃(r ,u,t)5P(t,m) f (r ,t,m)cos(mu) leads to the linear two-poinboundary-value problem,

1

r

]

]r F r S m̄22We2 m̄8p̄r2

m̄2 D f r G2 m2r 2m̄ f 50, ~66!f ~r 5R!51, ~67!

f→0, as r→`. ~68!The factorization ofp̃ was chosen so thatf (R)51, whichrequires

P~ t !5S m̂„We2/R~ t !2…1 1R~ t ! Ca~12m2! D N~ t !,for the perturbation pressure to satisfy the boundary cotion ~64!.

Equation~66! approaches its Newtonian counterpartm̄51, m̄850) asr→`, so we expect

f r~r !'2m

rf ~r !, ~69!

to hold for r @1. We impose Eq.~69! at a large, but finiteradius r 5r out, instead of Eq.~68! when solving the two-point boundary value problem~66!–~68! numerically. To-gether with the kinematic boundary condition, this complethe formulation of the problem, and the growth ratesm isgiven by

sm51

R F2Rt2S m̄122 We2 m̄18p̄r2~R!m̄12 D3S m̄11 1R Ca~12m2! D f r~R!G , ~70!

where m̄15m̄(We2 p̄r„R(t)…

2), m̄185m̄8(We2 p̄r„R(t)…

2),and f r is obtained through numerical integration of Eqs.~66!,~67! and ~69!.

C. Discussion

In Fig. 4 we show results of linear stability analysis fthe general shear-thinning fluid. First, in Fig. 4~a! decreasingthe shear-thinning parametera leads to an increased growtrate of the wave number of maximum growth, and its shtoward lower wave numbers, as predicted by the weanon-Newtonian model. Comparing Fig. 4~a! with Fig. 4~b!,we see that an increase of We reduces the range of unswave numbers considerably; one might expect increasedbility of short wavelengths for large We. Another pointnote is that, contrary to Fig. 4~a!, decreasinga does notnecessarily lead to an increase of the growth rate of the mof maximum growth. Still, strongly shear-thinning fluids dshow increased growth rates in the range of We mostly csidered in this paper (We,0.5). The comparison of Figs4~a!,~b! with Figs. 4~c!,~d! show the role of Ca; an increasof Ca makes shorter wavelengths unstable, increasesgrowth rate of unstable wave numbers, and augmentseffect of shear-thinning.


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de

n-

thehe

Increased wavelength selection, resulting from shethinning, as well as the stabilization of short wavelengtencourages the idea that shear-thinning might lead tosuppression of tip-splitting. However, due to the intrinsnonlinearity of the problem, we prefer not to make any denite conclusions based on linear theory alone. Linear stabanalysis does, however, provide us with the basic undstanding of the problem and guidance in performing funonlinear time dependent simulations of an expandbubble. This is the subject of the next section.

IV. NUMERICAL SIMULATION

In this section, we discuss the discretization and numcal solution of the full evolution problem~53!–~54! of a gasbubble expanding into a non-Newtonian fluid. As initial dawe take the interior interfaceG i as a circle perturbed with asingle azimuthal mode, and the outer boundaryGe as acircle.

The kinematic condition~54! can be viewed roughly asan ODE for the boundary of the bubble, with the right haside a complicated and nonlocal function of the boundarythe domain. The numerical solution of~53!–~54! using anexplicit time-integration scheme can then be outlined aslows.

~1! Given the boundary position, solve for the pressure fr~53!.

~2! Find the velocity from the pressure using Darcy’s la~45!.

~3! Find the new boundary position according to~54!.

The full evolution problem is much harder to solve nmerically than the corresponding problem for a Newtonfluid, where the pressure is harmonic. In that case, boundintegral methods coupled with the ‘‘small-scale decompotion’’ ~Hou, Lowengrub and Shelley6! make it possible tosolve the problem efficiently. In the non-Newtonian case,pressure satisfies the nonlinear BVP~53!, and must be solved

FIG. 4. The growth rates for general shear-thinning fluid~see the text!.


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for in the whole domain. Since the problem is driven by tcurvature of the boundaries, high spatial resolution isquired. Further, there is a severe stability constraint ontime-step, leading to a computationally intensive probleFor efficiency, we impose a four-fold symmetry on the initbubble shape, and the solution.

Methods for solving the equation for the pressurepresented in Sec. IV A. Issues related to evolving the bouaries are addressed in Sec. IV B.

