Numerical modelling of foam Couette flows

Transcription

1 EPJ manuscript No. (will be inserte by the eitor) Numerical moelling of foam Couette flows I. Cheai, P. Saramito, C. Raufaste 2 a, P. Marmottant 2, an F. Graner 2 CNRS, INRIA an Laboratoire Jean Kuntzmann, B.P. 53, 384 Grenoble Ceex 9, France 2 CNRS an Laboratoire e Spectrométrie Physique, B.P. 87, F-3842 St Martin Hères Ceex, France February 6, 28 hal-24536, ersion - 6 Feb 28 Introuction Abstract. In this paper, a numerical simulation base on a tensorial elastoiscoplastic moel is compare to experimental measurements on liqui foams for a biimensional Couette flow, both in stationary an transient cases. The main features of of the moel are elasticity up to a plastic yiel stress an iscoelasticity aboe. The numerical moelling bases on a small set of stanar material parameters that coul be fully characterise. Shear localisation as well as fine transient obserations are reprouce an are foun to be in goo agreement with experimental measurements. The effect of the friction of plates is taken into account. The plasticity appears to be the funamental mechanism of the localisation of the flow. Finally, the present approach coul be extene from liqui foams to similar materials such as emulsions, collois or wet granular materials, that exhibits localisations. PACS Bc Foams an emulsions 83.6.La Viscoplasticity; yiel stress 83.6.Df Nonlinear iscoelasticity 2.6.Cb Numerical simulation; solution of equations 7.5.Tp Computer moelling an simulation The biimensional Couette flow of a foam has been wiely stuie, from experimental, theoretical an numerical point of iew. Many stuies focus on the elocity profile that coul localise near the moing walls [ 3]: the measurements exhibits the coexistence between a flowing state an a soli state, similar to that obsere for bi- or triimensional shear flows of emulsions [4], collois [5] or wet granular materials [6 9]. Most of these stuies reeal either a continuous [6 8,] or iscontinuous [4, 5, 2] transition between the flowing an soli state. These ifferences haen t been unerstoo yet an continue to excite ebate [2, ]. A scalar elastoiscoplastic moel incluing iscous rag was presente [, 2]; it successfully reprouce the exponential ecay of elocity that was obsere in the biimensional plane [] an cylinrical [] Couette flow of a foam between two glass plates. In both cases, the localisation was interprete as the competition between the internal iscosity of the foam an the external friction from the glass plates. Recently, the ata presente in [] were re-analyse in [3] an, in aition to the elocity fiel, two tensorial informations were extracte both in stationary an transient regimes: the statistical elastic strain tensor an the plastic a Present aress: C. Raufaste, Physics of Geological Processes, Uniersity of Oslo, Sem Selans ei 24 NO-36 Oslo, Norway Corresponence to: pierre.saramito@imag.fr rearrangements rate tensor which measures the plasticity. Up to now, these measurements haen t been compare to a numerical moel. The aim of this paper is to compare the measurements on a biimensional Couette flow of a liqui foam between two glass plates presente in [,3] with the present numerical simulations base on a recent general tensorial an triimensional elastoiscoplastic moel [4] that combines iscoelasticity an iscoplasticity in the same framework. This moel, which obeys by construction to the secon principle of thermoynamics, leas to numerically stable equations an robust resolution algorithms. It is general enough to apply to both bi- an triimensional geometries an to seeral materials. Here, it is applie to biimensional foams using three parameters foun in the literature (with only minor ajustments) an fin a goo agreement with the publishe experiments. This implies we can also use it to preict flows in seeral geometries an conitions, incluing transient, steay an oscillatory flows, een if no experiment is aailable. The biimensional Couette flow experimental set-up is briefly recalle in the first section. The secon section presents the numerical moelling. The numerical results are then analyse an compare with ata measurements in the thir section for the transient case an in the fourth section for the stationary case. Finally, the last section explores the mechanism of localisation.

