Kelvin–Helmholtz instability in a Hele-Shaw cellF. Plouraboué and E. J. Hinch Citation: Phys. Fluids 14, 922 (2002); doi: 10.1063/1.1446884 View online: http://dx.doi.org/10.1063/1.1446884 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v14/i3 Published by the American Institute of Physics. Related ArticlesStability of flowing open fluidic channels AIP Advances 3, 022121 (2013) Longitudinal and transverse disturbances in unbounded, gas-fluidized beds Phys. Fluids 25, 023301 (2013) Finite amplitude vibrations of a sharp-edged beam immersed in a viscous fluid near a solid surface J. Appl. Phys. 112, 104907 (2012) Instability in evaporative binary mixtures. II. The effect of Rayleigh convection Phys. Fluids 24, 094102 (2012) Stability of a viscous pinching thread Phys. Fluids 24, 072103 (2012) Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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PHYSICS OF FLUIDS VOLUME 14, NUMBER 3 MARCH 2002

Kelvin–Helmholtz instability in a Hele-Shaw cellF. PlouraboueDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, CambridgeCB3 9EW, United Kingdomand Institut de Me´canique des Fluides de Toulouse, Toulouse, 31400, France

E. J. HinchDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, CambridgeCB3 9EW, United Kingdom

~Received 20 June 2001; accepted 7 December 2001!

A linear stability analysis is presented for the Kelvin–Helmholtz instability in a Hele-Shaw cell, ananalysis based on the Navier–Stokes equation to improve on the previous Euler–Darcy study thatGondret and Rabaud@Phys. Fluids9, 3267 ~1997!# made of their own experiments. ©2002American Institute of Physics.@DOI: 10.1063/1.1446884#

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I. INTRODUCTION

Recently Gondret and Rabaud1 studied the Kelvin–Helmholtz instability in a Hele-Shaw cell. Two verticasheets of glass 1.2 m long were held with a 0.35 mmseparating them. The edges were sealed so as to retain ilower half a viscous silicon oil and in the upper half nitroggas, with a viscosity ratio of a few thousand. Two holesboth ends allowed liquid and gas to be injected at one ena pressure 10% above atmospheric and removed at theend, so achieving a horizontal gas flow of several m s21 andin the same direction a liquid flow of several mm s21. Abovea critical flow, the flat interface was unstable and waves gto a finite amplitude. A small sinusoidal variation in the ijection pressure gave a critical flow for different wave nubers. The reduced Reynolds number for the gas flow, appriate to the nearly unidirectional flow, was about 7 and wvery small for the liquid.

In addition to the experiments, Gondret and Rabaud pformed a simple stability analysis that successfully predicthe onset of instability. Their analysis adopted the normdescription of flow in a Hele-Shaw cell that uses a gaaveraged velocity. To the Darcy equation governing the flthey added the inertia term of Euler, again using onlygap-averaged velocity. Recognizing a little difficulty hereaveraging nonlinear terms, Gondret and Rabaud suggestan Appendix that a correction factor of6

5 should multiply theadvective derivative, which is appropriate for the averagethe product of two velocity fields with parabolic profileacross the gap.

Our purpose in this paper is to replace the gap-averadescription with an asymptotic analysis of the Navier–Stoequation, exploiting the thinness of the gap compared towavelength of the instability, which is of the order of thcapillary length 2pAg/Drg.1 cm, with the surface tensiog, density differenceDr, and gravityg. We start in Sec. IIwith a quick review of the Gondret and Rabaud Euler–Dastability analysis, before proceeding to our Navier–Stoanalysis in Sec. III. Both stability analyses are linear. T

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results are compared to experiments and discussed inIV.

II. EULER–DARCY ANALYSIS

If inertia is ignored, the flow in a Hele-Shaw cell iproportional to the pressure gradient in access of the hystatic balance,

^u&52h2

3m~“p2rg!.

Here ^u& is the velocity averaged across the gap, 2h is thegap thickness,m the viscosity,p the pressure,r the density,andg the gravitational acceleration. Some inertia can betroduced by first rewriting the Darcy-flow equation as a forbalance and then adding the density times the material aceration of the gap-averaged velocity:

rS ]^u&]t

1^u&•“^u& D52¹p1rg23m

h2 ^u&. ~1!