A. Pressure solver

The solution of the pressure requires solving a nonlinelliptic PDE in a complicated evolving geometry. Althougsolution methods for problems of this type have been csidered in the literature, for example by Concus,45 they havetypically been for steady state calculations, where efficiewas not as critical as in the present problem.

We use a Lagrangian grid which conforms to the intfaces and moves with the fluid. The fluid domain is mapponto an annulus, where the the nonlinear BVP is discretiusing finite differences, and the resulting system of equatiis solved. We introduce on the annulus the coordinates~z,h!,where z is a ‘‘radial’’ coordinate, andh is a 2p-periodicazimuthal coordinate, so that~z,h! is mapped to the poin(x,y) in the fluid domain according to

H x5x~z,h!,y5y~z,h!, with Jacobian J5F xz yzxh yhG . ~71!The inner boundaryG i corresponds toz51, and the outerboundaryGe to z52. Formulas are modified when expressin the new coordinates, for example,“p(x,y)5J21“̃p(z,h), where“̃ is the gradient with respect to thannular variables. In particular, we solve

N~p!5“•H “pm̄~We2u“pu2!J 50 in V,and

~72!p5 f on the boundary]V,

for a givenf , with N expressed in the annular variables. Wpresent the numerical methods in the original variables (x,y)for clarity, but in practice, our computations are carried oin the annular variables.

We use two iterative methods to solve the BVP~72!.Both iterations reduce the nonlinear problem~72! to a se-quence of linear elliptic BVP’s. The solution of these lineproblems is an issue in itself.

Typically, we use Newton’s method,47 for which a lin-earization, or Fre´chet-derivative, ofN has to be calculatedOne then solves at each iteration the linear, variable cocient elliptic BVP,

N8~pn!pn115N8~pn!pn2N~pn!, in V,pn115 f on ]V,

for the approximationpn11 . Here the Fre´chet-derivativeN8is given by


-e.

l

ed-

r

-

y

-dds

t

r

fi-

N8~p!q5“•H m̄22 We2 m̄8“p“pTm̄2 “qJ.Since this linear problem is solved approximately

practice, the convergence of the resulting scheme is typicless than quadratic.48 Newton’s method can be quite senstive to the choice of the initial guess, and can diverge if tinitial guess is poor. We encounter this problem in the simlations when the interface begins to develop structure. Incase, we switch to the projection-iteration scheme.

The projection-iteration scheme49 for the solution of~72!is defined by

Dpn115Dpn2kN~pn!, in V, pn115 f on ]V,

wherek.0 is a parameter of the method. The iteration cbe shown to converge for any initial guessp0 , provided thatthe parameterk is chosen appropriately. However, sufficieconditions that guarantee convergence for a range ofk ap-pear to be far from tight, and we find that the convergencethe method can be enhanced considerably by choosingvalue of k dynamically. This value may lay outside of thrange of theoretically guaranteed convergence.

Finite differencing of the linear BVP produces a sparbut nonsymmetric linear system of equations for the uknown pressure at the grid points. After a comparison witnumber of iterative schemes,50 we chose to use the biconjugate gradient method with a diagonal preconditioner to sothe linear equations.

B. Moving the interface

As is typical for curvature driven free boundary flowthe computational problem is exceedingly stiff. The size otime-step is strongly constrained by numerical stability. Wfind that the stability constraint is always more restrictithan say resolving the time-scale of the fastest growing linmode. As is known for the Newtonian case,6 and is sug-gested by our weakly non-Newtonian linear stability analy~Sec. III!, the step-sizeDt for an explicit scheme shouldsatisfy

Dt,C~Dsmin!3, ~73!

whereDsmin is the minimum spacing of mesh points on thinterfaces, andC is a constant. We observe and enforce tconstraint in our code, using an empirically determined vafor C. For time-evolution, we use an explicit, two-stagRunge–Kutta method with repeated Richardson extraption. The step-size is sometimes reduced after takinghalf-steps and comparing the error with that obtained aftefull step. An implicit time-stepping scheme would presumably ameliorate the stability constraint, but the implemention of such a scheme in the present context is difficult. Tsevere stability constraint~73! is a primary obstacle to longtime simulation of the evolution problem~53!–~54!.