2 2 I. Cheai et al.: Numerical moelling of foam Couette flows Presentation of the experimental set-up y V r e θ e r x τ ǫ (p) η µ σ hal-24536, ersion - 6 Feb 28 r e Fig.. Experimental set-up: efinition of the geometrical an kinematic parameters ; picture of the confine biimensional liqui foam (from []). The experimental set-up is represente on Fig.. It consists of an inner shearing wheel of raius r an an outer fixe one of raius r e. Let = r e r enote the cyliner gap. The cyliner bounaries are tooth shape an there is no slip. The liqui fraction of the foam is 5.2% an the bubble size is of the orer 2 mm. The foam is confine between two transparent glass plates separate by an interal h=2 mm. The inner wheel rotates at V =.25 mm s. Two experimental runs are aailable: Run is relate to the transient case an measurements are aailable in [3]. The internal raius is r =7 mm an the external raius is r e =2 mm. To prepare the foam, the inner isk is rotate counterclockwise, until a stationary regime is reache; then, at an arbitrary time chosen as the origin (t = ), the shear irection is switche to clockwise, the experiment begins an measurements are mae using image analysis. In [3], the measure quantities are aerage oer eight equispace orthoraial circular boxes corresponing to positions r j = r ( (j.5)) for j = to 8. Run 2 focuses on the stationary case an measurements are presente in [,3,5]. The internal raius is r =7 mm, as for the preious run, an the external raius is r e =22 mm, which iffers from the preious run. The preparatory rotation is clockwise. Then, at an arbitrary time, it is switche to counterclockwise. Pictures are recore only after a full 2π turn. 2 The numerical moelling The strain tensor ǫ is suppose to split into two contributions: ǫ = ǫ (p) +, () where ǫ (p) an enote respectiely the plastic an the elastic strain tensors. The moel consiere in this pa- ǫ Fig. 2. The elastoiscoplastic moel. per is presente on Fig. 2. The total Cauchy stress tensor writes: σ = p.i + τ, where p is the pressure, I is the ientity tensor in two imensions, τ = 2µ is the elastic stress tensor, an µ is the elastic moulus of the foam. When the stress is lower than a yiel alue, the material behaes as an elastic soli: the plastic strain ǫ (p) is equal to zero an the stress is 2µǫ. Otherwise, the foam is suppose to behaes as a Maxwell iscoelastic flui with a relaxation time λ = η/µ, where η is the iscosity. This moel is a simplifie ersion of the moel introuce in [4]: the secon solent iscosity of the original moel is here taken to zero. This choice is coherent with experimental obserations [3, 3]: at slow rates of strain, when no plasticity occurs, the foam behaes as a soli elastic boy while the moel consiere in [4] escribes a more general Kelin-Voigt iscoelastic soli. This choice in the moelling is also justifie a posteriori in the present paper by comparisons between numerical simulations an experimental measurements. The tensor τ satisfies the following nonlinear ifferential constitutie equation: λ Dτ ( Dt + max, τ ) τ = 2ηD(), (2) τ where is the elocity fiel, D() = ( + T )/2 is the rate of strain tensor an τ > is the yiel stress constant. We enotes also τ = τ (/2)tr(τ)I the eiatoric part of τ an τ its matrix norm. The upper conecte eriatie of tensors writes : Dτ Dt = τ t + (. )τ τ T τ The problem is complete by the conseration of momentum: ( ) ρ t +. i σ = β, (3) h where ρ enotes the constant ensity. The right-han sie expresses the external force ue to the friction of the plates: β is a friction coefficient an h the istance

