J. Fluid Mech. (2013), . 714, pp. doi:10.1017/jfm.2012.476

J. Fluid Mech. (2013), vol. 714, pp. 258–282. c© Cambridge University Press 2013 258doi:10.1017/jfm.2012.476

Response of a two-dimensional liquid foam toair injection: swelling rate, fingering and fracture

Imen Ben Salem, Isabelle Cantat and Benjamin Dollet†

Institut de Physique de Rennes, UMR 6251 CNRS/Universite de Rennes 1, Campus Beaulieu,Batiment 11A, 35042 Rennes CEDEX, France

(Received 29 March 2012; revised 29 June 2012; accepted 24 September 2012)

The response of a two-dimensional liquid foam to a localized air injectionis investigated experimentally and theoretically. The experiments show a richphenomenology, with two essentially distinct behaviours, depending on the injectionconditions. At low flux, the injected air forms a central bubble that grows inside thefoam and induces plastic rearrangements, without film rupture. This ‘pure swelling’regime is reminiscent of ductile fracture. In this regime, the central bubble showsfingering patterns beyond a certain velocity. The dependence among the swelling rate,the injection overpressure and the other control parameters, namely cell gap, bubblesize and foam area, is captured by a simple balance between the pressure drop andbubble/wall friction under a radial assumption. Fingering is successfully modelled bythe linear stability analysis of an azimuthal perturbation of the radial model; yieldstress becomes an important parameter to determine the finger width. At high injectionrate, films are broken and narrow cracks form rapidly through the foam, reminiscentof brittle fracture. Criteria for the transition between ductile and brittle behaviours areinvestigated, both at the local and global scales.

Key words: complex fluids, fingering instability, foams

1. IntroductionThe interest in foam or emulsion flows in confined geometry, initially strongly

motivated by the oil industry (Hirasaki & Lawson 1985), has been recently rekindledby various other applications in digital microfluidics (lab-on-a-chip technology(Marmottant & Raven 2009; Baroud, Gallaire & Dangla 2010)), in biology (lungsurfactant flows (Goerke 1998)) and in civil engineering (soil remediation (Chowdiahet al. 1998)). The internal structure of foams is easy to observe, at least in twodimensions, and their mechanical properties at small scale are usually better knownthan the ones of other complex fluids. For these reasons, foam rheology is an activefield of research (Hohler & Cohen-Addad 2005), and foams are also consideredas good model systems for a broader class of complex fluids, all sharing similarrheological properties (Larson 1999). A classical set-up for confined flow studies is theHele-Shaw cell, made of two glass plates separated by a small gap.

In this paper, we study a foam radially pushed in a Hele-Shaw cell by injectionof gas through a small hole in the bottom plate. The gap between the plates is

† Email address for correspondence: [email protected]

mailto:[email protected]

Response of a two-dimensional liquid foam to air injection 259

small enough so that the bubbles arrange as a monolayer, the so-called quasi-two-dimensional foam (Vaz & Cox 2005). We will call our system a two-dimensional foamfor simplicity, but reminding readers that it should not be confused with foams thatare molecularly thin in one direction, as found in Langmuir monolayers (Dennin &Knobler 1997; Mann & Primak 1999; Courty et al. 2003), nor with idealized systemswhich are translationally invariant in one direction (Princen 1983; Kraynik & Hansen1987). Depending on the foam properties and on the imposed pressure (or injectionrate), two different behaviours are observed. At small pressure, the central bubblegrowth induces bubble rearrangements around it. This transformation conserves thetotal number of bubbles. After a short axisymmetric transient, fingering occurs. Athigher pressure, soap films burst around the large bubble and the foam is partiallydestroyed. Narrow cracks propagate quickly from the injection point to the externalfoam boundary.

Injection of air into a liquid foam belongs to the vast class of experimentswhere a viscous fluid is pushed by a less viscous one in a porous medium, orin a Hele-Shaw cell. As a general feature, these experiments display rich patternformation and fingering (Bensimon et al. 1986; Homsy 1987). This process is thewell-known Saffman–Taylor instability (Saffman & Taylor 1958). For a Newtonianfluid, the pattern properties depend on the fluid viscosity, which tends to narrow thefingers, and on the surface tension, which tends to broaden them (Paterson 1981).Non-Newtonian fluids lead to a large class of possible fingering behaviours (Lemaireet al. 1991; Lindner et al. 2002), not yet entirely elucidated. For shear-thinning fluids,several authors have analytically and numerically quantified the differences of fingermorphology, especially the finger width, with the Newtonian case (Wilson 1990; BenAmar 1995; Bonn et al. 1995; Kondic, Palffy-Muhoray & Shelley 1996; Kondic,Shelley & Palffy-Muhoray 1998; Ben Amar & Corvera Poire 1999). Yield-stress fluidshave been modelled with a tensorial Herschel–Bulkley constitutive law by Coussot(1999), and this theory successfully predicts the fingering patterns observed in agel (Lindner, Coussot & Bonn 2000). However, liquid foams, another yield-stressfluid with a priori qualitatively similar macroscopic rheological properties, behavedifferently (Park & Durian 1994; Lindner et al. 2000). Actually, in a Hele-Shaw cell,liquid foams easily slip on the walls. The velocity profile in a plane transverse to theplate is essentially a plug flow outside the thin lubrication layer localized betweenthe first layer of bubbles and the wall. For two-dimensional foams, made of a singlelayer of bubbles, the pressure drop is mainly governed by the flow in this lubricationlayer. This peculiarity, making the model developed by Coussot not directly applicableto foams, was first noticed by Park & Durian (1994) and Lindner et al. (2000) forthree-dimensional foams, i.e. with bubbles much smaller than the gap between theplates. The description of the foam flow thus requires taking into account the specificexternal viscous friction between the sample and the wall, and the usual yield-stressfluid properties that govern the friction between bubbles.

The transition from viscous fingering to a fracture regime at higher pressurehas been reported in several complex fluids which show both an elastic responseand viscous dissipation, especially dense granular suspensions (Lemaire et al. 1991;Chevalier et al. 2009; Holtzman & Juanes 2010; Sandnes et al. 2011). These studiessuggest that a transition from fingering and fracture arises at high deformation rateand/or high concentration, when viscous dissipation cannot relax elastic stress. Inother contexts, but related to this argument, transitions from a viscous behaviour andbrittle fracture above a given deformation rate have been shown to happen in bubblerafts under tension (Arciniaga, Kuo & Dennin 2011), during the pinch-off of drops

260 I. Ben Salem, I. Cantat and B. Dollet

of viscoelastic fluids (Tabuteau et al. 2009) or when studying the motion of objectsthrough micellar solutions (Gladden & Belmonte 2007). Foams in Hele-Shaw cells areespecially well suited to address this question, because the distinction between the tworegimes, usually relying on the observation of the macroscopic pattern, is based onthe direct observation of the foam structure and on the presence of film bursting. Thistransition has already been observed in two-dimensional foam in a straight channelgeometry (Hilgenfeldt, Arif & Tsai 2008; Arif, Tsai & Hilgenfeldt 2010, 2012).

