Numerical modelling of foam Couette flows

Transcription

1 Author manuscript, publishe in "European Physical Journal E 27, " DOI :.4/epje/i 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 Lab. Jean Kuntzmann, Univ. J. Fourier, CNRS an INRIA, 5, rue es Mathématiques, 384 Saint-Martin Hères, France 2 Lab. Spectrométrie Physique, UMR 5588, CNRS an Univ. J. Fourier, B.P. 87, 3842 Saint-Martin Hères Ceex, France April, 8 28 hal-24536, version 2 - Jul 28 Introuction Abstract. A numerical computation base on a tensorial visco-elasto-plastic moel base on continuous mechanics is compare to experimental measurements on liqui foams for a biimensional Couette flow between two glass plates, both in stationary an transient cases. The main features of the moel are elasticity up to a plastic yiel stress, an viscoelasticity above it. The effect of the friction of the plates is taken into account. The numerical moelling is base on a small set of stanar material parameters that are fully characterise. Shear localisation as well as acute transient observations are reprouce an agree with experimental measurements. 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 exhibit localisation. PACS Bc Foams an emulsions 83.6.La Viscoplasticity; yiel stress 83.6.Df Nonlinear viscoelasticity 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 points of view. Many stuies focus on the velocity profile that can localise near the moving walls [,2]: the measurements exhibit the coexistence between a flowing region an a region moving as a whole, similar to what has been observe for bi- or triimensional shear flows of emulsions [3], collois [4] or wet granular materials [5 8]. Most of these stuies reveal either a continuous [, 5 7] or iscontinuous [2 4] transition between the flowing an non-flowing regions. These ifferences have not been unerstoo yet an continue to excite ebate [2, 9]. A scalar visco-elasto-plastic moel incluing viscous rag [, ] successfully reprouce the exponential ecay of velocity that was observe in the biimensional plane [9] an cylinrical [] Couette flows of a foam between two glass plates. In both cases, the localisation was interprete as the competition between the internal viscosity of the foam an the external friction from the glass plates. Recently, the ata presente in [] were re-analyse in [2] an, in aition to the velocity fiel, two tensorial informations were extracte both in stationary an transient regimes: the statistical elastic strain tensor an the plastic rearrangea Present aress: C. Raufaste, Physics of Geological Processes, University of Oslo, Sem Selans vei 24 NO-36 Oslo, Norway Corresponence to: pierre.saramito@imag.fr ments rate tensor. Up to now, these measurements have not been compare yet to a preiction from 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 [, 2] with the present numerical computations base on a recent general tensorial an triimensional visco-elasto-plastic moel base on continuous mechanics [3] that combines viscoelasticity an viscoplasticity in a unifie framework. This moel, which obeys by construction the secon principle of thermoynamics, leas to numerically stable equations an robust resolution algorithms. It is general enough to apply to both bian triimensional geometries an to several materials. Here, it is applie to biimensional foams using three parameters foun in the literature with only minor ajustments an agrees with the publishe experiments. This implies that we can also use it to preict flows in several geometries an conitions, incluing transient, steay an oscillatory flows, even if no experiment is available. The biimensional Couette flow experimental set-up iscusse here 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 fifth 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 a b e θ e r x τ ε p η µ ε e σ r e ε hal-24536, version 2 - Jul 28 Fig.. Experimental set-up: a efinition of the geometric an kinematic parameters ; b picture of the confine biimensional liqui foam from []. Georges Debrégeas courteously provie ata from the experimental set-up 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 5.2% is homogeneous an the bubble size is of the orer of 2 mm. The foam is confine between two transparent glass plates separate by an interval h=2 mm. The inner wheel rotates at V =.25 mm s, well in the low velocity limit. Two experimental runs are available: Run is relate to the transient case, measurements are available in [2]. 