3 I. Cheai et al.: Numerical moelling of foam Couette flows 3 hal-24536, ersion - 6 Feb 28 between the two plates. Following [], a linear friction moel is assume. Finally, since the ensity is suppose to be constant, the mass conseration reuces to : i =. (4) The three unknowns of the problem are (τ,, p) an the corresponing equations are (2), (3) an (4). The problem is close by some bounary conitions for at r = r an r e for any t > an some initial conitions for τ an. In the biimensional polar coorinate system (r, θ), the solution is assume to be inepenent of θ: the raial component elocity r is zero an θ is simply written as. The problem reuces to a time-epenant one-imensional system of partial eriatie equation with four unknowns, τ rr, τ, τ that epen upon t an r. The problem is then sole using an operator splitting algorithm an a finite element metho that are similar to those use for iscoelastic flui flow problems [6]. The etails of the numerical algorithms will be etaile in a separate paper. In orer to perform computations an compare it to the experimental ata, in aition to the numerical alues of the kinematic an geometric parameters (V,, h) of the experimental set-up, the quantification of the material parameters (µ, η, τ, β) of the moel is also require. Let, V, /V an ηv/ be respectiely the characteristic length, elocity, time an stress. The problem inoles only three imensionless numbers, the Bingham, the Weissenberg an a friction numbers, efine by: Bi = τ η V, We = η V µ an C F = β2 ηh Notice that this choice of imensionless numbers correspons to a escription of the foam as a flui: the characteristic stress is a iscous stress. An alternatie choice woul be to choose the elastic moulus µ as the characteristic stress in orer to emphasise the soli-like behaiour of the foam; the corresponing imensionless numbers woul be the yiel eformation ε (e) = Bi We = τ µ instea of the Bingham number, βv the ratio µh = C F We of the friction of the plates with the elastic moulus instea of C F, an the same Weissenberg number. Howeer, with this choice, we cannot reach the iscoplastic limit. Thus, the first choice (Bi, We, C F ) is more general an we will present the numerical results as a function of these parameters. From [5], the biimensional elastic moulus in N m expresses µ=µ/h 2 2. The estimation of the elastic moulus µ in N m 2 is µ N m 2. From [3] the relaxation time λ=η/µ of the foam on this experiment shoul be of the orer of s an then η Pois. The imensionless quantity is ealuate from the experiments as the maximal alue of an then.26. It shoul be pointe out that, in the moel, is the alue of at which plasticity occurs an thus (5) can excee. Thus τ 5.2 N m 2. Also from [5], the friction force at the plates is estimate in N m 2 for any elocity as f =3.64. Since the maximum alue of the elocity is V =.25 mm/s, the maximum alue of the friction force is 3 (.25 3).64. Following [], a linear friction moel is assume: f =β/h, where β is chosen such that the maximal alue of the friction is the same with the two expressions: β 3 (.25 3).64 /(.25 3 )=63 Pois m. After a first computation base on this set of parameters, a little ajustment was necessary to compare better with the experimental ata. The following set of parameters leas to a goo agreement of the moel with both transient an stationary cases: µ=.9 N m 2, η=3. Pois, τ = β = 63 Pois m. All the comparisons between numerical computations an experimental measurements presente in this paper are performe with this set of parameters. This ajustment leas to the set of imensionless numbers presente in Table, as efine in (5). Since the nonlinear inertia term. run Bi We C F. transient stationary Table. The choice of imensionless numbers as efine in (5). The material parameters of the foam are ientical. Since the cyliner gap iffers between the two runs, the imensionless numbers are slightly ifferent between the two runs. in (3) anishes for simple shear flows such as the Couette flow, the Reynols number Re = ρv /η 5 5 has a negligible influence on the transient problem an no influence on the stationary one. Thus, Re is taken as zero for the simulations. For the comparison with experiments, the results at time t an raius r is be expresse in a imensionless form, respectiely, as the total applie shear γ = V t/ an as the istance to the inner cyliner normalise by the size of the gap (r r )/. 3 The transient flow simulation The transient numerical simulation is performe in the following way: first, the simulation is one with the foam initially at the rest an, for t >, with (r ) = V an (r e ) =. When a stationary regime is reache, the bounary conitions are change to (r ) = V an (r e ) = an computations are store at each time. Fig. 3.a plots the elocity profile ersus r at ifferent fixe times. Conersely, the elocity is represente on Fig. 3.b ersus the times at ifferent fixe raius, from r = r to r 8. At the beginning of the transient, when γ =., the shear irection has just switche an the elocity profile is roughly exponential. After a short transition, γ., the elocity profile becomes quasi linear up to γ.3. Then, at γ =.3 an r=r, i.e. near the