This paper presents new experimental and theoretical results on several aspects. Inthe swelling regime, i.e. in the absence of film bursting, we provide the first detailedinvestigation of the swelling rate as a function of the injection overpressure, the cellgap, the bubble size and the initial foam area. All experimental dependences aresuccessfully captured by a continuum model of the foam based on a radial assumptionthat takes into account the specific law that characterizes the friction between thefoam and the plate. We significantly extend the experimental characterization of thefingering process by Park & Durian (1994) and Lindner et al. (2000) by varying amuch larger set of control parameters. We discuss the similarities and differences withthe results by Park & Durian (1994). We develop an original model, based on a linearstability analysis, that adds the foam/wall friction into the model of Coussot (1999).We show that this external viscosity dominates and we discuss the corrections inducedby the yield stress to the fingering. This model predicts a characteristic finger widthin good agreement with our experimental observation. Finally, we determine the cross-over between swelling and rupture in the whole phase diagram, corresponding to ourmultiple control parameters (bubble size, liquid fraction, pressure, foam size, cell gapthickness). Our theoretical prediction for the cross-over relies on a local geometricalcriterion for film bursts, and differs from the predictions proposed in Hilgenfeldt et al.(2008) and Arif et al. (2010, 2012), for a reason that we identify.

2. Materials and methods2.1. Foam production and characterization

The flow cell is made of two horizontal circular glass plates of radius D = 30 cmseparated by three small spacers of controlled thickness e varying between 0.4 and1.5 mm. The top plate is simply set on the spacers and the lateral boundary of thecell is open. Beside the flow cell, a flat open container is entirely filled with a freshfoaming solution of sodium dodecyl sulphate (SDS) at concentration 10 g l−1 andglycerol 10 % wt in ultra-pure water. The viscosity of this solution is η = 1.2 mPa s,its surface tension is σ = 36.8 ± 0.3 mN m−1, and its surface modulus ED quantifyingits interfacial viscoelasticity (Denkov et al. 2005, 2009b) is below 1 mN m−1 (Dollet& Cantat 2010). The foam is prepared by blowing air at a flux controlled by a syringepump (PHD2000, Harvard Apparatus) through a needle in the container. Bubbling wasalways performed at a rate low enough to ensure that bubbles are monodisperse. Foamaccumulates at the free surface of the liquid. We take the top plate, and place it justabove the container, in contact with the foam. Most of the foam is then transferredto the plate after withdrawal. The top plate is then set on the spacers, and the foamis squeezed between the two plates. No significant bubble coalescence nor burstingwas observed during this squeezing stage, and the bubbles quickly relaxed towards anequilibrium, non-prestressed state. The bubble size and the spacer height are chosenin order to obtain a single monolayer of bubbles. The foaming solution reservoiris weighed before and after the foam collection by the top plate; the difference isa measure of the mass of the foam, from which we deduce the total volume of


(a) (b)

(c) (d )

FIGURE 1. Response of a two-dimensional liquid foam to a localized air injection, withincreasing flux: (a) quasi-static response, with a central bubble of regular, elliptic-like shape.Notice the elastic stress concentration at the bubble tip; (b) intermediate regime, where thecentral bubble grows fingers with a characteristic length scale; (c,d) high-speed regime, wherethe injected air has broken a series of soap films to form a narrow crack (c) or several cracksby a branching process (d). All scale bars are 1 cm. Control parameters: (a) P0 = 11 Pa,L = 0.21 cm, Atot = 111 cm2, e = 0.1 cm; (b) P0 = 1.1 kPa, L = 0.22 cm, Atot = 100 cm2,e = 0.06 cm; (c) P0 = 1.4 kPa, L = 0.24 cm, Atot = 117 cm2, e = 0.06 cm; (d) P0 = 60 kPa,L = 0.13 cm, Atot = 344 cm2, e = 0.06 cm. Movies of these four cases are available assupplementary information at journals.cambridge.org/flm.

solution Vsol in the collected foam. The relative uncertainty in this quantity comesfrom the precision of the weighing balance, and is at most 4 %.

The foam structure is lit with a circular neon tube put between the bottom plateand a black plate, and is recorded with a high-speed camera (Photron APX-RS). Thebubble edges appear in white on a black background (figure 1). Image segmentationallows us to identify all the bubbles in our system, at each time, and to measure theirindividual area, seen from above. The foam, i.e. the assembly of all the bubbles exceptthe large bubble produced by the gas injection in the centre, is made of N bubbles ofaverage area S. We define Atot = NS as the initial area occupied by the foam, beforeair injection. If there is no film rupture, this parameter is constant with time, as wellas N and S. The bubble size is quantified by its equivalent radius L =

√S/π. The

area of the central bubble is denoted by A(t) and is directly extracted from the image

http://www.journals.cambridge.org/flm



























processing without any assumption on its shape. The equivalent radius of the centralbubble is defined by R=√A/π. The liquid fraction is then obtained as φ` = Vsol/eAtot .In most of our experiments, it is close to 3 %. However, the relevant control parameterfor the theory is not directly the liquid fraction but the radius of curvature a of thePlateau borders. We determine the total length of Plateau borders in contact with oneplate L by the relation L = πNL. The capillary suction σ/a is of the order of 150 Pain our experiments, whereas the hydrostatic pressure difference between the top andbottom plates does not exceed 10 Pa, hence gravity effects are negligible. Top andbottom Plateau borders thus have the same shape, and their radius is s = (2 − π/2)a2

(Cantat et al. 2010). Hence, the total volume of the Plateau borders in contact with theplates is 2L s= 2π(2− π/2)NLa2. Moreover, each Plateau border perpendicular to theplate has a length e and a cross-section equal to (

√3 − π/2)a2 (Cantat et al. 2010);

there are on average six of them per bubble, and each of them is shared between threebubbles. Hence their total volume is (2

√3 − π)Nea2. Overall, the volume of liquid

contained in the foam obeys: Vsol = Na2[π(4− π)L+ (2√3− π)e], hence

a=√

Vsol

N[π(4− π)L+ (2√3− π)e] . (2.1)

This control parameter is measured a posteriori with relative uncertainty of 2 %.

2.2. Foam flowThe bottom plate is drilled in the middle with a 2 mm hole to allow gas injection (seefigure 1). We perform our experiments at controlled pressure; for this, we use a largebuffer tank of gas that we connect to the cell to blow gas at the centre of the foam. Anelectrovalve allows a rapid opening of this gas reservoir, within 5 ms. As the typicalpressure is of the order of a few thousand Pascals or less, the gas can be considered asincompressible. Hence, in the absence of film rupture, the flux is directly related to thearea variation of the central bubble: Q(t) = e × dA/dt. If film rupture occurs startingfrom the central bubble, it affects its area and the flux is determined from the externalarea instead. Such a measurement of the flux is possible as long as the central bubbleremains completely surrounded by the foam.

The overpressure in the gas is measured by a differential pressure transducer(DP103, Validyne Engineering Corp.), with an accuracy of 5 Pa. The tube lengthbetween the cell and the pressure sensor must be larger than the distance between thebottom plate and the black plate, and with a small enough diameter in order not tospoil the images. Given these constraints, we cannot neglect the pressure drop Pcor(Q)along this tube. It has been calibrated in the absence of foam for each value of the gasflux in the experimental range. The pressure P0 in the central bubble is then given byP0 = Psens−Pcor(Q) with Psens the value measured by the pressure sensor. The referencepressure is the external pressure. The pressure variations during each experiment, theuncertainty in Pcor(Q), and the accuracy of the pressure sensor combine to give theerror bars in figure 4 below.