We note that this treatment of the inertial forces is a simpfication because some fluid will be moving faster that the gaverage and so will have larger accelerations. To take saccount of this, Gondret and Rabaud suggested multiplythe ^u&•“^u& term by 6

5, which would be appropriate if thevelocity profile across the gap were parabolic. But the prois not exactly parabolic when inertia forces act differenacross the gap, so we will not make their suggested mocation.

In the base state, let the gas be inz.0 and the liquid inz,0. We use subscriptsg and l on quantities to denote gaand liquid. The base flow is horizontal withmgUg5m lUl

because the two flows are driven by the same pressuredient.

Now consider a small-amplitude perturbation of theterface fromz50 to z5z0eik(x2ct), with real positive wavenumberk and complex wave speedc. We thus start by ex-amining the temporal stability. The small-amplitude appromation requires a small slopekz0!1. The perturbation flow

© 2002 American Institute of Physics

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923Phys. Fluids, Vol. 14, No. 3, March 2002 Kelvin–Helmholtz instability in a Hele-Shaw cell

is potential, and so has a spatial variationeik(x2ct)7kz in z:0. Satisfying the kinematic boundary condition, we fithe perturbation flow

^u&5~61,i !~U2c!kz0eik~x2ct!7kz.

The Euler–Darcy momentum equation~1! gives a perturba-tion pressure,

p57S r~U2c!22 i3m

kh2 ~U2c! D kz0eik~x2ct!7kz.

When imposing the pressure boundary condition, we neeadd to the pressure perturbation the basic hydrostatic psure evaluated at the perturbed surface,2rgz. The jump inpressure across the boundary is set equal to the capipressure2(p/4)gk2z, where we have included the Park anHomsy2 correction factorp/4, which takes into account thvariation of the principal radii of curvature across the gapa liquid that is perfectly wetting. Gondret and Rabaud didinclude thisp/4 factor. Thus, we obtain the dispersion retion

S 2rg~Ug2c!2k1 i3

h2 mg~Ug2c!2rggD2S r l~Ul2c!2k2 i

3

h2 m l~Ul2c!2r lgD52p

4gk2. ~2!

In the experimental conditions of Gondret and Rabaudsmall viscosity and density ratio,mg!m l and rg!r l withrg /r l@mg

2/m l2 so thatrgUg

2@r lUl2, and small reduced Rey

nolds number in the liquid flow,r lUlkh2/m l!1, the disper-sion relation~2! reduces to

rgUg2k2 i

3

h2 m l~2Ul2c!2r lg5p

4gk2, ~3!

where we have usedmgUg5m lUl . From this simplified dis-persion relation, we extract the phase velocitycr and thegrowth ratekci ,

cr.2Ul , kci.k2h2

3m l

S rgUg22

r lg1p

4gk2

kD . ~4!

It is useful for later developments to understand the splified physics in the experimental conditions. Becauseliquid is more viscous and has a higher inertia, it movslowly. The gas thus flows past an effectively stationary luid surface atz5z0eikx. In order to accelerate over the peaand to decelerate into the troughs of the perturbed interfthere must be a low pressure at the peaks and a high prein the troughs. The magnitude of this suction pressure atpeaks is given by Bernoulli asrgUg

2kz0 . This destabilizingsuction is to be offset against the stabilizing effects ofliquid hydrostatic pressurer lgz0 plus a capillary pressure(p/4)gk2z0 . The stabilizing effects win at all wave numbeif the gas velocity is below a critical value: stable ifrgUg

2

,Ar lgpg.Above the critical gas velocity, there is a range of u

stable wave numbers with a net suction pressure at the pin the liquid,

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The pressure gradientkpl in the Darcy flow of the liquidinduces an upward velocity of the peakskciz0

5h2kpl /3m l , with the result~4! for the growth rate.The phase velocity of the wave comes from the visco

pressure droppv53mgUgz0 /h2 across a peak in the gaflow past the stationary perturbed liquid surface. The prsure gradient kpv drives a perturbation Darcy flowh2kpv/3m l5Ulkz0 in the liquid, which propagates the inteface atUl relative to the base liquid velocityUl ; hence theresult ~4! for the phase velocity.