4 4 I. Cheai et al.: Numerical moelling of foam Couette flows r = r.8.8 r = r 2 r = r 3 (r ).6.4 γ =. γ =. γ =.25 (r ).6.4 r = r 4 r = r 5 r = r 6.2 γ =.5.2 r = r 7 r = r 8 γ = r r γ = V t hal-24536, ersion - 6 Feb 28 Fig. 3. Transient case: elocity profile ersus r at ifferent times ; elocity profile ersus t for r = r to r 8. moing isk, it starts to ecrease rather abruptly. Far from the moing isk, the ecrease starts later, at γ =.35. At γ =, the foam has reache a stationary regime in which the flow is strongly localise near the moing isk. Fig 4 shows the cross component of =τ /(2µ) ersus time at fixe raius r = r to r 8. After a short transition of about.5, reaches a first regime where it aries linearly (.5 γ.3 for r = r an.2 γ.4 for r = r 8 ). Notices that the imensionless time γ interprets also as the applie strain. Thus, the stress τ epens linearly upon γ i.e. the material behaes as an elastic soli in this regime. Then, after a secon transition, the flow reaches a secon regime that is the stationary one. For a fixe raius r, the characteristic alue of γ associate to this secon transition is enote by γ c (r). It is efine by the intersection between the first linear regimes an the secon asymptotic one, as shown in the inset of Fig. 4. Then, γ c is represente ersus the raius an compare to experimental ata from [3] on Fig. 5.a. Remarks that all the alues preicte by the numerical moelling are in the error margin of the experiment. Both for the experiment an the numerical simulation, the alue of γ c increases graually with r an this ariation is linear. A linear regression proceure leas to γ c (r) with α=.49. Here, γ c () α r r + γ() c is associate to the first transition regime an epens upon the initial conitions. Recall that when the slope α is not anishing, the region close to the moing isk saturates earlier than the region far from it. Let us stuy how the slope α is influence by the alues of the parameters. As the foam behaes like an elastic soli uring transient, we fin more aequate to stuy this epenency with respect to (Bi We, C F We, We) as has been alreay iscusse in section 2. On the one han, when either Bi aries in the range [ : 37] or We in the = Fig. 6. Transient case: from r=r to r 8..2 r = r r = r 2 increasing r γ = V t ersus t for ifferent alues of r range [ :.25] an the prouct C F We =.287 is maintaine constant, the relatie ariation of α is less than 6%. On the other han, when both Bi an W e are fixe as in Table (), an only the prouct C F We alone aries from to.6 then the slope aries linearly as α =.42 C F We, as shown on Fig. 5.b. Clearly, the slope α is goerne by βv µh the prouct C F We = : it is proportional to the rag coefficient β an inersely proportional to the elastic moulus µ. Finally: γ c (r) ( ) r r.42 C F We + γ c ()

5 I. Cheai et al.: Numerical moelling of foam Couette flows increasing r hal-24536, ersion - 6 Feb γ c γ = V t Fig. 4. Transient case: cross component of the elastic strain s t for r=r to r8. In the inset, the ertical ashe lines mark the cross-oer γ c between the transient an the stationary regimes, efine by the intersection of the asymptotic alue of an the linear regression of in the first regime. Fig. 6 plots the norm of the eiatoric part of the elastic strain tensor ( ( ) 2 = 2 ( ) ) 2 + rr 2. 