3. Experimental results3.1. Phenomenology

The response of a foam to a localized air injection was observed to be very differentas the injection flux varies. At very low flux, the central bubble grows by movingbubbles in the foam apart by a succession of plastic rearrangements, which are


(d)

lc

lc

lc

(a) (b) (c)

FIGURE 2. (Colour online) Superposition of the inner boundary of the foam at differenttimes, showing the development of the different patterns: (a) quasi-static regime, with analmost elliptical central bubble (time interval 5 min between successive locations of theboundary); (b) intermediate regime, where fingering occurs (time interval 0.24 s); and (c)high-speed regime, with film rupture (time interval 4 ms). In the fingering regime, each fingerwhich tip flattens then splits into two new fingers; we measure its width at the onset ofsplitting, and identify it as the distance lc discussed in the text, here for three cases. (d) Therectangular box in (b) is zoomed, to show the formation of a new finger (time interval 0.08 s).

well separated in space and time: the frequency of rearrangements is much lowerthan the inverse of the relaxation time following a rearrangement. It is thereforea quasi-static, elasto-plastic regime, and viscous dissipation between rearrangementsis negligible. The shape of the central bubble is very regular, and resembles anellipse (figure 1a), which inflates with time (figure 2a). At higher flux, the centralbubble growth still proceeds by plastic rearrangements, but it shows fingers with awell-defined characteristic wavelength (figures 1b and 2b). As we will explain in thepaper, this marked change of shape is a signature of the domination of dissipationover elastic effects. We call these low-flux regimes ‘pure swelling’. Above a certainthreshold, films begin to burst; if the flux is large enough, bursting events percolatethrough the foam, and a narrow single crack is opened between the injection point andthe external boundary of the foam (figures 1c and 2c), whereas the rest of the foamremains unaffected by the localized rupture of films. At very high flux, multiple cracksdevelop and branching occurs (figure 1d).

This overall scenario is reminiscent of ductile fracture in the slow regime, and ofbrittle fracture in the fast regime (Freund 1990; Bouchbinder, Fineberg & Marder2010). Among this rich phenomenology, we will focus in this paper only on theintermediate regime characterized by ductile fracture and fingering, and on the onsetof the brittle regime. We will show that such a regime is dominated by bubble/wallfriction. The very slow, quasi-static regime and the very fast one will be discussed infuture studies, since their underlying physics is essentially different.


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FIGURE 3. (Colour online) Examples of the time evolution of the central bubble area A(t)in the pure swelling regime. •, P0 = 403 Pa, L = 0.33 cm, Atot = 95 cm2, e = 0.06 cm,a = 0.019 cm; �, P0 = 630 Pa, L = 0.15 cm, Atot = 110 cm2, e = 0.04 cm, a = 0.021 cm;N, P0 = 750 Pa, L = 0.13 cm, Atot = 136 cm2, e = 0.06 cm, a = 0.022 cm. The dashed linesare affine fit, with t = 0 corresponding to the beginning of swelling. Inset: time evolution ofthe rescaled area A(t)/Atot , fitted by (4.18).

3.2. Pressure value in the pure swelling regimeWe consider here the foams flowing without rupture. They can be unambiguouslydistinguished from the cases where ruptures occur, by direct observation or from thedetermination of the total number of bubbles as a function of time; N(t) is constantduring the pure swelling process, whereas it decreases when film ruptures occur. Atimposed pressure P0, the flux is almost constant during the whole experiment, asshown on figure 3: experiments at controlled flux or at controlled pressure are thusequivalent. This particular behaviour is predicted by the theory, for the first stage ofswelling, when the central bubble is still quite circular, and small in comparison withthe foam area (see § 4). For each experiment, we thus plot the central bubble area as afunction of time and we measure the slope dA/dt.

Our experimental control parameters are the bubble size L, the initial foam areaAtot , the cell gap e and the radius of curvature of the Plateau borders a. The latterparameter is difficult to vary significantly: large values induce drainage and a Plateauborder size difference between the upper and lower plates, and very small values leadto very fragile foams. We thus choose to keep this parameter as constant as possibleand for figure 4, corresponding to our four control parameters, a= 0.024± 0.002 cm.

We first fix all control parameters to study the variation of P0 as a function of dA/dt(figure 4a): it shows that P0 increases at increasing dA/dt. Fitting with a power-lawdependence yields P0 ∝ (dA/dt)0.7±0.1 at fixed L, e and Atot . We have checked thatchoosing another set of control parameters gives a similar exponent.

We then turn to the dependence on each of the three control parameters L, Atot ande, fixing the other two, and also keeping either P0 or dA/dt as constant as possible.


2.8

2.9

3.0

3.1

3.2

2.3 2.4 2.5 2.6 2.72.0

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–1.4 –1.2 –1.0

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3.0

–0.8 –0.6 –0.4

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1.7

1.8

1.5 2.0 2.5

(a) (b)

(c) (d )

FIGURE 4. (Colour online) (a,b,c) Pressure in the central bubble, for three different sets ofparameters. (a) P0 versus dA/dt, other parameters fixed: e = 0.06 cm, L = 0.132 ± 0.001 cm,Atot = 149 ± 14 cm2; (b) P0 versus e, other parameters fixed: dA/dt = 264 ± 11 cm2 s−1,L = 0.160 ± 0.004 cm, Atot = 73 ± 13 cm2; (c) P0 versus L, other parameters fixed:dA/dt = 200 ± 17 cm2 s−1, e = 0.06 cm, Atot = 173 ± 9 cm2. (d) Growth rate of the centralbubble as a function of the foam area, other parameters fixed: P0 = 294 ± 2 Pa, e = 0.06 cm,L= 0.182± 0.002 cm. The solid lines are power laws of exponent, respectively, (a) 0.7± 0.1,(b) −1.1± 0.2, (c) −0.51± 0.05 and (d) −0.21± 0.02. The dotted lines indicate the marginalexponents.

In practice, it is difficult to vary these control parameters independently and over awide range, but our measurements clearly show the following trends: at fixed dA/dtthe pressure is an increasing function of the cell gap (figure 4b) and of the bubble size(figure 4c). To study the influence of the foam area, P0 was kept fixed, and dA/dt isshown to decrease with increasing foam area (figure 4d). Fitting these curves by powerlaws gives: P0 ∝ e−1.1±0.2 at fixed L, Atot and dA/dt; P0 ∝ L−0.51±0.05 at fixed e, Atot anddA/dt; and dA/dt ∝ A−0.21±0.02

tot at fixed L, e and P0.

3.3. Transition between swelling and ruptureDepending on the various parameter values, the foam may evolve by swelling orrupture. Around the transition, we observed that these two different processes arisesuccessively: some films burst around the central bubble and then a pure swelling isobserved, as depicted on figure 5. We therefore define three cases: (i) pure swelling;(ii) bursts and then swelling; (iii) bursts percolating through the foam, inducing a crackreaching the foam boundary. In order to identify the transition between these different


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345

6 45

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45

6 45

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56 4

56

FIGURE 5. (Colour online) Detail of a foam at different times, first experiencing rupture(t = 0–0.012 s), then swelling (t > 0.012 s). Bubbles 1, 2 and 3 burst, then the central bubblegrowth proceeds by plastic rearrangements, its tip moving bubbles 4, 5 and 6 apart.

regimes, we explored the parameter space identified in the previous paragraph in threeindependent planes, as shown in figure 6.