We note that our explanation of the mechanism ofKelvin–Helmholtz instability in a Hele-Shaw cell is nofraught with the dangers of using Bernoulli suction to eplain the original Kelvin–Helmholtz instability between twinviscid fluids. For inviscid fluids, time reversibility impliethe existence of two modes, one growing and one decayand a correct argument must explain both. Obviously sucat the peak explains the unstable mode. Less obviouslexplains the stable mode. The suction force produces, ininviscid fluid, an upward acceleration. An upward acceletion of a downward moving peak is a decaying mode.

III. NAVIER–STOKES ANALYSIS

A. Formulation

We now move on from the Gondret and Rabaud EuleDarcy analysis that used a gap-averaged velocity^u&. In thissection we calculate the variation of the velocity acrossgap using the full Navier–Stokes equation. The linearizNavier–Stokes equation for the perturbation velocu(x,y,z,t) and pressurep(x,y,z,t) is

rS ]u

]t1U"“u1u"“UD52“p1m¹2u, ~5!

where the base flow has a parabolic profile across the g

U5U3

2 S 12y2

h2D ~1,0,0!.

We again use subscriptsg andl for the gas inz.0 and liquidin z,0.

From our examination of the simplified physics in thexperimental conditions of Gondret and Rabaud, we takegas to flow over an effectively stationary perturbed liqusurfacez5z0eikx. The wavelength of the instability is muclarger than the thickness of the gap. As in lubrication theand in boundary layer theory, the pressure is asymptoticconstant across the thin gap and the pressure-driven gasis in the plane of the walls. In these circumstances, we fithat a possible solution of the linearized Navier–Stokequation~5! is a potential pressure field,

p5rgUg2kz0eikx2kzP,

with nondimensional pressure amplitudeP, along with a flowin the direction of the pressure gradient,

u5~1,0,i !Ugkz0eikx2kzf ~y/h!,

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924 Phys. Fluids, Vol. 14, No. 3, March 2002 F. Plouraboue and E. J. Hinch

with a variation across the gap given by the nondimensioprofile f. We define the nondimensional coordinate acrossgaph5y/h. The above ansatz for the pressure and velosatisfy the linearized Navier–Stokes equation~5! if the ve-locity profile f and pressure amplitudeP satisfy

32 ~12h2! f 52P1

1

i Ref 9, ~6!

with reduced Reynolds number Re5rgUgkh2/mg for the gasflow. The no-slip boundary condition on the sidewalls is

f 50, at h561.

The kinematic boundary condition on the effectively stionary liquid surface does not match the velocity profilef.Realizing that there will be an adjustment zone near theniscus with a height equal to the width of the gap, we neonly require the total normal component of the mass-flomatch. Hence we normalize the solution to have net flowunity in the nondimensionalization,

E0

1

f dh51.

The dynamic boundary condition equates the capillpressure to the jump between the pressure in the Darcyof the liquid, including a liquid hydrostatic contribution athe perturbed surface, and the dynamic gas pressurescribed by the pressure amplitudeP,

rgUg2kP1 i

3m l

h2 ~Ul2c!1r lg52p

4gk2,

corresponding to the simplified dispersion relation~3! in theEuler–Darcy analysis. Solving for the phase velocitycr andthe growth ratekci , we find

cr5Ul~11 13RePi ,! kci5

k2h2

3m l

S 2rgUg2Pr2

r lg1p

4gk2

kD ,

~7!

whereP5Pr1 iPi and where we have usedm lUl5mgUg tosimplify the phase velocity.

B. Numerical solution

The velocity profilef was found by integrating Eq.~6!numerically with a fourth-order Runge–Kutta scheme. A slution was first found to the non-normalized problem, settP51 and shooting fromh50 with symmetric conditionf 8(0)50 and a guess for the complex value off (0) that wasadjusted by linear extrapolation to givef (1)50. The solu-tion pair f andP were then both divided by*0

1 f dh in orderto normalize the solution.

Results for the velocity profilef (h) at Reynolds num-bers Re51, 3, 10, 30, 100 are given in Fig. 1. The real aimaginary parts are plotted separately. At low Reynolds nubers, Re<3, the velocity profile is parabolic and mostly reaAt high Reynolds numbers, Re>30, the velocity decreases ithe center of the gap and peaks near the walls.