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 [2], the measure quantities are average over 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 [,2,4]. The internal raius is r =7 mm, as for the previous run, an the external raius is r e =22 mm, which iffers from the previous 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 ε is suppose to split into two contributions: ε = ε p + ε e, where ε p an ε e enote respectively the plastic an the elastic strain tensors. The moel consiere in this Fig. 2. The visco-elasto-plastic moel. paper is presente on Fig. 2. The total Cauchy stress tensor writes: σ = p.i + τ, 2 where p is the pressure, I is the ientity tensor here in two imensions, τ = 2µε e is the elastic stress tensor, an µ is the elastic moulus of the foam. When the stress is lower than a yiel value, the material behaves as an elastic soli: the plastic strain ε p is equal to zero an the stress is 2µε. Otherwise, the foam is suppose to behave as a Maxwell viscoelastic flui with a relaxation time λ = η/µ, where η is the viscosity. This moel is a simplifie version of the moel introuce in [3]: the secon solvent viscosity of the original moel is here taken to zero. This choice is coherent with experimental observations [2, 5]: at slow strain rate an when no plasticity occurs, the foam behaves as a soli elastic boy while the moel consiere in [3] escribes a more general Kelvin-Voigt viscoelastic soli. This choice in the moelling is also justifie a posteriori in the present paper by comparisons between numerical computations an experimental measurements. The tensor τ satisfies the following nonlinear ifferential constitutive equation: λ Dτ Dt + max, τ τ = 2ηDv, 3 τ where v is the velocity fiel, Dv = v + v T /2 is the strain rate tensor an τ > is the yiel stress. We enote also by τ = τ /2trτI the eviatoric part of τ an by τ its matrix norm. The upper convecte erivative of tensors writes : Dτ Dt = τ t + v. τ τ vt v τ. The set of equations is close by the conservation of momentum: v ρ t + v. v iv σ = β v, 4 h

3 I. Cheai et al.: Numerical moelling of foam Couette flows 3 hal-24536, version 2 - Jul 28 where ρ enotes the ensity, which is constant [6]. The right-han sie expresses the external force ue to the friction of the plates: β is a friction coefficient an h the istance between the two plates; following [], a linear friction moel is assume for the sake of simplicity. A iscussion on the effect of this approximation at low velocity can be foun in [7]. Finally, since the ensity is constant, the mass conservation reuces to : ivv =. 5 The three unknowns of the problem are τ,v, p an the corresponing equations are 3, 4 an 5. They are complete by some bounary conitions for v at r = r an r e for any t >, an some initial conitions for τ an v at t =. In the biimensional polar coorinate system r, θ, we assume the solution to be inepenent of θ: the raial component velocity v r is zero an v θ is simply written as v. Detaile equations presente in appenix A show that the problem reuces to a time-epenent oneimensional system of partial erivative equations with four scalar unknowns v, τ rr, τ, τ θθ that epen upon t an r. We solve it with a secon orer time splitting algorithm, similar to what has been use for viscoelastic flui flows problems [8]; this algorithm allows to simplify the highly non-linear initial problem into a sequel of i elliptic problems resolutions an of ii constitutive equation resolutions. Problems i an ii can then be solve by classical methos. A mixe finite element metho is implemente for the iscretisation with respect with r; the gap [r, r e ] is iscretise with 5 points. The time iscretisation is performe with a time step t of the orer of.2 s, which is small consiering that the motion is very slow. The stationary solutions are obtaine by running the transient algorithm until the relative error between two time steps ivie by t is less than 3. In orer to perform computations an compare them to the experimental ata, we nee to specify the numerical values of the kinematic an geometric parameters V,, h of the experimental set-up, an the material parameters µ, η, τ, β. Let, V, /V an ηv/ be the characteristic length, velocity, time an stress, respectively. We