2 Recall that = τ/(2µ) an that the plasticity occurs when τ τ. Thus the plasticity occurs when is aboe = τ /(2µ) = Bi We/2 =.25. Fig. 6 shows that is situation appears only close to the moing cyliner, for r = r, an only for small times, uring the first transition regime, an for large time, uring the stationary regime. At initial time, is aboe near the moing isk. As the isk moes, the elocity an its graient change of sign an ecreases rapily until < eerywhere in the foam. Therefore there is no plasticity in the foam an the constitutie equation D writes also: = D(). This relation expresses that Dt all the elocity graient is loae into. The foam behaes like an elastic boy an increases linearly with the applie strain γ. Because of the cylinric geometry an the friction of the plates, is higher near the inner moing isk. When reaches near the inner isk, plasticity occurs an the foam starts to flow. Then saturates eerywhere in the gap een though it is aboe the yiel eformation only in a region near the moing isk. Insie this region, the foam is flowing while outsie, there is no flow an the foam is at the rest. Let us summarise the situation for the transient case. After a brief transition, the material behaes as an elastic soli: there is no plasticity yet an the elocity profile is quasi linear. The eformation saturates first near the moing isk, an then graually throughout the gap. The friction of the plates is responsible for this graual saturation. Finally, after this secon transition, the stationary regime is reache: the material is flowing an the flow is strongly localise near the moing isk. 4 The stationary flow simulation The stationary numerical simulation is performe in the following way: at t =, the foam is at rest, all ariables are set to zero. For t > the inner cyliner moes an the bounary conitions are (r )=V an (r e )=. The computation is performe until the stationary regime is reache. Fig. 7.a compares the numerical results to measurements mae in []. The elocity is strongly localise near the moing isk, as it was alreay obsere in the transient case on Fig. 3.b. Both the compute an the experimental elocities hae the same initial slope, but the transition to zero is more abrupt in the case of the numerical resolution. Let us enote r c the raius where the elocity rops to zero. The computation gies r c =77.6 mm while r c =84. mm for the experimental ata. The r c preiction error is of about 2 % of the gap size.

6 6 I. Cheai et al.: Numerical moelling of foam Couette flows.6.8 Slope α of γ c γ c r r C FWe hal-24536, ersion - 6 Feb 28 Fig. 5. Transient case: γ c s r. The points with error bars are experimental ata while the are numerical preiction ata an the soli line is a linear regression on these numerical ata. Influence of C F on the slope α of γ c. The are ata from numerical simulation an the soli line is a linear regression on these numerical ata (r ) (r ) ǫ..2 r r.3..2 r r.3 Fig. 7. Stationary case: Velocity ersus r. The cure is the compute elocity while the points are experimental ata from []. Shear strain rate ǫ s r. Fig. 7.b represent the total shear strain rate ǫ. Howeer, the compute rate of strain exhibits an interesting feature: it strongly localise near the moing isk an also iscontinuous at the transition between the flowing state at the jamme state. There is no experimental ata aailable for the comparison. Finally, the stationary elastic strain is compare to experimental measurements from [7] for both the shear component (Fig. 8.a) an the ifference of normal components rr (Fig. 8.b). The compute ǫ(e) is slightly oerestimate. Neertheless, it presents qualitatiely the same behaiour as the experimental one: it oes not localise near the moing isk an it aries smoothly with r. The compute ifference of the normal components rr presents a iscontinuity at the point at r = r c, where the compute elocity an plasticity rop to zero. The sign of the experimental ata changes in the mile of the gap. Experimental ata coul isplay measurement artefacts [3], but anyway it is not surprising