Results for the pressure amplitudeP as a function of theReynolds number Re are given in Fig. 2. At low Reyno

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numbers, the pressure amplitude varies asP;3i /Re. At highReynolds numbers, the pressure amplitude is complexdecreases.

C. Low Reynolds numbers

The low Reynolds number behavior can be studiedmaking an asymptotic expansion,

f ; f 01Ref 1 ,

P;Re21 P01P1 .

The governing equation~6!, boundary conditions, and normalization are all expanded. At leading order we find tPoiseuille profile of inertialess Darcy flow,

f 05 32 ~12h2!, P053i .

The inertia-induced correction is

f 15 i 3280 ~25133h2235h417h6!, P152 54

35 .

The numerical results in Fig. 1 for the velocity profif (h) are consistent with these asymptotic results. At Rnolds numbers 1 and 3, the real part is parabolic in shwith a centreline value of 1.5. The imaginary part, with vaishing integral from the normalization constraint, is positinear the walls and negative in the center, with a centervalue of20.054 Re. Figure 2 for the pressure amplitude aexhibits the asymptotic behavior ofPi;3/Re andPr;2 54

35

521.543.At low Reynolds numbers we have found the press

amplitude from the Navier–Stokes equation is

P;3i Re212 5435. ~8!

The corresponding nondimensional expression fromEuler–Darcy analysis of Sec. II, valid for all Reynolds numbers, is

P;3i Re2121.

Thus, the destabilizing inertial effects, the real part ofP, aremore than 50% larger than estimated by the Euler–Daapproximation. The Gondret and Rabaud correction facto65 is seen to be insufficient, at least at low Reynolds numb

We have already noted that the value of the Reynonumber in the experiments for the flow of gas was 7. Tnumerical results in Fig. 2 show that the low Reynolds nuber asymptotics will provide a reasonable estimate forpressure amplitude at this less-than-small value, to witerrors of 15%.

D. High Reynolds numbers

At high Reynolds numbers, the gas flow has a differebehavior in the center of the gap and near the walls. Incenter, the pressure perturbation accelerates the gas asadvected along by the mean flow,

f ;2P/ 32~12h2!.

Thus, the velocity profile increases away from the centerlisee Fig. 1 for Re530 and 100, because moving with slowmean flow off the centerline it has longer to be accelerat

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searn-

925Phys. Fluids, Vol. 14, No. 3, March 2002 Kelvin–Helmholtz instability in a Hele-Shaw cell

FIG. 1. The velocity profilesf (h) at various Reynoldsnumbers, Re51, 3, 10, 30, and 100. In the real partf r ,the centerline valuef r(0) decreases as the Reynoldnumber increases. The value of the imaginary part nthe wall h51 increases as the Reynolds number icreases.

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r,

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W

he

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eedre-se-

thghhef itsthe

Near the walls viscosity becomes important. A balanbetween viscosity and advection is possible within a bouary layer of thicknessd5Re21/3, where to match with theinterior f 5O(P/d). To exhibit this boundary layer behaviowe have plotted in Fig. 3,f rd/P as a function of (12h)/d, at Re530, 100, 300, and 1000. We see that with threscaling the peaks in the velocity occur at the same posin the boundary layer and with essentially the same amtude.

The normalization integral has a leading-order contribtion *0

12dP/3(12h)dh; 13P ln d, along with O(1) correc-

tions from the boundary layer and the central region.have evaluated theseO(1) contributions numerically as

E0

1

f dh;2P~ 19 ln Re10.4021 i0.175!.

Equating this to unity gives the asymptotic behavior of tpressure amplitude at large Reynolds numbers,

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e-

ni-

-

e

P;21/~ 19 ln Re10.4021 i0.175!. ~9!

These asymptotic results are plotted in Fig. 2.At high Reynolds numbers the pressure amplitudeP de-

creases slowly with increasing Reynolds number. Thiscrease has its origin in the slow speed of the base flow nto the walls. Taking longer to be advected at this slow spthrough a wavelength, a smaller pressure perturbation isquired to achieve the same velocity perturbation. The conquences of the smaller pressure amplitudeP in dispersionrelation ~7! are a smaller phase velocity and lower growrate: it is as if the base flow is moving slower. Thus at hiReynolds numbers the instability is controlled not by tgap-averaged value of the base velocity nor the average osquare, giving greater weight to the peaks and leading toGondret and Rabaud suggested factor of6

5, but is more influ-enced by the slower moving parts of the base flow.