can efine a Reynols number Re = ρv R/η, but we will see below that we can neglect it. We choose three imensionless numbers: the Bingham an Weissenberg numbers, an a friction coefficient, efine as, respectively: Bi = τ η V, We = η V µ an C F = β2 ηh. 6 This choice of imensionless numbers Bi, We, C F correspons to a escription of the foam as a flui: the characteristic stress is a viscous stress ηv/. An alternative choice woul be to choose the elastic moulus µ as the characteristic stress, in orer to emphasise the elasticlike behaviour of the foam; the corresponing imensionless numbers woul be the yiel eformation ε e = /2Bi We = τ /2µ, the ratio βv /µh = C F We of the friction of the plates with the elastic moulus, an the same Weissenberg number. We will see in the sequel that the latter choice is appropriate for the transient flow escription section 3, whereas the former choice is appropriate for the stationary flow escription section 4. From [9], the biimensional elastic moulus in Nm is expresse as µ=µ/h 2 2 Nm. The estimation of the elastic moulus µ is thus µ Nm 2. From [5] the relaxation time λ=η/µ of the foam on this experiment shoul be of the orer of s, an thus η Pa s. is imensionless; it is evaluate from the experiments [2] as the maximal value of ε e Fig. : ε e.26, an thus τ 5.2 N m 2. It The yiel strain ε e shoul be pointe out that in the moel, ε e can excee ε e at non zero velocity. Again from [9], the fric- tion force of the plates is estimate in N m 2 for any velocity v as f =3 v.64. Since the maximum value of the velocity is V =.25 mm s, the maximum value of the friction force is As we have chosen a linear friction moel for the sake of simplicity, f =βv/h, we choose β such that the maximal value of the friction is the same with both expressions: β /.25 3 =63 Pa s m. After a first computation base on this set of parameters, we foun a goo agreement of the moel with both transient an stationary cases with slightly ajuste parameters which we use throughout the paper for comparison with experiments: µ =.9 N m 2, η = 3. Pa s, τ = 5.47 N m 2. β = 63 Pa s m. This ajustment leas to the set of imensionless numbers 6 presente in Table. run Bi We C F. transient stationary Table. The choice of imensionless numbers as efine in 6. The material parameters of the foam are ientical in both runs, but since the cyliner gaps iffer, the imensionless numbers iffer as well. Since the nonlinear inertia term v. v in 4 vanishes 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 computations. For the comparison with experiments, the results at time t an raius r are expresse in a imensionless form, respectively, 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 /.

4 4 I. Cheai et al.: Numerical moelling of foam Couette flows a r = r b.8.8 r = r 2 r = r 3 v vr.6.4 γ =. γ =. γ =.25 v vr.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, version 2 - Jul 28 Fig. 3. Transient case: a velocity profile versus r at ifferent times ; b velocity profile versus t for r = r to r 8. ε e r = r r = r 8 ε e γ c γ = V t Fig. 4. Transient case: cross component of the elastic strain ε e versus t for r =r to r8. Inset: the vertical ashe line mark the cross-over γ c between the transient an the stationary regimes, efine by the intersection of the tangent at the point of minimum curvature an the asymptotic value of ε e, here plotte for r = r4. 3 The transient flow The transient numerical computation is performe as follows: first, the computation is one with the foam initially at rest with vr = V an vr e =. When a stationary regime is reache, the bounary conitions are change to vr = V an vr e = at a time chosen as the origin t = an results are store at each time step. Fig. 3a plots the velocity profile versus r at ifferent times. Conversely, the velocity is represente on Fig. 3b versus time at ifferent raii, from r = r to r 8. When γ =., the shear irection has just been switche an the velocity profile is roughly exponential. After a short transient, γ., the velocity profile becomes quasi linear up to γ.3. Then, at γ =.3 an r = r, i.e. near the moving isk, it starts to ecrease rather abruptly. Far from the moving isk, the ecreasing starts later, at