Remarks. ~1! Number of grid points. The length of theinterface increases by more than two orders of magnit


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during the simulations. A large number of points woulddesirable for resolution, whereas a small number of powould be preferred to alleviate the stability constraint.strike a balance, we begin typically with 64 points on a quter of an interface in the azimuthal direction, with the reslution increased as needed up to 512 points. In the radirection, we use a fixed number of points, typically 100150. A lower resolution leads to a rapid loss of accuracy

~2! Time extrapolation. The pressure solver, in particulaNewton’s method, is quite sensitive to the initial guess. Sithe time steps are relatively small because of the stabconstraint, we can find a good initial guess for the solutionthe next time step by extrapolating the results from two pvious time steps. However, when the bubble develops mstructure, this initial guess might not be good enough tosure the convergence of Newton’s iteration. In this case,switch to the projection-iteration method.

~3! Clustering of grid points. The Lagrangian discretization tends to move grid points away from the tips of tforming fingers, and into the fjords. This clustering is undsirable; the flow near the tips is left underresolved, andfjords are overresolved. The unnecessary clustering of poin the fjords also worsens the stability constraint. Conquently, regridding to equally spaced points in the azimutvariable is performed when needed. An alternative approwould be to impose this dynamically by adding an azimutvelocity component to the velocity of the mesh points soto keep the grid points equally spaced~see Houet al.6!.

~4! Second order accuracy. Our numerical scheme isecond order accurate in time and space. Evaluating thelocity from a pressure field through~45! requires special attention at the boundaries. We find that calculating a findifference approximation to a radial~or z! derivative of thepressure at the boundaries by extrapolating from two intelevels of points, as is commonly done, leads to a nonsmoradial error in the velocity: Although the one-sided appromation used at the boundaries and the centered approxtion used away from the boundaries are both second oaccurate, the one-sided approximation has a much laconstant multiplying the leading order error. To avoid thproblem, we have devised an improved extrapolation schwhich uses three layers of points in the interior to calculthe derivatives at the boundaries. The new scheme yisecond order accuracy, but with a smooth error in the veity field. We have verified the second-order accuracyspace and time of our code by varying the spatial and tporal resolutions, and estimating the numerical errors.

V. DISCUSSION OF THE RESULTS

In this section we discuss the simulational results. Fia study of the influence of shear-thinning fluid behaviorpattern formation is given. A more detailed analysis of tinfluence of nondimensional parameters in the problemlows, as well as some comments on the effect of usingferent effective viscosity functions. Finally, we addressquestion of the dependence of emerging length-scales onflow and fluid characteristics.


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A. The effect of shear-thinning on the dynamics ofthe interface

From experiment,46 theory51–53,7and simulation,6 the ba-sic elements of pattern formation are well understood fogas bubble expanding into a Newtonian fluid in a radHele–Shaw cell. Very roughly, a perturbation of the bubbinterface grows outwardly into an expanding petal. Whthis petal’s radius of curvature exceeds the wavelength ounstable mode, it ‘‘tip-splits’’ into two nascent petals, whicthemselves broaden and split. This repeated process yieldinterface described by a population of branches and fjoand whose evolution is characterized by strong competiamong the branches, with some branches being ‘‘shieldand retracting, and others advancing farther into the fluClearly, if tip-splitting can be suppressed a much differepattern morphology will follow.

The beginnings of the pattern formation scenario foNewtonian fluid are seen in Fig. 5~a!, which shows the simu-lation of an expanding bubble, plotted at equal time intervaThe initial shape is a circle perturbed by anm54 cosinusmode of amplitudea, where a/R050.1. In Fig. 5~a! weobserve the unstable mode growing into a petal~say, aboutu50!, which widens, and then splits into two as its radiuscurvature increases.~Again, much more developed patterncan be computed with higher accuracy using boundary ingral methods.6!

The bubble evolution in a strongly shear-thinning fluidstrikingly different, as is illustrated in Fig. 5~b!. This simu-lation has the same capillary number as the Newtonian si

FIG. 5. The snap-shots of the evolving bubble interface for~a! Newtonianfluid and ~b! strongly shear-thinning fluid~Ca5480 for both simulations,a50.15, We50.15 for shear-thinning one!.

FIG. 6. ~Color! Contour plot of the viscosity of the driven fluid~We50.15, a50.15, Ca5480!.


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FIG. 7. ~a! The viscosity of the fluidalong the interface for a nontip-splitting finger ~S is the arc lengthmeasured from the tip!; ~b! the timeevolution of the viscosity at the tip;~c!the time evolution of the curvature othe tip.