Following the idea proposed by Arif et al. (2010), the rupture threshold has beencharacterized by a local criterion, in terms of the maximal velocity vmax that a film cansustain without bursting. In the case of ruptures followed by swelling, we measuredthe film velocity around the central bubble, on the images following the last burst,for the films that were touching the last bursting bubble. This velocity is the mostaccurate estimate of vmax that we can make. This local parameter appears to be verywell correlated to the ratio a/L, independently of the other parameters, as shown onfigure 7. Here, unlike figure 4, we varied a, between 0.019 cm and 0.033 cm, and Lvaried between 0.15 cm and 0.52 cm.

3.4. FingeringThe fingering process is difficult to quantify precisely. The tip splitting onset ischaracterized by the change of convexity of the finger tip. On the image just before theonset of tip splitting, the finger has a flat tip and thus a relatively well-defined widththat we use as a definition for lc, as depicted on figure 2(b,d). The discrete nature ofthe underlying network of bubbles leads to a significant uncertainty in the instant ofonset of tip splitting, and we estimate a relative uncertainty in this measure of 20 %.We did not obtain sufficient statistics to perform the same parametric study as in § 3.2.On the other hand, on given experiments, we observed that lc increases as the centralbubble area increases, as shown in figure 8.

4. Modelling and comparison with experiments4.1. Forces acting on the foam

When the foam is pushed in the Hele-Shaw cell, it is subjected to several forces:the pressure gradient ∇p, the external viscous friction between the bubbles and theconfining plates, and the elastic and viscous stresses within the foam, τ . Although


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FIGURE 6. (Colour online) Phase diagram of the foam behaviour for three parameter sets.◦, ruptures; O, ruptures followed by a swelling phase; M, pure swelling. (a) P0 versus L,fixed parameters: e = 0.06 cm, N = 291; (b) P0 versus L, fixed parameters: e = 0.06 cm,Atot = 160.39 cm2; (c) P0 versus e, fixed parameters: L = 0.153 cm, Atot = 93.24 cm2. Thesolid lines are the predictions of (4.22).

the total number of bubbles is not large (of order 103), we model the foam as acontinuum. One important consequence of this assumption is that, in the modelling ofthe instability leading to fingering (§ 4.4), wavelengths smaller than a bubble diameterwill be non-physical. Under such an assumption, conservation of momentum yields:

ρfdvdt=−∇p+ f ′v +∇ · τ , (4.1)

where ρf is the foam density and f ′v is a volumetric force accounting for the externalfriction: f ′v = Fv/e with Fv the external friction per unit area of the foam.

4.1.1. External friction: viscous force on the platesSince SDS gives ‘mobile’ interfaces (Denkov et al. 2005, 2009a), the external

friction arises from the transition regions between the Plateau borders in contact


20

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0.05 0.10 0.15 0.200

80

FIGURE 7. (Colour online) Maximal velocity of the films vmax , recorded just after the end ofthe rupture phase, near the last burst, as a function of a/L. The solid line is the predictiongiven by (4.20). The error bars on vmax come from the uncertainty in the location of the films,and the error bars on a/L come from the uncertainty in a and the standard deviation of thebubble size.

–0.1

0.1

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log R (cm)–0.2 0.6

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l c (

cm)

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0.4

FIGURE 8. (Colour online) Finger width as a function of the central bubble effectiveradius R = √A/π. Other parameters: �, e = 0.06 cm, L = 0.148 ± 0.011 cm, dA/dt =234 ± 21 cm2 s−1 (fitted by the solid line, of slope 0.50 ± 0.15); ◦, e = 0.06 cm, L =0.334± 0.004 cm, dA/dt = 196± 1 cm2 s−1 (fitted by the dashed line, of slope 0.44± 0.08).

with the plates and the thin lubrication films between the bubbles and the plates.It is quantified as a force per unit length of Plateau border, fv. It is not the onlysituation where viscous friction occurs in a transition region of length ` between a thinwetting film of thickness h and a ‘macroscopic’ volume of liquid, of typical radius ofcurvature a. This also occurs in the motion of a single bubble through a tube, wherea is the tube radius (Bretherton 1961), and in the context of coating of flat plateswithdrawn from a bath, where a is the capillary length (Landau & Levich 1942). In allthese situations, viscous friction and capillary effects are the two significant effects inthe transition regions, and this balance yields the following scalings: h ≈ aCa2/3, and`≈ aCa1/3 under the assumption Ca1/3� 1, where Ca= ηv/σ is the capillary number(de Gennes, Brochard-Wyart & Quere 2002). Consequently, the friction force per unitlength in the transition region scales as: fv ≈ ηv`/h ≈ σCa2/3 and, importantly, does


not depend on a (the Plateau border radius in our geometry). Hence, for a Plateauborder moving in normal motion:

fv = KσCam, (4.2)

with m = 2/3 and K a dimensionless constant. Since ED � σ (see § 2.1), we neglectthe deviation of surface tension from its equilibrium value.

Bretherton’s calculation yields K = 4.7; however, experiments and simulations ondry foams in various geometries have lead to different values (Ratulowski & Chang1989; Wong, Radke & Morris 1995; Terriac, Etrillard & Cantat 2006; Raufaste, Foulon& Dollet 2009). Systematic measurements at varying liquid fraction showed that fvdepends on the ratio a/L as K = κ (a/L)ma , which is not predicted by the Brethertontheory, and remains up to now theoretically unexplained. We found empiricallyK = 6.8 (a/L)−0.5 in Raufaste et al. (2009), or K = 9 (a/L)−0.4 in Terriac et al. (2006).We will use the empirical correction proposed in Raufaste et al. (2009): κ = 6.8, andma =−0.5.

Finally, for Plateau borders with arbitrary orientations, only the projected Plateauborder length in the direction perpendicular to the velocity should be considered(Cantat, Kern & Delannay 2004). We will use the total projected length of Plateauborder per unit area (including the top and bottom plate contributions) for anhexagonal bubble, averaged over all possible orientations. This length is αh/L, withαh ' 1.31 (Raufaste et al. 2009). An average force per unit foam area can be deduced:

Fv = αh

Lfv =−αh

Lκ(a

L

)maσCamv

v. (4.3)

4.1.2. Internal foam stressFoams store elastic stress when bubbles deform. The corresponding elastic stress

scales as σ/L; hence, from (4.1), if R is the typical size of the central bubble, elasticityovercomes external friction below a velocity v′ the order of magnitude of which obeysσ/LR= Fv/e.

Let us estimate v′ based on our experimental parameters. From figure 7, a/L ≈ 0.1,hence from (4.3), Fv ≈ 29σCa2/3/L. Therefore, v′ = σ (e/29R)3/2 /η. Taking e≈ 1 mm,we compute v′ ≈ 6 mm s−1 for R ≈ 1 cm at the beginning of the blowing process,and v′ decreases rapidly with increasing R. Hence if v � v′, the foam respondselasto-plastically (figure 1a), whereas if v� v′, the foam response is dissipative. Allexperiments presented in the current paper are in the dissipative regime.