5 I. Cheai et al.: Numerical moelling of foam Couette flows 5.6 a.6 b C F We = C F We =.287 C F We =.2 γ c.4 γ c.4 C F We =. C F We = r r r r.8 hal-24536, version 2 - Jul 28 Fig. 5. Transient case: a γ c versus r with parameters as in table. : experimental ata with error bars; soli line: numerical ata. b γ c versus r with Bi an We as in table an C FWe =,.,.2,.287,.4. γ =.35. At γ =, the foam has reache a stationary regime in which the flow is strongly localise near the moving isk. Fig 4 shows ε e = τ /2µ versus time for r = r to r 8. After a short transient of about.5, ε e reaches a first regime where it varies linearly.5 γ.3 for r =r an.2 γ.4 for r =r 8. Thus, the stress τ epens linearly upon γ, i.e. the material behaves as an elastic soli in this regime. Then, after a secon transition, τ saturates an reaches a stationary regime. For a given raius r, the characteristic value of γ associate to this secon transition is enote by γ c r, efine as the intersection between the tangent at the point of minimal curvature in the transient regime an the horizontal line corresponing to the asymptotic value in the stationary regime, as shown in the inset of Fig. 4. Figure 5a shows the inhomogeneity in yieling accross the gap. The numerical results agree with the experiment [2] within error bars. Both numerical an experimental results show that the regions close to the moving wall saturate earlier than the regions far from it. We take avantage of the numerical computations to stuy how this inhomogeneity epens on the physical parameters. As mentione in section 2, the appropriate parameters here are BiWe, C F We, We. We fin that when C F We =.287 is maintaine constant an BiWe an We are varying, the curve γ c r is only translate upwars or ownwars, the inhomogeneity throughout the gap remains the same. Conversely, when BiWe an We are fixe as in Table, the inhomogeneity varies with C F We see Fig. 5b. We plot in Fig. 6 γ c r e γ c r versus C F We; the epenence is almost linear: γ c r e γ c r.65 C F We γ c r e γ c r C FWe Fig. 6. Transient case: γ cr e γ cr versus C FWe. : numerical ata; ashe line: linear fit tothese ata. In particular, when C F We =, γ c r = for all r ; the inhomogeneity is thus an effect of the friction of the plates. Fig. 7 plots the norm of the eviatoric part of the elastic strain tensor ε e = [ 2 ε e ] 2 ε e rr 2 εe θθ.

6 6 I. Cheai et al.: Numerical moelling of foam Couette flows hal-24536, version 2 - Jul 28 ε e =.25 ε e Fig. 7. Transient case:.2 ε e r = r r = r 2 r = r 3 r = r γ = V t versus t for r from r=r to r 8. We recall that the plasticity occurs when τ τ, an thus when ε e is above εe = τ /2µ = Bi We/2 =.25. Fig. 7 shows that it occurs only close to the moving cyliner, for r = r ; an only for small times, uring the first transition regime, or for large time, uring the stationary regime. At initial time, ε e is above εe near the moving isk. As the isk moves, the velocity an its graient change sign an ε e ecreases rapily until ε e < εe everywhere in the foam. Therefore there is no plasticity in the foam an the constitutive equation 3 reuces to: Dε e /Dt = Dv. This relation expresses that all the velocity graient is loae into ε e. The foam behaves like an elastic boy an ε e increases linearly with the applie strain γ. Because of the cylinric geometry an the friction of the plates, ε e is higher near the inner moving isk. When ε e reaches εe near the inner isk, plasticity occurs an the foam starts to flow. Then ε e saturates everywhere in the gap, even though it is above the yiel eformation only in a region near the moving isk. Insie this region, the foam is flowing, while outsie there is no flow an the foam is at rest. Let us summarise our finings for the transient case. After a brief transient, the material behaves as an elastic soli: there is no plasticity yet an the velocity profile is quasi linear. The eformation saturates first near the moving isk, an then graually throughout the gap. The friction of the plates is responsible for this graual saturation. Finally, after this secon transient, the stationary regime is reache: the material is flowing an the flow is strongly localise near the moving isk. 4 The stationary flow The stationary numerical computation is performe as follows: as in the experiment, there is a preparatory clockwise rotation the bounary conitions are vr = V an vr e = after which the inner cyliner is rotate counterclockwise the bounary conitions become vr = V an vr e =. As in the