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lation, and again has initial data unstable to the SaffmaTaylor instability. The first and plainest effect of sheathinning is to suppress the tip-splitting of the outwardgrowing petal. As the petal expands outwards, it appearnear a splitting, but then ‘‘refocuses,’’ leaving behind ‘‘sidbranches,’’ and continues to grow outwards. This refocusoccurs twice during the shown course of the evolution, wthe larger~and later! side-branches themselves beginninggrow outwards and giving the impression of a trifurcationthe petal, rather than the bifurcation associated withsplitting in the Newtonian flow. We note that the presencea single mode (m54) at t50 necessarily influences thshape of evolving patterns by imposing a symmetry whichnot present in a physical experiment. By performing adtional simulations, characterized by different modesm andalso by a combination of differentm’s, we have verified thatthe main results~in particular, the phase diagrams of SeV B, and the length-scale results of Sec. V D! are not modi-fied by this assumption.

Figures 6, 7 and 8 provide us with some intuitive undstanding of the source of suppression of tip-splitting. Fig6 shows the viscositym̄a(We

2 u“pu2) in the fluid external tothe bubble, at the final time shown in Fig. 5~b!. As expected,we see that the lowest viscosity appears at the ends opetals. The viscosity increases sharply as one moves afrom the tips, and is highest within the fjords, where itnearly a constant unity~recall that the ‘‘zero shear’’ viscosityis normalized to one!. Figure 7~a! shows the viscosity alongthe bubble interface at several times, includingt50, andshows that the viscosity is always lowest in a fairly localizregion around the tip.


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It is this phenomena that results in the narrowed peobserved from the nonlinear development of the SaffmaTaylor instability: The fluid velocity is locally accentuateby the non-Newtonian effect, which pulls the interface owards at the tips. Thus, a tip remains a tip, and therebyconditions for a lower local viscosity are maintained. Ocourse, this effect is limited by capillarity, which seekslower the length to area ratio, and which is also likely relatto the production of ‘‘side-branches’’ left behind the advaning tip. As is shown in Sec. V C, one can actually induce tformation of fingers~rather than narrowed petals! by a dif-ferent choice of viscosity function, even in the open radgeometry.

More information on the production of side-branchesfound in Figs. 7~b! and 7~c!, which show, respectively, thetime evolution of the viscosity and curvature at a petal~about u50!. In the viscosity, we observed an early timbehavior characterized by only small changes in absovalue, but having fast, irregular oscillations. We find theoscillations curious, and have no explanation for them,cept to note that they persist under refinement in bothspace and time resolution. In particular, even if it is not ovious from Fig. 7~b!, these small oscillations are smootThere are approximately 1500 computational time steps100 data points presented in this oscillatory region. Durthis period, the curvature shows little change. These osctions are followed by a period of monotonic increase inviscosity, as the radius of curvature likewise increases~thepetal spreads!. At somewhere less thant53, the velocity atthe petal tips increases relative to the surrounding partsthe interface~a suppression of tip-splitting!, which leads to

FIG. 8. ~Color! The pressure contours and velocity vectors of the driven fluid at the final time:~a! Newtonian fluid;~b! shear-thinning fluid~the parametersare the same as in Fig. 5!.


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the shedding of a side-branch, and an ensuing decreaseviscosity and radius of curvature. Somewhat later thecrease in tip viscosity slows, the radius of curvature agdecreases, but this is followed by yet another sheddingside-branch, a decrease in the radius of curvature, and arapid decrease in tip viscosity.

It is worth re-emphasizing that the side-branches didoriginate at the sides of the petal, but rather formed neartip during the growth of the radius of the curvature, and wleft behind the propagating tip. This observation points tosimilarity of the pattern formation mechanism in this systeto the formation of dendrites in solidification,54 even thoughour system lacks any imposed directionality.

Finally, Fig. 8 shows the pressure distributions atfinal times for the Newtonian and non-Newtonian simutions, overlaid by their respective velocity vector fields. Wdo not observe flattening of the pressure in the shear-thinfluid, in contrast to what the linear stability analysis in Dacord and Nittmann55 and Nittmann, Daccord and Stanley56

suggests. Based on this predicted flattening, it was con

FIG. 9. The pressure in front of the growing tip as a function of raddistance. The data are taken at nondimensional timet55.0 both for New-tonian and shear-thinning case. The parameters are as in Fig. 5.