The internal viscous stress equals, for a three-dimensional foam of liquid fractionφ` between 2 % and 20 % and for sufficiently soluble surfactants (Denkov et al.2009a): τv = 1.16σCa′0.47

(1− φ`)5/6 (0.26− φ`)0.1 φ−0.5` /L, where Ca′ = ηγL/σ with

γ the deformation rate within the foam: here, γ ≈ v/R. Since we are looking at anorder of magnitude of the viscous stress, we neglect the influence of confinement andwe apply directly the three-dimensional formula; hence with our typical liquid fractionφ` = 3 %, τv ≈ 6σCa′0.47

/L, whence the ratio of this viscous stress to the externalfriction: (τv/R)/(Fv/e) ≈ 0.2eCa′0.47

/RCa2/3. At a velocity of 10 cm s−1, typical inour experiments, this ratio equals 0.04, and it depends very weakly on the velocity.Hence, viscous stress is generally negligible in our experiments, and dissipative effectscome mainly from bubble/wall friction. This is a major difference with the model ofCoussot (1999) of a radial flow of a yield stress fluid in a Hele-Shaw cell, wheredissipation was only coming from viscous stresses arising from velocity gradientswithin the cell.


4.1.3. InertiaInertia scales as ρfv

2/R in (4.1), hence the ratio of inertia to external friction is:(ρfv

2/R)/(Fv/e) ≈ 6 × 10−3 for the highest velocity reached, v = 0.8 m s−1 (figure 7).Therefore, inertia remains negligible.

In summary, in our experiments, the foam motion results mostly from a balancebetween a pressure gradient and the external friction. However, the other contributionsmay become significant for a different foam or another flow parameter range, and wewill show that yield stress is relevant to understand fingering (§ 4.4). For the sakeof generality we thus write the flow equations for the general case. Comparison withexperiments will be performed using the relevant simplified cases.

4.2. Pressure dropIn this section, we determine the pressure in the central bubble assuming axisymmetry.The notation X(0) refers to this approximation. The correction induced by fingering willbe the subject of § 4.4.

4.2.1. General equationsAssuming that the motion is purely radial, p(0) is function of r only, and v(0) =

v(0)r (r)er in polar coordinates (r, θ), with er the unit radial vector. Axisymmetry andincompressibility entirely determine the velocity field: v(0) = qer/r, with q = Q/2πe.The external viscous force (4.3) can be written:

Fv =−eξvm−1v, (4.4)

with:

ξ = αh

eLσκ(a

L

)ma( ησ

)m. (4.5)

Therefore, f ′v(0) =−ξ (q/r)m er.

Stresses in foams are well captured by the Herschel–Bulkley constitutive law, eitherin its scalar version for simple shear (Hohler & Cohen-Addad 2005), or in its tensorialversion for more complex flow geometries (Cheddadi et al. 2011). The tensorialHerschel–Bulkley law is written (Coussot 1999):

τ ={

τc√Tr(D2)/2

+ 2nKp

[Tr(D2)/2](1−n)/2

}D, (4.6)

with τc the yield stress, D = (∇v+t∇v)/2 the rate of deformation tensor, and Tr thetrace. In polar coordinates, the components of the rate of deformation tensor are:

Drr = ∂vr

∂r, (4.7)

Drθ = 12

[1r

(∂vr

∂θ− vθ

)+ ∂vθ∂r

], (4.8)

Dθθ = 1r

(∂vθ

∂θ+ vr

), (4.9)

hence D(0)rr =−D(0)

θθ =−q/r2 and D(0)rθ = 0. Inserting in (4.6), we get the stress at order

0: τ (0)rr =−τ (0)θθ =−τc − Kp (2q/r2)n, and τ (0)rθ = 0.


The r-component of (4.1) is written:

0=−∂p

∂r+ f ′v,r +

∂τrr

∂r+ 1

r

∂τrθ

∂θ+ τrr − τθθ

r, (4.10)

hence at order 0:

∂p(0)

∂r=−ξ

(q

r

)m − 2τc

r− 2(1− n)Kp

(2q)n

r2n+1. (4.11)

The external boundary condition for the pressure is p(0)(r = rext) = 0. At the innerinterface r = R, there is a Laplace pressure jump between the central bubble and thefoam, due to surface tension: P0 − p(0)(r = R) = σC , with C the interface curvature.We assume that the inner interface is smooth at the scale of the individual bubblesin the foam, hence in the radial geometry: C (0) = 1/R. This approximation, whichconsists of neglecting the actual curvature of the bubbles of the foam in contact withthe central bubble induces an offset in the pressure, but it will enable us to accountcorrectly for stabilizing effects in the subsequent analysis of fingering. Similarly, at theexternal boundary of the foam, the boundary condition is p(0)(r = rext) = σ/rext . Withthese boundary conditions, we get the pressure field:

p(0)(r)= ξqm

1− m(r1−m

ext − r1−m)+ 2τc lnrextr

+ 1− n

nKp (2q)n

(1

r2n− 1

r2next

)+ σ

rext, (4.12)

whence the prediction of the pressure in the central bubble:

P0 = ξqm

1− m(r1−m

ext − R1−m)+ 2τc lnrextR

+ 1− n

n2nKpqn

(1

R2n− 1

r2next

)+ σ

(1R+ 1

rext

). (4.13)

4.2.2. Comparison with experimental resultsWe now compare the predictions of the axisymmetric model to the experiments.

According to § 4.1.2, we need retain only external friction in the right-hand side of(4.13), hence:

P0 = ξqm

m− 1(R1−m − r1−m

ext ), (4.14)

or, for the case m= 1:

P0 = ξq ln(rext/R). (4.15)

By definition, q = A/2π with A = dA/dt the rate of variation of the centralbubble area, hence A = 2π [(1− m)P0/ξ(r1−m

ext − R1−m)]1/m. Moreover, R = √A/π andA + Atot = πr2

ext , hence we get an ordinary differential equation (ODE) for the timeevolution of A:

A= 2π(1+m)/2m

{(1− m)P0

ξ [(A+ Atot)(1−m)/2−A(1−m)/2]

}1/m

. (4.16)


2000 4000

Exp

erim

enta

l P0

(Pa)

0

500

1000

1500

6000

Theoretical P0 (Pa)

FIGURE 9. (Colour online) Experimental value of the pressure, as a function of its theoreticalvalue given by (4.14). The best linear fit gives a slope 0.25± 0.01.

In the early stages of blowing, A� Atot , hence:

A' 2π(1+m)/2m

[(1− m)P0

ξA(1−m)/2tot

]1/m

= β. (4.17)

Therefore, we predict that the central bubble area grows linearly with time, which isobserved well in experiments (figure 3). Furthermore, expanding (4.16) to the nextorder in A/Atot , gives A' β[1+ (A/Atot)

(1−m)/2 /m], hence with m= 2/3:

A' βt

[1+ 9

7

(βt

Atot

)1/6]. (4.18)

This correction explains the slight increase of A observed in the experiments (inset offigure 3).