previous section, after a transient, the flow evolves asymptotically an monotonously towars a stationary flow. Fig. 8a compares the numerical results to measurements mae in []. The velocity is strongly localise near the moving isk, as in the transient case Fig. 3a. Both the compute an experimental velocities have the same initial slope, but the transition to zero is more abrupt in the case of the numerical resolution. We enote by r c the raius at which the velocity rops to zero. The computation yiels r c =77.6 mm while r c =84. mm for the experimental ata. The r c preiction error is about 2 % of the gap size. The origin of this abrupt behaviour is iscusse in the sequel. Fig. 8b represents the total shear strain rate ε : it strongly localises near the moving isk, an is iscontinuous at the transition between the flowing an non-flowing regions. There is no experimental ata available for the comparison. The stationary elastic strain ε e Fig. 9 is compare to experimental measurements from [2]. The compute shear component ε e Fig. 9a is slightly overestimate. Nevertheless, it presents qualitatively the same behaviour as in the experiment: it oes not localise near the moving isk an varies smoothly with r. The compute ifference of the normal components ε e rr ε e presents a iscontinuity at the point at r = r c, where the compute velocity an plasticity rop to zero Fig. 9b. Conversely, the sign of the experimental ata changes in the mile of the gap. This strong ifference is not surprising since ε e rr ε e not so geometrically constraine as is ε e, an epens θθ θθ is on the foam preparation. This is especially true in the region of the foam which is not sheare much beyon the plasticity limit. Anisotropic trappe elasticity [2] may be initially present in the experimental setup [2]. Similarly, in the moel, the stationary normal components epen on the initial conitions Cf. appenix B. Finally, we plot ε e versus r Fig. ; this quantity is iscontinuous as well; max, τ / τ is zero for r = r c + but non zero for r = r c. Such an abrupt transition has been observe on bubble rafts [2] an various experimental systems [3, 4]. In the present numerical computation, this iscontinuity is an effect of the moel an not a numerical artefact. Calculations presente in appenix B show that in the stationary regime the moel analytically preicts either continuous or iscontinuous quantities, separate by a critical strain rate: ε c = τ r c 2η τ 2 τ r c /2. + λ2 η τ 2 r c 2 8

7 I. Cheai et al.: Numerical moelling of foam Couette flows 7.8 v vr a 8 vr ε b r r.3..2 r r.3 hal-24536, version 2 - Jul 28 Fig. 8. Stationary case: a Velocity versus r. Soli line: computation; : experimental ata from []. b Shear strain rate ε versus r. There is no experimental ata available for the comparison ε e a ε e rr ε e θθ b r r r r.8 Fig. 9. Stationary case: Soli line: numerical resolution; : experimental ata from [2]. a Shear component ε e versus r. b Difference of normal components ε e rr ε e θθ versus r. If we assume that the normal stress ifference is small compare to the shear stress in the non-flowing, elastic region, then τ r c is of the orer of τ / 2 an 8 yiels a more practical formula that oes not involve τ r c : ε c 2τ 4η + λ2 2η τ 2 2 / yiels /vr ε c =.72, in excellent agreement with the compute value Fig. 8b /vr ε c =.73. If we set λ = in 3, the moel reuces to the Bingham moel; if we set τ = in 3, the moel reuces to the Olroy-B moel. None of these moels exhibits a critical strain rate whereas their combination oes. When λ is not zero, the term τ v t vτ in the objective erivative introuces normal components in the off-iagonal term of the constitutive equation Cf. appenix A which enables the moel to have iscontinuous solutions even though the term max, τ / τ τ is continuous. Thus, the iscontinuous behaviour observe with the present moel