7 I. Cheai et al.: Numerical moelling of foam Couette flows rr r r r r.8 hal-24536, ersion - 6 Feb 28 Fig. 8. Stationary case: Soli line is the numerical resolution while points are experimental ata from [3]. Shear component s r. Difference of normal components ǫ(e) rr s r. that the two ata sets coul not be compare quantitatiely: since rr is not so geometrically constraine as is, this quantity epens on the initial conitions, especially in the unshake soli region. Actually anisotropic trappe elasticity may be initially present in the experimental setup [8], which is not consiere in the moel. Compare to experimental measurements, the compute ata exhibit a rather iscontinuous transition from flowing to jamme behaiour, in particular a iscontinuity in the rate of strain an in the ifference of the normal stress components. The origin of this abrupt preicte behaiour is also not yet clearly eluciate. We recall that such an abrupt transition has been obsere on bubble rafts [2] an arious experimental systems [4, 5]. This problem will be aresse in future works. 5 The mechanism of localisation In orer to probe the effect of the friction of the plates on the localisation in the stationary regime, a stationary numerical resolution with both Bi = 85. an We = unchange, an C F set to zero, is performe. The result is compare on Fig. 9.a to the numerical solution with the reference set of parameters C F = 6.7 from Table (). Obsere that the elocity profile is almost the same: the friction of the plates oesn t seem to be responsible of the localisation in this experiment. The epenence of the localisation point r c upon We was foun to be negligible. More precisely, for any fixe alue of Bi in the range [ : 53], the ariation of r c / upon We in the range [ :.96 2 ] was foun to be less than 3%. Thus, the localisation point r c coul be approximate by using the Bingham moel. Fig. 9.b plots the localisation point ersus Bi for the Bingham moel (We=C F =). Obsere that r c r behaes as Bi /2. An explicit computation for the Couette flow of a Bingham flui (see e.g. [9] or [2, p. 24]) expresses the ratio r c /r as the solution of: ( rc r ) 2 2 ln ( rc r ) = + 2 ( ) 2 Bi A Taylor expansion for small r c r leas to: r c r r ( 2 /4 r ) /2 Bi /2. (6) This explicit approximate formulæ is of practical interest for the preiction on the localisation point, as shown on Fig. 9.b. In [], the authors shear a bubble raft in a plane Couette geometry an see no localisation at all. Aing a plate on the top, a localisation is obsere. The authors conclue that the plate is responsible for the localisation. This result seems to be in contraiction with the preious conclusion, where the friction was foun to hae a minor influence on the localisation. Howeer, the geometry is not the same. We performe the numerical resolution of our moel in the case of a plane Couette flow in orer to check if our moel was able to reprouce the behaiour obsere in []. A gap with of 4 mm an a moing bounary with a elocity of.25 mm/s were use for the numerical resolution, so that we hae geometrical parameters similar to []. The imensionless numbers Bi = 85 an We = are fixe an only C F is arying. Fig. shows the results for of C F =, 6.7 an 6.7. For C F =, all the foam is flowing an the elocity profile is

8 8 I. Cheai et al.: Numerical moelling of foam Couette flows.8.6 (R ).4 r c r / r R.3 R R Bi hal-24536, ersion - 6 Feb 28 Fig. 9. Stationary case: elocity profile for Bi=85, We= an both C F = (soli cure) an C F =6.7 (ashe cure). Localisation point s Bi for the stationary Bingham flui (We = C F = ): points are obtaine by an exact computation while the soli line represents the approximate formulae (6)..8.6 (r ).4.2 C f = C f =6.7 C f = =.25 C f = C f =6.7 C f = r r r r.8 Fig.. Stationary plane Couette case with Bi=85, We= an C F arying: elocity profile ;. linear (Fig..a) while is constant an aboe (Fig..b). For C F =6.7, all the foam is flowing but the elocity profile is no more linear while is non uniform but aboe throughout the gap. For C F = 6.7, the elocity is localise in the 42 % left region of the gap while is not uniform: it is aboe ǫ(e) ajacent to the moing plane an uner in the region otherwise. As a main conclusion of this paragraph, we o not explain the localisation by the competition between an internal iscosity an an external friction as in [2], but rather by the non uniform stress an the simple picture that part of the system is aboe the yiel stress an part of the system is below the yiel stress. In the case of the plane Couette flow, the friction of the plates is responsible for the non uniform stress, whereas in the case of the cylinrical Couette flow, it is mostly the geometry. In aition, in this case, the localisation position is mainly goerne