Inserting (4.5) into (4.17), we get A ∝ P1/m0 A−(1−m)/2m

tot e1/mL(1+ma)/m, hence withm= 2/3 and ma =−0.5:

P0 ∝ A2/3A1/6tot e−1L−0.5. (4.19)

The four scalings are found to hold in experiments, which lead to P0 ∝A0.7±0.1e−1.1±0.2L−0.51±0.05 at fixed A and A ∝ A−0.21±0.02

tot at fixed e, L and P0 (figure 4),the latter exponent being in agreement with A ∝ A−1/4

tot from (4.19). This fully confirmsthat in our range of control parameters, the foam flow is dominated by the frictionbetween bubbles and plates.

As a further comparison between experiments and theory, we plot the experimentalpressure data versus the prediction (4.14) in figure 9. It shows that the theoreticalprefactor has been overestimated by a factor 4. This mainly arises from the fact thatthe axisymmetry is rapidly lost, as discussed in § 3.4: as only a part of the foam ismoving, the pressure is overestimated. Overall, this comparison shows that the scalingsof the pressure with respect to the other parameters are independent of the precisegeometry of the swelling process, in contrast to its quantitative value.

4.3. Rupture thresholdFor confined foam, the rupture of a film between the plates occurs when the film,moving at a certain velocity, is unable to pull its Plateau borders at the same velocity.


As the maximal force per unit length the film can exert is 2σ , the rupture threshold isgiven by 2σ = fv(v) (Dollet & Cantat 2010). From (4.2), this leads to a very simplecriterion of maximal velocity:

vrup =(

2K

)1/mσ

η. (4.20)

This law is plotted on figure 7, with K = 6.8 (a/L)−0.5 and m= 2/3 (see § 4.1.1). It isindeed an upper bound of the highest velocities experimentally measured.

Another criterion for maximal velocity before rupture has been proposed inequation (4.2) in Hilgenfeldt et al. (2008), which is, using our notation, vrup =(σ/η) [4 a/(4.70L)]3/2. However, it is based on the following expression for thepressure jump across one straight film moving perpendicularly to itself (i.e. a straightfilm whose normal vector is parallel to the velocity), equation (3.1) in Hilgenfeldtet al. (2008): δP = 2 × 4.70σCa2/3/a, which we believe is wrong when a < e/2, acondition fulfilled in our experiments and in the ones of Arif et al. (2010). Indeed,a force balance on such a film of surface e × L, expressing that the pressure jumpcompensates viscous friction, can be written: eLδP = 2Lfv = 2KLσCa2/3 using (4.2);hence, the geometric factor in δP is 1/e, and not 1/a. In a more recent paper, Arifet al. (2012) develop a new model for the transition from the rupture to the swellingprocess, based on the fact that the fracture velocity cannot be smaller than the soundvelocity in the foam.

In the framework of the axisymmetric model, the criterion (4.20) can be expressedin terms of a maximum pressure in the central bubble; requiring that A = 2πRvrup in(4.17), we get:

P0,rup/swell = 6σαh

eLR2/3

(Atot

π

)1/6

. (4.21)

The pressure threshold thus increases with the central bubble radius. So, for a givenrange of pressure, P0 can be above the threshold at the beginning of the experimentand below the threshold once the central bubble area reaches a certain value: this isin agreement with the intermediate regime depicted on figure 6, where film bursting isfollowed by plastic rearrangements. No rupture are expected to occur if P0 < P0,rup/swell

from the beginning, i.e. when R= L, which leads to the more restrictive threshold:

P0,rup/swell = 6σαh

eL−1/3

(Atot

π

)1/6

. (4.22)

This pressure is a prediction for the transition between the pure swelling andthe rupture/swelling processes, and is plotted on the three graphs of figure 6.The prediction is still in fair agreement with the experiments, although it is lessaccurate than for the critical local velocity. This may be because the model relieson an assumption of axisymmetry. Another potential cause of discrepancy betweenexperiment and model may be the existence of a short transient at the valve opening,which lasts about 5 ms, see § 2.2. If the pressure and the flux reach their steady valueafter 5 ms, the central bubble radius is already a few L. Replacing R by L between(4.21) and (4.22) is thus questionable.


4.4. Fingering

4.4.1. Field equationsWe now investigate the response of the foam to a small azimuthal perturbation, such

that the internal boundary of the foam is now defined by the equation F(r, θ; t) = 0with F = r − R(t) − ε(θ; t) where ε = εeikθ+ωt, with k an integer > 2. The goal of thisstability analysis is to predict the dependence of the growth rate ω on the integer k andon the parameters of the foam.

Following Coussot (1999), we set: vr = v(0)r + v(1)r with v(1)r = εφ(r)/r, andp= p(0) + p(1) with p(1) = εf . Mass conservation implies:

0=∇ ·v= 1r

∂

∂r(rvr)+ 1

r

∂vθ

∂θ, (4.23)

hence at order ε1, since dε/dθ = ikε, v(1)θ =−εφ′/(ik).Similarly, the equation of momentum conservation has an r-component given by

(4.10), and the following θ -component:

0=−1r

∂p

∂θ+ f ′v,θ +

∂τθr

∂r+ 1

r

∂τθθ

∂θ+ 2

τrθ

r. (4.24)

Now f ′v =−ξvm−1v and, at order ε1:

vm−1 = (v2r + v2

θ )(m−1)/2 '

(q

r

)m−1[

1+ (m− 1)εφ

q

]. (4.25)

Hence f ′v = f ′v(0) + f ′v

(1) with:

f ′v(1) =−εξ

(q

r

)m−1(

mφ

rer − φ

′

ikeθ

). (4.26)

Next, ∂v(1)r /∂r = ε(φ′/r − φ/r2) and ∂v(1)r /∂θ =−εφ′′/ik, hence from (4.9):

D(1)rr =−D(1)

θθ = ε(φ′

r− φ

r2

), (4.27)

D(1)rθ =

12ε

(−φ

′′

ik+ φ′

ikr+ ikφ

r2

). (4.28)

From these components, lengthy but straightforward computations yield:

τ (1)rr =−τ (1)θθ = 2nKpqn−1nεr1−2n

(φ′ − φ

r

), (4.29)

τ(1)rθ = iε

τc

2q

[1+ Kp

q

(2q

r2

)n](kφ − rφ′

k+ r2φ′′

k

), (4.30)

and:

(∇ · τ (1))r = −τckr

2q

(kφ

r2− φ

′

rk+ φ

′′

k

)+ Kpr

q

(2q

r2

)n [(2n2 − 2n− k2

2

)φ

r2

+(−2n2 + 2n+ 1

2

)φ′

r+(

n− 12

)φ′′], (4.31)


(∇ · τ (1))θ = iτcr2

2q

[2kφ

r3+ φ

′(k2 − 3)kr2

+ 3φ′′

kr+ φ

′′′

k

]+ i

Kpr2

q

(2q

r2

)n

×[

kφ

r3+(

k

2+ 2n− 3

2k− kn

)φ′

r2+ 3− 2n

2k

φ′′

r+ φ

′′′

2k

], (4.32)

whence the two ODEs for φ and f :