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all

926 Phys. Fluids, Vol. 14, No. 3, March 2002 F. Plouraboue and E. J. Hinch

FIG. 2. The pressure amplitudeP as a function of theReynolds number Re. The dashed curves are the smReynolds number asymptotic result~8!, while the dot-ted curves are the high Reynolds number result~9!.

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IV. COMPARISON WITH EXPERIMENTS

A. Threshold of instability

In this section we compare our theoretical predictiowith the experimental observations of Gondret and Raba1

We start with the minimal gas velocity for the appearancethe instability, and in the following section look at the phavelocity.

In their experiments, Gondret and Rabaud added a soscillatory perturbation to the pressure of the liquid asentered the device. Varying the frequency of this excitatithey could measure the threshold velocity of the gas forstability at different observed wave numbers. A more caremeasurement of the decay length of stable waves suggethat the true critical velocity might be 2% higher than thereported threshold velocities. We plot in Fig. 4 their thresold gas velocitiesUg has a function of the observed wavnumber k. We have nondimensionalized these observ

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quantities using the natural capillary lengthl * and the gasvelocity V* whose Bernoulli pressure has the capillary presure of this length,

l * 5A pg

4r lgand V* 5A pg

2rgl *.

In the experiments,g52.06 1022 N m21, r l5965 kg m23,g59.81 m s22, and rg51.28 kg m23, so that l * 51.3131023 m andV* 54.40 m s21.

Using these scalings, the predictions of the Euler–Daanalysis of Sec. II for the threshold gas velocity maywritten as

S Ug

V*D 2

511~kl* !2

2kl*, ~10!

which is also plotted in Fig. 4, along with our results frothe Navier–Stokes analysis of Sec. III,

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f

r

927Phys. Fluids, Vol. 14, No. 3, March 2002 Kelvin–Helmholtz instability in a Hele-Shaw cell

FIG. 3. The boundary layer scaling othe velocity profile at high Reynoldsnumbers. The curves increasing fromthe bottom are for Reynolds numbeRe530, 100, 300, and 1000.

a

rit

orntidthv

antion-tothes.

hentionchat

newca-.e.,

S Ug

V*D 2

511~kl* !2

2kl*

1

2Pr~Re!, ~11!

where Pr is the real part of the pressure amplitude. It isfunction of the reduced Reynolds number Re5Ugkh2/n, seeFig. 2, and so varies with the gas velocityUg and the wavenumberk. In the range of interest,Pr'21.5. Being largerthan unity, the Navier–Stokes analysis predicts a lower ccal gas velocity, lower by about 1/A1.5, i.e., about 82% ofthe Euler–Darcy value.

Figure 4 shows that the improved Navier–Stokes thedoes not produce improved predictions of the experimeresults. In view of these poor predictions, we have consered a number of differences between our theory andexperimental conditions. First we have assumed that the

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cosity and the density of the liquid are very much larger ththe values of the gas, so that the liquid appears to be staary as far as the gas flow is concern. It is not difficultgeneralize our approach to the Navier–Stokes problem tocase of two fluids with arbitrary viscosity and density ratioWe find that if the experiments had used air and water tthere would have been a 5% overestimate by the assumpof infinity ratios. The experiments, however, used a mumore viscous liquid, and for this larger ratio we find ththere would be no detectable change in Fig. 4.

Another simplification of our theory is the assumptiothat the gas flow is incompressible. With gas velocities a fpercent the speed of sound, one would anticipate modifitions of the gasdynamics by the square of that ratio, i

y

--e

FIG. 4. The critical velocity of the gasUg for the appearance of an instabilitas a function of the wave numberk,nondimensionalized by the capillarylengthl * and the associated gas velocity V* . The points are the experimental results of Gondret and Rabaud. Thdashed curve is the prediction~10! ofthe Euler–Darcy analysis, while thecontinuous curve is the prediction~11!of the Navier–Stokes analysis.

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928 Phys. Fluids, Vol. 14, No. 3, March 2002 F. Plouraboue and E. J. Hinch

totally negligible. There is, however, an effect of the copressibility on the base state of the gas, because thepressure was typically 10% above the atmospheric pressThis means that the density of the gas near the inlet is tcally 10% higher. This in turn reduces the value of our vlocity scaleV* by 5%. The experimental conditions nearthe inlet are relevant, because it is there that one obsewhether the waves are growing or decaying.