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tured that shear-thinning would not influence considerathe instability structure. Figure 9 shows the pressure indriven fluid in front of the growing tip versus radial distancfor a Newtonian and a non-Newtonian fluid. The data psented in Fig. 9 are taken at the same time for Newtonianshear-thinning simulations. The tip of the finger expandinto the shear-thinning fluid has propagated out farther tthe tip of the splitting petal growing into the Newtoniafluid. Despite this, the pressure in front of the tips is quatatively similar in both cases. Away from the tips, the diffeent shapes of the interfaces for the Newtonian and the shthinning fluid modify the pressure distribution considerab~Fig. 8!.

B. Parametric dependence

Here we explore the role which the three dimensionlparameters,a, Ca and We, play in the bubble evolution.

Figures 10–12 summarize the results of simulationsdifferent regions of this parameter space.~In all cases theinitial bubble size is the same. The patterns are enlargedpresentational purposes.! In each of these ‘‘phase diagrams,there is a region~B! of the parameter space where splittingthe finger tips is suppressed. Figure 10 illustrates some oeffects of strong shear-thinning (a50.15). For small We weobserve ‘‘Newtonian’’ patterns—i.e., widening petals th

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FIG. 10. Phase diagram for pattern formation in the strongly shear-thinnfluid, a50.15, for small values of We,0.25. In A one gets wide ‘‘New-tonian’’ petals, inB tip-splitting is suppressed, and inC narrow~relative toA!, but tip-splitting petals are observed.

a
FIG. 11. Phase diagram for fixed C5240. Part~a! shows the results forrather strongly shear-thinning fluid,a,0.25, with small values of We,0.25. InA one gets wide ‘‘Newton-ian’’ fingers, in B tip-splitting is sup-pressed, and inC narrow ~relative toA!, but tip-splitting fingers are ob-served. Part~b! gives the results for awide range ofa and We.

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split ~regionA!. Increasing We brings us to regionB wheresplitting is suppressed. Even higher values of We yield nrowed petals, which tip-split~see also Fig. 11!.

Note that these general observations agree with whaseen by comparing Figs. 1~a! and 1~b! with Figs. 1~c! and1~d!. For a fixed capillary number, the Weissenberg numdecreases by 2.5 in moving from Figs. 1~a! and 1~b! to Figs.1~c! and 1~d!. In this decrease the re-emergence of a mNewtonian pattern is observed~i.e., more tip-splitting!.

An increase of Ca leads to the same consequences aNewtonian fluids: Shorter wavelengths become unstawhich induces tip-splitting~see also Fig. 12!. The increase ofCa leads also to a narrowing of regionB, where tip-splittingis suppressed. In Fig. 10, the size of the window of Wewhich tip-splitting is suppressed is decreased for Ca.500.Also, increasing Ca shifts this window towards lower We.an experiment this would mean that if one uses a higpumping pressure, the fluid should have a shorter relaxatime if nonsplitting tips are to be observed. This effect hasfact, been observed by Buka, Kertesz and Viscek11 in experi-ments with nematic liquid crystals,11 where the driving pres-sure was varied. At low driving pressures, the pattern wNewtonian~corresponding here to small Ca and We—regA in Fig. 10!. At intermediate driving pressures, the tips dnot split ~as in regionB!, and finally, high driving pressure~large Ca and We! resulted again in a tip-splitting phase~asin regionC!. These experimental observations agree remaably well with our results.

Figure 11 shows the phase diagram asa and We arevaried while Ca5240 is fixed.~We cannot explore the regiowhere a,1/9, where the production of slip layers in thdriven fluid might be expected.18! We focus first on Fig.11~a!, wherea and We are rather small, and where thesulting patterns depend quite sensitively on changes ofparameters. As in Fig. 10, a larger We leads to tip-splittand narrow petals, in contrast to the ones produced for sWe. The role of We is to determine which part of the vcosity curve~Fig. 3! governs the viscous response of tfluid. For small values of We, the viscosity in the neighbohood of the tip does not change very much, and the resulpatterns are close to Newtonian~A!. In Figs. 10–12, thepatterns in region~B!, where nonsplitting fingers are ob

FIG. 12. Phase diagram for fixed We50.15, and for a range ofa,0.50,and Ca,1000.