0 = −f ′ − mξ(q

r

)m−1φ

r− τckr

2q

(kφ

r2− φ

′

kr+ φ

′′

k

)+ Kpr

q

(2q

r2

)n

×[(

2n2 − 2n− k2

2

)φ

r2+(−2n2 + 2n+ 1

2

)φ′

r+(

n− 12

)φ′′], (4.33)

0 = − ikf

r+ ξ

ik

(q

r

)m−1φ′ + i

τcr2

2q

(2kφ

r3+ k2 − 3

k

φ′

r2+ 3φ′′

kr+ φ

′′′

k

)+i

Kpr2

q

(2q

r2

)n [kφ

r3+(

k

2+ 2n− 3

2k− kn

)φ′

r2+ 3− 2n

2k

φ′′

r+ φ

′′′

2k

]. (4.34)

4.4.2. Boundary conditionsWe start with the kinematic boundary condition, which states that the internal

boundary of the foam is a material interface: 0 = dF/dt = ∂F/∂t + v · ∇F at F = 0.Now, ∂F/∂t =−R−ωε, and ∇F = er − ikεeθ/r, hence: −R−ωε+ vr − ikεvθ/r = 0 atF = 0. Using Taylor’s expansion at order ε1:

vr(F = 0)' vr(r = R, θ; t)+ ε ∂vr

∂r(r = R, θ; t), (4.35)

we get at order ε1: −ωε + v(1)r + ε∂v(0)r /∂r = 0 at r = R, hence:

ω =− q

R2+ φ(R)

R. (4.36)

We now turn to the continuity of stress at the internal boundary of the foam, thegeneral expression for which is (Edwards, Brenner & Wasan 1991):

[(p− P0)1− τ ] ·n=∇s · τ s, (4.37)

where the normal unit vector at the interface is n = ∇F/|∇F| = er − ikεeθ/R, theinterfacial stress is τ s = σ1s, and 1s and ∇s are the identity tensor and gradientoperator projected on the interface:

1s = 1− n⊗ n= eθ ⊗ eθ + ikεR(er ⊗ eθ + eθ ⊗ er), (4.38)

and:

∇s = 1s ·∇ = eθR

(1− ε

R

)∂

∂θ+ ikε

R

(er

R

∂

∂θ+ eθ

∂

∂r

). (4.39)

From these expressions we compute:

∇s · τ s = σ{−er

R+ ε

R2[ikeθ − (k2 − 1)er]

}, (4.40)

hence stress continuity yields:

(p− P0)n− τ ·n= σ{−er

R+ ε

R2[ikeθ − (k2 − 1)er]

}. (4.41)


Hence, continuity of normal stress is written: p − P0 − τrr − ikε(τrθ + τθr)/R =−σ/R − σ(k2 − 1)ε/R2 at F = 0, thus at order ε1 and at r = R, ε∂p(0)/∂r + p(1) −ε∂τ (0)rr /∂r + τ (1)rr =−εσ (k2 − 1)/R2, and:

0=−2nKp (2q)n R−2n−1 + 2nKpqn−1nR1−2n

(φ′ − φ

R

)− (1− n)2n+1KpqnR−2n−1

− 2τc

R− ξ( q

R

)m + f (R)+ σ k2 − 1R2

. (4.42)

Moreover, continuity of tangential stress gives: τrθ + ikε(τrr − τθθ)/R = 0 at F = 0,hence at r = R:

0= k

R

[−τc − Kp

(2q

r2

)n]+ τc

2q

[1+ Kp

q

(2q

r2

)n](kφ − rφ′

k+ r2φ′′

k

). (4.43)

Similarly, we could write the boundary conditions at the external boundary of thefoam, but for simplicity, we will neglect the finite size of the foam in the stabilityanalysis, and consider that all quantities of order ε1 must vanish as r tends towards+∞.

To summarize, we have to solve for the unknown functions φ and f satisfying thecoupled ODEs (4.33) and (4.34), the boundary conditions (4.42) and (4.43), and:

limr→+∞

φ(r)= 0, limr→+∞

f (r)= 0. (4.44)

Inserting the computed value of φ(R) in (4.36) then yields the growth rate of theperturbation as a function of k. For a given set of parameters, this problem can besolved numerically, but not analytically.

However, we have proven in § 4.1.1 that internal viscous stresses are negligiblein our experiments, hence we consider the case Kp = 0. Furthermore, linearizingthe external friction, i.e. setting m = 1, reduces (4.33) and (4.34) to Euler–Cauchyequations and allows an analytical resolution.

4.4.3. Analytical solution with yield stress and linear frictionWe set m = 1 and ξ = ξl in (4.33) and (4.34), with ξl the friction coefficient

corresponding to a linear viscous force, discussed below. We get:

2q f ′

τc=−(γ + k2)

φ

r+ φ′ − rφ′′, (4.45)

2q f

τc= 2φ + rφ′

k2(k2 − 3− γ )+ 3r2φ′′

k2+ r3φ′′′

k2, (4.46)

with γ = 2q ξl/τc. The boundary conditions are:

ω = φ(R)R− q

R2, (4.47)

0=−4qk

R+ kφ(R)− Rφ′(R)

k+ R2φ′′(R)

k, (4.48)

f (R)= 2τc + ξlq

R− σ k2 − 1

R2. (4.49)

The solutions f and φ are generically power laws of r, so we set φ = b(q/R) (r/R)µ

and f = cτcφ/2q, with b and c dimensionless parameters. Equations (4.45) and (4.46)


leads to

c= 1µ(−γ − k2 + 2µ− µ2), (4.50)

0= µ4 + 2µ2(

k2 − γ2− 2)+ k2(k2 + γ ). (4.51)

From the four roots of (4.51), only two have a negative real part:

µ1,2 =−(

2+ γ2− k2 ±√∆

)1/2, (4.52)

with ∆= γ 2/4+ 4− 2γ k2 − 4k2 + 2γ .The solution is thus φ(r) = b1φ1 + b2φ2, with φi = qrµi/Rµi+1 for i = 1, 2, and

f = τc(c1φ1 + c2φ2)/2q. The coefficients ci are directly given by (4.50) and thecoefficients bi are determined from the boundary conditions (4.48) and (4.49). Finally:

b1 =4+ γ − δ(2k2 − 2)− 4k2c2

k2 − 2µ2 + µ22

c1 − c2k2 − 2µ1 + µ2

1

k2 − 2µ2 + µ22

, (4.53)

b2 = 4k2 − b1(k2 − 2µ1 + µ21)

k2 − 2µ2 + µ22

, (4.54)

with δ = σ/τcR.The instability growth rate is given by (4.47):

ω(k)= q

R2(b1 + b2 − 1)= q

R2F (γ, δ, k). (4.55)

The sign of the growth rate, and thus the stability of the flow, only depends onthe wavenumber k and on two dimensionless parameters, γ = 2qξl/τc and δ = σ/τcR,through the complicated but explicit function F . From (4.15), we have qξl = P0 upto logarithmic corrections, and thus γ = 2P0/τc. An important consequence is thatthe stability of the different modes only depends on P0/τc and σ/τcR. For a givenpressure the unstable modes do not depend on the cell thickness. Note that consideringthe general case m 6= 1 leads to the same conclusion. A similar scaling result canindeed be obtained in which the two dimensionless numbers governing the stability areξqm/(τcRm−1)= (1− m)P0 (R/rext)

1−m /τc, and σ/τcR.A numerical estimation of ω(k) in the range τc/2qξl = 1/γ ∈ [5 × 10−3; 0.2] and

σ/2qξlR = δ/γ ∈ [10−6; 10−1] has been performed using MATLAB. It shows that thefront is always unstable. The destabilization by the viscous effect increases withk, but the surface tension eventually stabilizes the short wavelengths, leading, fornegligible yield stress, to a parabolic profile for ω(k) (figure 10a). The yield stressplays a negligible role in the reference flow, as discussed in § 4.1.2, but becomesimportant in predicting the stability of the flow, since the parameter comparingsurface tension and yield stress, δ = σ/τcR ≈ L/R, is of order 0.1 in our experiments.Numerical simulations with σ = 0 show that the yield stress does not stabilize theshort wavelengths. However, from figure 10(a), we see that it strongly reduces thegrowth rate of every mode, and displaces the maximum of ω, as seen on figure 10(b).