The 10% increase in the base pressure near the inlea second effect. The gas velocity was measured in theperiments by applying Darcy’s law to the basic pressure gdient, which was assumed to be linear and given by theference between the pressure at the inlet and outlet divby the distance between them. A linear pressure gradgives a gas velocity that is constant along the channeconstant gas velocity does not give a constant masswhen the density varies. In order to have a constant mflux, the square of the pressure must vary linearly downchannel. This means that the gas velocities are 5% lonear the inlet and 5% higher near the outlet. Henceshould reduce the observed gas velocityUg by 5%. As thevelocity scaleV* should also be reduced by 5%, there wbe no effect on the plotted ratioUg /V* . The fact that theratio Ug /V* does not change value is an unforeseen advtage of making the nondimensional plot.

Our theoretical study also assumed that the basewas horizontal, i.e., we ignored the radial flow out of tsource and into the sink at the ends of the channel. Tradial flow introduces an extra pressure drop, rougequivalent to extending the length of the channel for ehole by (H/2p)ln(H/2pR), whereH is the height of the gasflow and R is the radius of of the hole. In the experimenH55 cm andR53.5 mm, so we find that each hole adds 6mm to the length of the 1.2 m channel, i.e., a correctionjust over 1%. As the experimental velocities were calculafrom an assumed Darcy flow, this 1% extra pressure dreduced the observed velocities by 1%, which is in the crect direction but too small in magnitude. We have not ajusted the experimental data for this small effect.

A further assumption in our theory that affects tthreshold velocity is the assumption that the basic flow haparabolic profile. At the reduced Reynolds number of57, the base flow only becomes fully established after owavelength, i.e., about 1 cm. We note, however, that inexperiments a splitter plate of 10 cm divides the streamgas and liquid before they meet. Hence, at the separatiothe plates and at the velocities used, the base flow shoulfully established at the place where the instability wasserved.

Finally, our analysis made a long wave approximatiokh!1. This resulted in the neglect of the slightly differeflows near the meniscus, modifications of the flow thattend away from the meniscus a distance of about a gap thness. In the experiments the value of this small paramwaskh50.13, which may not be so small. We are currentrying to develop a theory for this correction, but at this timwe have no estimate of the magnitude of the correction.

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B. Phase velocity

The phase velocity was measured by Gondret aRabaud1 to becr51 mm s21. We discuss the phase velocitin a nondimensionalized form by scaling with the velocitythe liquid when the gas has the velocityV* , i.e., scaled withVl* 5mgV* /m l . In the experiments,V* 54.4 m s21, mg

51.7531025 Pa s and m l50.1 Pa s, so that Vl*50.77 mm s21. Thus, the nondimensional measure of tobserved velocity iscr /Vl* 51.30.

The Euler–Darcy theory of Sec. II predicts a phaselocity that is a little higher, cr /Ul52.00, i.e., cr /Vl*52.00. The Navier–Stokes analysis of Sec. III predicts atReynolds number Re55.84, corresponding to the marginwaves withUg /V* 50.816 andkl* 50.95, a pressure amplitude Pi50.637, which gives a phase velocity by~7!, cr /Ul

52.24, i.e., cr /Vl* 51.83. The two theories thus significantly overestimate the phase velocity of the marginastable waves. The omissions of the theories discussed inprevious section would obviously lead to some minor adjuments to these values, but would not give the significcorrection required.

One feature of the our simplified treatment of the mencus does, however, have a significant effect on the phvelocity without modifying the gas velocity at the threshoof the instability. The two theories apply a pressure conditat the interface with a jump of the capillary pressu(2p/4)gk2z. This takes into account the curvature of thinterface in thexz plane. There is, however, a much largcurvature 1/h in the perpendiculary direction across the gapThe large capillary pressureg/h associated with this curvature does not alter the analysis because it is constant, ipendent of the displacement of the interface. A problarises, however, from relatively small corrections to thlarge constant pressure jump, because the corrections caof the same size as the term (2p/4)gk2z, and they can alsovary with the displacement of the interface.