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tained, correspond to the situation where viscosity varconsiderably along the interface—this seems to be a nesary condition for the suppression of tip-splitting. At evehigher values of We, the viscosity at the tips moves tolower plateau of the viscosity curve. One might be tempto explain the observed nonsplitting, narrow, fingers in terof a local capillary number by using the viscosity at the tiinstead of a constant capillary number. However, this wolead to small values of this ‘‘effective’’ Ca, which wouldpredict larger length-scales in region~C! than in region~A!,contrary to our results. The important point here is that thare still very low values of the pressure gradient not odeep in fjords, but also on the finger sides~see Figs. 6 and 8!:The flow still sees the steep part of the viscosity curHigher viscosity for low pressure gradients further supresses the motion of the finger sides, leading to the decrof the resulting length-scales.

Figure 11~b! shows a larger range ofa and We of thephase diagram in 11~a!. Large values ofa and small Weyield Newtonian patterns. On the other hand, smalla andlarge We lead to petals which split, but which are narrowthan those in regionA. In this case, the boundary betweethe regionsA and C is not very sharp; there is a transitioregion for large values ofa and We. An interesting case ia50.40, We50.15, where the effect of shear-thinningstrong enough to prevent splitting~at least at this stage of thgrowth of the bubble!, but not strong enough to producnarrow pointed fingers, such as those formed at smallerues ofa. The inspection of Fig. 11 clearly shows that dcreasinga leads to the decrease of the resulting lengscales. This effect was observed in experiments with wabased muds,15 which were performed in a channel geometrwhere the increase of colloid concentration led to stronshear-thinning~i.e., a decrease ofa!, and to the decrease ofinger width. Similarly, the recent experiments57 with hy-droxypropyl methyl cellulose~HPMC! solutions in a radialHele–Shaw cell showed the decrease of the resulting lenscales with the increase of the concentration of HPMwhich corresponds to stronger shear-thinning.

In Fig. 12, where We50.15, we observe again narropointed fingers for smalla and Ca~regionB!; narrow, split-ting petals for smalla, and larger values of Ca~region A!,and patterns resembling the Newtonian case for larger vaof a ~regionC!.

Figures 10–12 demonstrate that a strongly shethinning fluid is required in order to prevent tip-splitting. Wdo not observe narrow, nonsplitting fingers fora.acrit50.35. Also, larger values of Ca typically lead to tipsplitting. Finally, there is a window of We, where tipsplitting is suppressed: This window is shifted towards lowvalues of We as Ca is increased, becomes narrower for lavalues of a, and disappears completely whena.acrit50.35.

Another effect of tip-splitting is to modify the velocityof the finger tip. Figure 13 shows the tip velocity for twchoices of parameters which lead to tip-splitting~Newtoniananda50.15, We50.15, and Ca5600!, and one choice forwhich tip-splitting is suppressed~a50.15, We50.15 andCa5240!. ~Specifically, the tip velocity is calculated atu


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50, whereu is an azimuthal angle measured from a positx axis.! The velocity of the splitting petal is continuousdecreasing, as the tip of the petal is widening on its routebecome a fjord, whereas the velocity of a nonsplitting fingis roughly a constant@the arrows show the points where thcurvature of the~former! tip changes sign#. This effect hasbeen noted by Meiburg and Homsy58 in a theoretical study ofchannel flow of Newtonian fluid, where the curvature of tfinger tip was held constant artificially. The same study oserved also dendritic modes and side-branches.

Remark:There is an intriguing similarity in our resultto simulations of Newtonian Hele–Shaw flow with anistropic boundary conditions,59,10 where side-branching waalso observed, as well as to local solidification models wanisotropy.60 Further, power-law fluids in a rectangulaHele–Shaw cell were recently the subject of a theoretstudy ~Poiréand Ben Amar20,21!. Experiments with foams,16

where elastic properties might be of importance, and bmiscible5 and immiscible61,57 polymeric liquids, can alsoproduce structures quite similar to ours.

C. Different viscosity models

The form of the developing patterns in Hele–Shaw floof non-Newtonian fluids is very sensitive to a variation of tparameters which define the viscous response of the flSimilar sensitivity has been also widely observed in expments with polymeric fluids and clays~see, e.g., McCloudand Maher,5 Van Damme and Lemaire15!. Consequently, onealso expects that the choice of the non-Newtonian viscomodel would influence considerably the response ofdriven fluid and pattern formation.

While the use of the viscosity~40! is motivated by thefact that it follows from the well-established JSO model fviscoelastic fluids, it is also of interest to study the patteresulting from a different viscosity model. Figure 14 shothe evolution of the interface for a fluid with the effectivviscosity,

m̄a~We2u“pu2!5

11aWe2 u“pu2

11We2 u“pu2, ~74!

where 1/9,a,1. That is, we definem̄a(We2u“pu2) di-

FIG. 13. The velocity of tip propagation, along thex axis. The arrows showthe point where curvature of the tip changes sign. A dashed line showsimulation where fingers do not split.