9 I. Cheai et al.: Numerical moelling of foam Couette flows 9 hal-24536, ersion - 6 Feb 28 by the Bingham number while both the friction an the Weissenberg number hae only a minor influence. Conclusion For the first time, comparisons between an elastoiscoplastic moel an measurements both in stationary an transient cases are performe an foun to be in goo agreement. The irect numerical resolution allows a full comparison with all aailable ata from measurement. The comparison was performe with only one set of material parameters: it confirms the preictie character of the propose moel. This moel is fully parametrise by only three stanar imensionless numbers. These numbers epen on a limite set of material parameters that coul be quantifie by some measurements. The transition to the shear baning was analyse both in the transient an stationary cases: an elastic behaiour uring the transient flow is exhibite. The effect of the friction of plates on the propagation of the plastic rearrangements throughout the gap is also analyse in etails. In the stationary case, a non uniform stress throughout the gap leas to a flowing part aboe the yiel stress an a jamme part uner the yiel stress. It appears to be the funamental mechanism for the localisation of the flow. The mechanism of localisation is analyse with etails from two points of iew: the effect of friction on the plates an the plasticity. Finally, the plasticity appears to be the funamental mechanism of the localisation of the flow. In the future, we plan to compare the numerical resolution of our moel with more complex liqui foam flows like Stokes flows an flows aroun an obstacle [2,22]. We will also try to hae a better unerstaning of the abrupt jamming transition. Finally, we point out the fact that our approach is not specific to liqui foams: it coul well escribe emulsions, collois or wet granular materials that are known to eelop a similar behaiour. 3. A. Kabla, J. Scheibert, G. Debrégeas, J. of Flui Mech. 587, pp. 45 (27) 4. P. Coussot, J.S. Raynau, F. Bertran, P. Moucheront, J.P. Guilbau, H.T. Huynh, S. Jarny, D. Lesueur, Phys. Re. Lett. 88(2), 283 (22) 5. J.B. Salmon, A. Colin, S. Manneille, F. Molino, Phys. Re. Lett. 9(22), (23) 6. D. Howell, R.P. Behringer, C. Veje, Phys. Re. Lett. 82(26), 524 (999) 7. D.M. Mueth, G.F. Debregeas, G.S. Karczmar, P.J. Eng, S.R. Nagel, H.M. Jaeger, Nature 46, 385 (2) 8. W. Losert, L. Bocquet, T.C. Lubensky, J.P. Gollub, Phys. Re. Lett. 85(7), 428 (2) 9. N. Huang, G. Oarlez, F. Bertran, S. Rots, P. Coussot, D. Bonn, Phys. Re. Lett. 94(2), 283 ( 4) (25).. Wang, K. Krishan, M. Dennin, Phys. Re. E 73 (26). E. Janiau, D. Weaire, S. Hutzler, Phys. Re. Lett. 97(3), 3832 (26) 2. R.J. Clancy, E. Janiau, D. Weaire, S. Hutzler, Eur. Phys. J. E 2(2), 23 (26) 3. E. Janiau, F. Graner, J. Flui Mech. 532, 243 (25) 4. P. Saramito, J. Non Newtonian Flui Mech. 45(), (27) 5. C. Raufaste, Ph.D. thesis, Uniersité Joseph Fourier, Grenoble, France (27), 6. P. Saramito, J. Non Newtonian Flui Mech. 6, 99 (995) 7. P. Marmottant, C. Raufaste, F. Graner, submitte (27), 8. V. Labiausse, R. Höhler, S. Cohen-Aa, J. Rheol. 5, 479 (27) 9. M. Fortin, D. Côté, P.A. Tanguy, Comput. Methos in Appl. Mech. an Eng. 88, 97 (99) 2. N. Roquet, Ph.D. thesis, Uniersité Joseph Fourier, Grenoble, France (2), Roquet-these.pf 2. N. Roquet, P. Saramito, Comput. Appl. Meth. Mech. Engrg. 92(3-32), 337 (23) 22. B. Dollet, F. Graner, J. Flui Mech. 585, 8 (27) Acknowlegements We are extremely grateful to G. Debrégeas for proiing us with many recorings of his publishe an unpublishe experiments. We woul also like to thank S. Cox, B. Dollet, E. Janiau, an A. Wyn an the Foam Mechanics Workshop (Grenoble, January 28) an all its participants for fruitful iscussions. References. G. Debrégeas, H. Tabuteau, J.M. i Meglio, Phys. Re. Letter 87(7) (2) 2. J. Laurisen, G. Chanan, M. Dennin, Phys. Re. Lett. 93(), 833 (24)