The liquid used in the experiments wets the sidewaperfectly, and so to leading order the meniscus adoptsemicircular shape across the gap. As the wave crests prgate along the channel, the meniscus rises and falls. Asmeniscus falls it leaves a thin film on the sidewalls of thicness 1.34h(m lun /g)2/3, whereun is the normal velocity ofthe interface. The presence of these thin films increaslightly the curvature of the interface, contributing toslightly higher jump in the capillary pressure. Viscous dispation in the thin films leads to a similar additional contbution to the pressure drop. This problem was studiedBretherton.3 Adopting his results for a tube to our channehe found the pressure drop across the falling meniscus i

g

h F113.723S m lun

g D 2/3G ,and the pressure drop across a meniscus rising over aleft behind by the meniscus descending at the same spe

g

h F120.976S m lun

g D 2/3G .

cense or copyright; see http://pof.aip.org/about/rights_and_permissions

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929Phys. Fluids, Vol. 14, No. 3, March 2002 Kelvin–Helmholtz instability in a Hele-Shaw cell

The difference between these two pressures must be addour jump in the capillary pressure (2p/4)gk2z.

Now for the marginally stable surface displacemenz5z0 cosk(x2crt), the normal velocity of the interface isun

5z0kcr sink(x2crt), i.e., out of phase with the displacemenTo use in the linear stability analysis the new additional prsure drop, which is nonlinear, we break sin2/3 into its Fouriercomponents, in particular, sin2/351.0712 sin1higher har-monics. Adding the extra out-of-phase pressure jump toboundary condition, we find that the expression for the phvelocity ~7! becomes

cr

Ul511 1

3 Re Pi

21.67S m lUl

g D 21/3

~kh!2/3S h

z0D 1/3S cr

UlD 2/3

.

The expression for the growth rate is unaffected, becaususes the component of the pressure in phase with the sudisplacement, or, in more physical terms, the destabilizBernoulli suction only sees a stationary liquid surface.

Applying our new expression for the phase velocitythe experiments, we takeUl50.63 mm s21, so that the cap-illary numberm lUl /g53.031023. We do not know the am-plitude of the observed waves, but looking at the figuresGondret and Rabaud,1 we takez051 mm. The equation forthe phase velocity then becomes

cr

Ul11.63S cr

UlD 2/3

52.23,

with solutioncr /Ul50.82, i.e.,cr /Vl* 50.67. This value ismuch smaller than the observed value, whereas the origtheory gave much larger. While there are considerable un

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tainties in our analysis in this section, for example, ourlection of the amplitude of the wave, it is clear that the effeis significant. Some additional experiments could usefullymade measuring the pressure jump across a meniscusHele-Shaw cell that oscillates up and down with similar aplitudes to those of the propagating waves.

After their first paper on the subject, Gondret, ErMeignin, and Rabaud4 went on to study the transition fromconvective instability to absolute instability by following thevolution of an initial impulse. They found the transitionabsolute instability occurred in their experiments whengas velocity was 15% higher than the threshold for anystability. A numerical study of the Euler–Darcy dispersiorelation, now including thep/4 Park and Homsy factor andfactor of 6

5 multiplying the nonlinear terms, predicted thtransition to absolute instability occurred 7% above tthreshold for any instability. We have repeated their analyfor our Navier–Stokes dispersion relation and find thattransition occurs at 22% above threshold. We can now nthat these calculations of the transitions to absolute instaity involve the out-of-phase pressure termPi and so wouldbe modified significantly by including Bretherton’s extpressure jump. Further, the nonlinear dependence of the epressure jump is not the normal quadratic dependence.

1P. Gondret and M. Rabaud, ‘‘Shear instability of two-fluid parallel flowa Hele-Shaw cell,’’ Phys. Fluids9, 3267~1997!.

2C.-W. Park and G. M. Homsy, ‘‘Two-phase displacement in Hele Shcells: Theory,’’ J. Fluid Mech.139, 291 ~1984!.

3F. P. Bretherton, ‘‘The motion of long bubbles in tubes,’’ J. Fluid Mec10, 166 ~1961!.

4P. Gondret, P. Ern, L. Meignin, and M. Rabaud, ‘‘Experimental evidenof a nonlinear transition from convective to absolute instability,’’ PhyRev. Lett.82, 1442~1999!.

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