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rectly, instead of starting with a viscosity functionm andfinding the correspondingm̄ through~43!, ~46!. It is possibleto find the physical viscositym that yields the effective vis-cosity in Eq.~74!; see Appendix A 4.

In Fig. 14 we see that the growing fingers are munarrower and more elongated than the fingers obtainedviously. The oscillatory mode is still present as in Fig. 5~b!,even if rather strongly suppressed. We conjecture thatshear-thinning behavior of the driven fluid alone can leadthe suppression of tip-splitting. This feature is independof the particular model, although the choice of the viscosmodel is important if one is interested in the details of tpattern formation.

D. Emerging length-scales

A typical length-scale (l ) of patterns which develop in aradial Hele–Shaw flow for Newtonian fluids is determinby the capillary number Ca. For large Ca, linear stabilsuggests that a length-scale associated with the initial groof the patterns is given by46

lm52pR

mmaxNewt'2pRA 3Ca, ~75!

where R is the time-dependent radius of the expandibubble, and we have used the expression for the wavenber of maximum growth~61!. Such a scaling is observeapproximately in both simulation and experiment32,62,63 forNewtonian flows, and experimentally for non-Newtoniaflows.56,55 We look into our simulation results for a similalength scaling in shear-thinning liquids.

Figure 15 shows the length-scales emerging fromsimulation of a strongly shear-thinning fluid, as well as tresult of linear stability analysis and a fit of the forA Ca21/2. Here the length-scale was approximated by msuring the radius of curvature at the tip of a growing fingAs we have shown, the curvature can oscillate at the tipso we plot a representative value where the error bars sthe size of the fluctuations. Note that this length-scalemeasured in the strongly nonlinear regime, where the resof linear theory would not be expected to apply. The linestability result is obtained by assuming that the radius ofbubble is equal to its initial size, soR51 in ~75!. In theapparent form of its decrease with Ca, the results of linstability for the shear-thinning liquid is consistent with thfit. And while it is unclear that the simulational length-sca

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FIG. 14. The snap-shots of the evolving bubble interface for a differviscosity model. Herea50.30, We50.30, and Ca5240.


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behaves as Ca21/2, there is a reasonable agreement in mnitude between the simulations and the result of lintheory.

In experiments using a shear-thinning polymeric solutbeing displaced by water,56,55 emerging length-scales havbeen measured as the gap widthb is varied, apparently while

holding fixed the characteristic velocityṘ0 . These resultssuggest that the length-scale scales linearly withb. For New-tonian fluids this observation confirms the result of linestability, since Ca;1/b2 if the characteristic velocity is fixedindependently ofb. However, the flow also depends up thWeissenberg number, We, which is itself a function ofb. So,one should modify both Ca and We accordingly, in orderobtain a realistic comparison with experimental resuThese resulting length-scales measured in this way are gin Fig. 16. Since our simulational results~and the experimental observations56,55! suffer from relatively large uncertaintywe cannot conclude from this that scalingl;b is satisfied.Still there is a good qualitative agreement of the simulatioand the experimental results.

FIG. 15. Capillary number dependence of the length-scale (l ) of the mostunstable modes following from linear stability~dashed!, emergent length-scales from the simulations~dots! and the fit of the formA Ca21/2 ~solid!,whereA is taken from the first data point. Herea50.15 and We50.15.

FIG. 16. The dependence of the length-scale (l ) on plate separationb. Hereb0 is the plate separation which gives Ca5240 and We50.15 att50. Lin-ear stability results~dashed!, simulation results~dots! and fit l;b ~solid! areshown (a50.15). The constantk required for fitting lineL5kb is deter-mined from the data pointb5b0 .