8 8 I. Cheai et al.: Numerical moelling of foam Couette flows hal-24536, version 2 - Jul 28 ε e.3 ε e = r r.8 Fig.. Stationary case: ε e versus r. Soli line: numerical resolution; : experimental ata from [2]. arises from the tensorial formalism. A scalar moel may not be able to reprouce it. Let us summarise our finings in the stationary case. The flowing region is well preicte by the moel. The nonflowing region is very sensitive, especially to foam preparation, both in the experiments an in the moel. In between, the moel can preict a iscontinuous transition. In this case, we are able to preict the velocity graient jump. 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 computation with both Bi = 85. an We = unchange, an C F set to zero, is performe. The result is compare Fig. a with the numerical solution with the reference set of parameters C F = 6.7 from Table. The velocity profile is almost the same: the friction of the plates oes not seem to be responsible for 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 value of Bi in the range [ : 53], the variation of r c / upon We in the range [ :.96 2 ] was foun to be less than 3%. Therefore, the localisation point r c can be approximate using the Bingham moel. Fig. b plots the localisation point versus Bi for the Bingham moel We=C F =: r c r behaves as Bi /2. An explicit computation for the Couette flow of a Bingham flui see e.g. [2] or [22, p. 24] expresses the ratio r c /r as the solution of: 2 rc 2 ln r rc r = Bi A Taylor expansion of for small r c r leas to: r c r r 2 /4 r /2 Bi /2. This explicit approximate formula preicts well the localisation point, as shown on Fig. b. In [9], 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 observe. The authors conclue that the plate is responsible for the localisation. This result seems to be in contraiction with our conclusion that the friction has a minor influence on the localisation. Therefore, 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 behaviour observe in [9]. We use a gap with of 4 mm an a moving bounary with a velocity of.25 mm/s as in [9]. The imensionless numbers Bi=85 an We= are fixe an only C F is varying. Fig. 2 shows the results for C F =, 6.7 an 6.7. For C F =, all the foam is flowing an the velocity profile is linear Fig. 2a while ε e is constant an above ε e Fig. 2b. For C F =6.7, all the foam is flowing but the velocity profile is no longer linear, while ε e is non uniform but above εe throughout the gap. For C F = 6.7, the velocity is localise near the moving wall an ε e is not uniform: it is above εe in the region where the velocity is not zero an uner ε e otherwise. Let us summarise our finings. We explain the localisation by the non uniform stress: part of the system is above 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. In the case of the cylinrical Couette flow, it is mostly the geometry, an the localisation position is mainly governe by the Bingham number while both the friction an the Weissenberg number have only a minor influence. Conclusion For the first time, comparisons between an visco-elastoplastic moel base on continuous mechanics an measurements both in stationary an transient cases are performe an foun to be in goo agreement. The irect numerical resolution enables a full comparison with all available ata from measurement. The comparison performe with only one set of material parameters confirms the preictive character of the propose moel. This moel is fully parametrise by only three stanar imensionless numbers that epen on a limite set of experimentally measurable material parameters.

9 I. Cheai et al.: Numerical moelling of foam Couette flows 9 a b.8 v vr.6.4 r c r / r R.3 R R. Bi hal-24536, version 2 - Jul 28 Fig.. Stationary case: a velocity profile for Bi = 85, We = ; C F = soli line or C F = 6.7 ashe line. b Localisation point versus Bi for the stationary Bingham flui We = C F = ; : exact analytical result ; soli line: approximate numerical preiction. v vr C f = C f =6.7 a ε e ε e =.25 C f = C f =6.7 b C f =6.7 C f = r r r r.8 ε e. Fig. 2. Stationary plane Couette case with Bi=85, We= an C F varying: a velocity profile ; b The transition to the shear baning was analyse both in the transient an stationary cases. An elastic behaviour appears uring the transient flow. 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 above the yiel stress an a nonflowing part uner the yiel stress. It appears to be the funamental mechanism for the localisation of the flow. The mechanism of localisation is analyse in etails with respect to the effect of friction on the plates an to 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 flows aroun an obstacle. It can also be interesting to stuy moels with a ifferent escription of plasticity as in [4]. 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 evelop a similar behaviour.