We computed the value of kmax(γ, δ) leading to the largest value of ω

and we assumed that the observed finger thickness lc at the ramification stage


0

5

5 10 15 20 25

k0 30

0

20

40

60

80

0.05 0.10 0.150 0.20

(a) (b)

FIGURE 10. (a) Numerical value of ωR2/q as a function of k for different values of thereduced yield stress 1/γ = τc/2qξl, in the range [10−3; 0.4] from the top curve to thebottom. (b) Numerical value of kmax as a function of 1/γ = τc/2qξl, for different valuesof σ/(2qξlR)= δ/γ , in the range [10−6–10−1] from the top curve to the bottom curve.

satisfies lc = 2πR/kmax. The best power law fitting the numerical data shown infigure 10(b) corresponds, in our experimental range, to

lc = 2πR

0.58δ0.35γ −0.43. (4.56)

The dependence on τc is very weak (numerical exponent of 0.08) in the investigatedparameter range. To check our calculations, we can also notice that the limit τc = 0reduces the problem to the much simpler result already obtained by Paterson (1981):from (4.45), (4.46), (4.47) and (4.49) we deduce, in the limit ξlRQ/σe� 1,

lc = 2π

√6πeσR

ξlQ= 2π

√3σR

P0. (4.57)

Using the relation γ = 2P0/τc in (4.56) we deduce the scaling:

lc = 2πR

0.58

(σ

τcR

)0.35(τc

2P0

)0.43

. (4.58)

For a three-dimensional foam with a few tens of bubbles across the gap, Park &Durian (1994) found a finger width that obeys the relation lc ∼√σR/P0(π

√12+e/4L).

The scaling lc ∝ R0.5 observed is not very far from the prediction R0.65 of our model.However, our model is unable to predict the linear increase of the finger width withthe plate separation, and we do not observe it experimentally.

In order to express lc as a function of the injected flux, we need an expression for ξl

adapted to our nonlinear viscous friction. We set, from (4.4) and (4.5):

ξl = αh

eLσκ(a

L

)−0.5( ησ

)2/3v−1/3typ , (4.59)

with vtyp = Q/2πeR the typical velocity in the foam, close to the interface.The numerical expression of the relation (4.56) using (4.59) and σ = 37 mN m−1,

η = 1.2 × 10−3 Pa s, τc = σ/L, αh = 1.31, κ = 6.8 is, with all lengths expressed inmetres and the flux in m3 s−1:

lc = 14 e0.72Q−0.29L0.13a0.2R0.51. (4.60)


0

0.02

0.03

0.05

0.01 0.02 0.03

0.01

0.04

FIGURE 11. (Colour online) Experimental value of lc, as a function of its theoretical valuegiven by (4.60). The best linear fit corresponds to lc,exp = (1.22± 0.05)lc,th.

We compare our experimental measurements of the distance between fingers lc tothe theoretical prediction on figure 11. To do so, we measure the central bubblearea at our measured onset of the fingers, and take R to be the radius of a circleof the same area. We find a good linear correlation between the experimental dataand the prediction, the experimental values being slightly underestimated by thetheory. Such a good agreement is remarkable, given both the difficulty in makinga systematic and precise measurement of the finger spacing in experiments, and theapproximations of the theory. Indeed, the latter is based on a slight perturbation of anideal radial interface, whereas most of the fingers were measured at times where thecentral bubble was already clearly non-circular. Furthermore, there is not a large-scaleseparation between the individual bubble size and the finger spacing, as shown by thecorrugated boundary of the central bubble in figure 2. A possible reason for such agood agreement is that away from the immediate neighbourhood of the central bubble,the foam response to the blowing and fingering processes becomes insensitive to thefine geometric details of the internal boundary.

5. ConclusionsWe have performed a systematic study of the response of a two-dimensional liquid

foam to a localized air injection. We have found a rich variety of phenomena (figure 1)with two essentially different types of response: either the injected air enlarges acentral bubble that grows within the foam inducing successive plastic rearrangements,without film rupture; or it breaks soap films and forms cracks through the foam,like in brittle fracture. We have studied in detail the first, pure swelling regime; wehave shown that the scaling relations between air overpressure, swelling rate, cell gap,bubble size and foam area can be deduced from a simple dynamical model, underthe assumption of radial symmetry, relating the pressure drop within the foam to thefriction between the bubbles and the walls. In practice, at moderate velocities, thecentral bubble grows by fingering, which we quantitatively explain by a linear stabilityanalysis of the radial model. In this case, yield stress becomes important to set thecharacteristic finger width.


We have also investigated the transition between the pure swelling and the brittlefracture regimes. At the local scale, we have confirmed that the films have a maximalvelocity (Arif et al. 2010), and we provide a modification of the criterion previouslyproposed by Hilgenfeldt et al. (2008). We also transcribe this idea to the global scale,proposing a maximal pressure that a foam can sustain without breaking, and showingthat fracture and swelling can happen successively in a certain range of pressures.

Future perspectives include the study of the two extreme regimes depicted infigure 1. At very low speed of injection, the experiment is purely elasto-plastic. Sincetwo-dimensional foams enable access to elastic stress, displacement fields and plasticrearrangements at the local scale (Dollet & Graner 2007), this situation could be agood benchmark to further investigate some concepts in fracture beyond linear elasticfracture mechanics (Freund 1990; Bouchbinder et al. 2010), such as nonlinear elasticeffects (Livne, Bouchbinder & Fineberg 2008) and plastic events (Cortet et al. 2005) atthe crack tip. Conversely, at very high speed, it would be interesting to relate the onsetof crack branching that we observed to existing studies in other soft materials suchas hydrogels (Baumberger et al. 2008). Since liquid foams share many rheologicalsimilarities with other systems studied in fracture, namely an elastic response and ayield stress, like gels, and strong rate-dependent effects, like polymers and suspensions,they may constitute a model material to investigate many questions relevant to fracturemechanics.

AcknowledgementsWe thank S. Santucci and C. Raufaste for regular and insightful discussions, A.

Faisant and P. Chasle for help in designing and interfacing the setup, and A. Le Bouilfor experimental help.

Supplementary moviesSupplementary movies are available at journals.cambridge.org/flm.

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J. Fluid Mech. (2013), . 714, pp. doi:10.1017/jfm.2012.476