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The driving pressure is another control parameter whinfluence on emerging length-scales can be explored. Inperiments@air displacing water based muds15 and HPMC~polymeric! solutions61,57# increasing the driving pressurtypically decreases the observed length-scales.15,61,57 How-ever, these experimental data are not very precise in expring the length-scale dependence upon the driving pressFigure 17 compares the length-scales obtained fromsimulations to the results of linear stability, and to a fittinfunction of the form l;1/AdP, where dP is the drivingpressure. The motivation for this particular fit arises froanalogy with Newtonian fluids wherel;1/ACa, and Ca;dP ~see also Fig 15!. Here we observe that linear stabilittheory and simulational results agree rather well at smadriving pressures. For larger values ofdP, the length-scalesresulting from linear stability analysis saturate to a constawhile the results of the simulations fitl;1/AdP veryclosely. We hope to verify this prediction experimentally.64

Figure 18 shows the possible source of emerging lengscales for shear-thinning fluids. This figure presentsvariation of viscosity in radial direction in front of the tipm(r ), the viscosity in the fluid adjacent to the interfacem(S)

FIG. 17. The dependence of the length-scales on driving pressuredP. HeredPc is the the driving pressure which gives Ca5240 and We50.15. Linearstability results ~dashed!, simulation results ~dots! and fit l;k(dP/dPc)21/2 ~solid! are shown. The constantk is determined from thedata pointdP5dPc.

FIG. 18. The curvature of the interfacek(S), viscosity in the fluid along theinterfacem(S) and viscosity in the radial directionm(r ) are shown. Hereris radial distance from the tip andS is the arc-length measured from the tip


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and the curvaturek(S) along the interface, wherer is theradial distance from the tip andS is the arc-length along theinterface, measured from the tip. The data are taken fromlast time presented in Fig. 5. It is intriguing that the lengscales on which each of these quantities vary substantare comparable~of course, the curvature of the tip giveapproximately the length-scale on which curvature alonginterface changes sign!. In particular, we observe that thvariation of viscosity in the radial direction compares wwith the variation of viscosity along the interface~Fig. 6!.We conjecture that the length-scale associated with thecosity variation in a driven fluid relates closely to the lengscale of emerging patterns.

VI. CONCLUSION

In this paper we have shown that, under certain assutions, flow in a Hele–Shaw cell of a complex viscoelasfluid simplifies to that of a generalized Newtonian fluid. Funumerical simulations of the two phase~liquid/gas! flowshow that shear-thinning behavior of the driven fluid mofies significantly the morphology of the patterns, relativethose for Newtonian liquids, by suppressing tip-splittinThis can lead to structures of dendritic appearance resbling those occurring in quasistatic solidification. Thesesults are consistent with available experimental results. Fther, we provide morphological phase diagrams that shthe flow and fluid parameters required to suppresssplitting. Lastly, the varying of length-scales emerging froour simulations, as parameters are changed, is in reasonagreement with those observed in experiments. In particuwe observe in our simulations that the typical length-scalethe patterns scales with driving pressure asl;P0

21/2—thisprediction is still to be verified experimentally.

We have ignored in this work several potentially impotant aspects of these flows, that preclude us from havinfuller understanding of these problems. First, better comphension is needed of the flows close to the interface. Cortions to the simple Laplace–Young boundary condition habeen derived for Newtonian fluids~see Homsy65 and the ref-erences therein! that account for the presence of a meniscand of films wetted to the cell plates. This has been dona lesser degree for non-Newtonian fluids~Ro andHomsy65,37!. An elastic response is also likely to be impotant in the neighborhood of the meniscus, and an improunderstanding of the boundary flows would lead to a mquantitative understanding of the coil–stretch transition31

Second, in our scaling we do not allow for an elasticsponse in the bulk fluid, that is, we look at flow only at ordone Weissenberg number. To consider higher Weissennumbers would apparently require solving fully timdependent PDEs for the extra stress in the bulk fluidwould be of interest to find some scaling of the equatiothat would allow this to be done in a tractable way. Lastby our construction of the effective viscosities, we have ctainly not allowed for the possibility of slip-layer formatio~see Kondic, Palffy-Muhoray and Shelley18 and the refer-ences therein!. Though the origin of wall slip is stillcontroversial,23 the JSO equations do formally allow fo


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shearing flows with slip layers though having a nonmontonic stress/rate-of-strain relation~for a,1/9!.

Nonetheless, given that our present~relatively simple!approximation seems to capture many of the salient featof shear-thinning flows—in particular the suppression of tsplitting—there are some fundamental questions to beswered. A central one is understanding at a detailed lehow this system, without any explicit anisotropy, forms figering structures so reminiscent of directional solidificatioObviously, an effective anisotropy is being created nonlearly by the shear-thinning, and is intimately related tosuppression of tip-splitting. Understanding this will requimathematical analyses combined with refined experimeand accurate numerical simulation.

The analysis of Poire´ and Ben Amar20,21on finger selec-tion in weakly shear-thinning, power

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