10 I. Cheai et al.: Numerical moelling of foam Couette flows Acknowlegements Appenix hal-24536, version 2 - Jul 28 We are extremely grateful to G. Debrégeas for proviing us with many recorings of his publishe an unpublishe experiments. We woul also like to thank S. Cox, B. Dollet, E. Janiau, 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. Rev. Letter 87, J. Laurisen, G. Chanan, M. Dennin, Phys. Rev. Lett. 93, P. Coussot, J.S. Raynau, F. Bertran, P. Moucheront, J.P. Guilbau, H.T. Huynh, S. Jarny, D. Lesueur, Phys. Rev. Lett. 88, J.B. Salmon, A. Colin, S. Manneville, F. Molino, Phys. Rev. Lett. 9, D. Howell, R.P. Behringer, C. Veje, Phys. Rev. Lett. 82, D.M. Mueth, G.F. Debregeas, G.S. Karczmar, P.J. Eng, S.R. Nagel, H.M. Jaeger, Nature 46, W. Losert, L. Bocquet, T.C. Lubensky, J.P. Gollub, Phys. Rev. Lett. 85, N. Huang, G. Ovarlez, F. Bertran, S. Rots, P. Coussot, D. Bonn, Phys. Rev. Lett. 94, Wang, K. Krishan, M. Dennin, Phys. Rev. E 73, E. Janiau, D. Weaire, S. Hutzler, Phys. Rev. Lett. 97, R.J. Clancy, E. Janiau, D. Weaire, S. Hutzler, Eur. Phys. J. E 2, E. Janiau, F. Graner, J. Flui Mech. 532, P. Saramito, J. Non Newtonian Flui Mech. 45, P. Marmottant, C. Raufaste, F. Graner, to appear in Eur. Phys. J. E 28, 5. A. Kabla, J. Scheibert, G. Debrégeas, J. Flui Mech. 587, D. Weaire, S. Hutzler, The physics of foams Oxfor: Oxfor University Press, G. Katgert, M.E. Mobius, M. van Hecke, submitte 28, 8. P. Saramito, J. Non Newtonian Flui Mech. 6, C. Raufaste, Ph.D. thesis, Université Joseph Fourier, Grenoble, France 27, 2. V. Labiausse, R. Höhler, S. Cohen-Aa, J. Rheol. 5, M. Fortin, D. Côté, P.A. Tanguy, Comput. Methos in Appl. Mech. an Eng. 88, N. Roquet, Ph.D. thesis, Université Joseph Fourier, Grenoble, France 2, Roquet-these.pf A Detaile equations We write equations 3, 4 an 5 in cylinrical coorinates, using the fact that the velocity is of the form v =, vt, r an that the τ tensor components epen only on t an r. The tensorial equation 3 is equivalent to three scalar equations: τrr λ + 2 v t r τ + max, τ τ rr =, τ τ λ t + max v τ rr + v r τ θθ, τ τ η τ v v r 2a =, 2b τθθ λ 2 v t τ + max, τ τ θθ =. τ 2c with τ = 2τ τ rr τ θθ 2 /2. The vectorial equation 4 is equivalent to two scalar equations: p τ rr τ rr τ θθ r r 2 =, 3a r 2 τ = βv h. 3b The equation 5 is verifie for any v of the form, vr. B Analytical analysis of the stationary equations We show here that the moel can lea either to continuous or iscontinuous quantities in the stationary regime for the circular geometry. Similar evelopments can be mae for the plane geometry an yiel the same results. We make the reasonable assumptions that in the stationary regime, τ is ecreasing throughout the gap, an that there exists r c such that τ > τ for r < r c an τ < τ for r > r c. First of all, as there are no point forces at the interface between flowing an non flowing regions, the total stress τ pi has to be continuous. However, this is not necessarily the case for the components of τ: 3a shows that the normal components of τ can be iscontinuous if the pressure is iscontinuous; conversely, 3b shows that τ must be continuous as it is not balance with pressure. Then we write the constitutive equation with t =. We have:

11 I. Cheai et al.: Numerical moelling of foam Couette flows When r > r c : the plasticity term is zero, so 2a leas to 2λv/rτ =, an v = as τ. Eqn. 2c leas to 2λ v τ =, an thus v =. However, 2b is then equivalent to =, an the normal stress components are not etermine by the stationary equations only. In the transient problem, their value are etermine by the initial conitions. Finally, as v =, 3b yiels τ = C/r 2 where C is a constant. When r < r c : 2a-2c lea to 2λ v r τ + τ τ rr =, τ v λ τ rr v r τ θθ + τ v τ = η τ v, r 2λ v τ + τ τ τ θθ =. hal-24536, version 2 - Jul 28 We enote with a resp. a + the quantities evaluate in r = r c resp. in r = r + c ; v an τ are continuous, thus v = v + =, τ = τ+ = τ r c an we have: τ τ λ v τrr + τ τ 2λ v τ r c + τ τ τ rr =, τ r c = η v τ θθ =. If τ =, we fin v τ = = v + : there is no iscontinuity. If τ, we fin: τ τ rr =, τ θθ = 2λ η τ r c 2, v = τ η τ τ r c,, with now As v v /2 τ = 2τ r c λ2 η 2 τ r c 4. +, the strain rate is iscontinuous at r = r c an we can efine a critical strain rate: ε c = v 2 v r = 2η τ 2 τ r c /2 + λ2 η τ 2 r c 2 τ r c.