Slow flows of yield stress fluids: yielding liquids or flowing solids?

Transcription

1 Slow flows of yield stress fluids: yielding liquids or flowing solids? Philippe Coussot To ite this version: Philippe Coussot. Slow flows of yield stress fluids: yielding liquids or flowing solids?. Rheologia Ata, Springer Verlag, 2018, 57 (1), pp < < /s >. <hal > HAL Id: hal Submitted on 3 May 2018 HAL is a multi-disiplinary open aess arhive for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or private researh enters. L arhive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou privés.

2 Slow flows of yield stress fluids: yielding liquids or flowing solids? P. Coussot Université Paris-Est, Laboratoire Navier (ENPC-IFSTTAR-CNRS), Champs sur Marne, Frane Abstrat: Yield stress fluids (YSF) exhibit strongly non-linear rheologial harateristis. As a onsequene they develop original flow features (as ompared to simple fluids) under various boundary onditions. This paper reviews and analyze the harateristis of a series of slow flows (just beyond yielding) under more or less omplex onditions (simple shear flow, flow through a avity, dip-oating, blade-oating, Rayleigh-Taylor instability, Saffman-Taylor instability) and highlight some of their ommon original harateristis: i) a transition from a solid regime to a flowing regime whih does not orrespond to a true liquid state, the flow in this regime may rather be seen as a suession of solid states during very large deformation; ii) a strong tendeny to loalization of the yielded regions in some small region of the material while the rest of the material undergoes some deformation in its solid state; iii) the deformation of YSF interfae with another fluid, in the form of fingers tending to penetrate the material via a loal liquefation proess. Finally these observations suggest that for slow flows of YSF are a kind of extension of plasti flows for very large deformations and without irreversible hanges of the struture. This suggests that the field of plastiity and the field of slow flows of YSF ould benefit from eah other. 1. Introdution Although ertainly not realized by the man of the street, yield stress fluids (YSF) suh as foams, emulsions, paints, olloids, physial gels, et, play a major role in our everyday life, and arouse a growing interest in rheology [Coussot 2017]. These materials behave as solids under a ritial stress and yield, in the sense that flow in an (apparent) liquid regime, when the ritial (yield) stress is overome. The main fields of appliation, use, and researh, are oil industry and onstrution, both fields in whih the speifi yield stress behavior of the materials (ement paste, drilling fluids, foams, fresh onrete) is well reognized and taken into aount. In other fields suh as osmetis, foodstuffs, treatments of mining residues, slewage sludges, eramis, natural flows, et., the yielding behavior and some related parameters, are often used to haraterize the material and adjust the formulation. The flow harateristis of suh material type under speifi boundary and initial onditions are less often studied in these fields. Yet YSF flows exhibit original harateristis whih an have a strong impat on the proess effiieny. In fluid mehanis the beautiful and sientifially attrative features of flows under speifi boundary onditions has drawn for a long time the attention of people. This led to various websites, videos and books, an enthusiasm illustrated by the ompetition organized eah year by the Division of Fluid Dynamis of the Amerian Physial Soiety. With a similar motivation, in the field of non-newtonian fluid mehanis, D.V. Boger and K. Walters published a book essentially foused on flows of visoelasti liquids [Boger and Walters 1993] and the (Amerian) Soiety of Rheology started to organize in 2017 a Gallery of Rheology ontest. The starting point of the present paper was a modest ontribution in a similar spirit onerning yield stress fluid flows. We review a series of yield stress fluid flows under omplex boundary onditions and for whih we have a detailed information on the internal flow harateristis. In order to learly distinguish this field from that of simple fluids

3 we fous on slow flows, that we define as flow onditions for whih the yielding harateristis of the fluid are dominant, i.e. the (maximum) stress needed to maintain the flow is lose to the ritial stress needed to indue a flow in an apparent liquid regime. A more quantitative definition will be given in the paper. More tehnial and omplete reviews on different aspets of yield stress fluid behavior or flow harateristis may be found in reent publiations suh as [Coussot 2014, Balmforth et al. 2014, Bonn et al. 2017, Rheologia Ata Speial Issue on yield stress fluids 2017]. Here we will rather fous on the original flow features of yield stress fluids under various more or less omplex onditions: flow in a rheometer geometry, flow through a avity larger than the main onduit, displaement of a long objet through a bath of YSF, blade-oating (a proess widely used in ivil engineering for example for paints, mortars, et), tration of a YSF between two solid surfaes or as a result of gravity, destabilization of the interfae with another fluid (Saffman-Taylor instability, Rayleigh-Taylor instability). For eah of them, we will onsider the distintive harateristis of the flow with regards to what would be obtained with a simple fluid and, as far as possible, we will attempt to explain the physial origin of these differenes with simple arguments. During these review and analysis some original harateristis of YSF flows will be highlighted, whih will struture the plan of the paper: i) a transition from a solid regime to a flowing regime whih does not orrespond to a true liquid state, the flow in this regime may rather be seen as a suession of solid states during very large deformation (Setion 2); ii) a strong tendeny to loalization of the yielded regions in some small region of the material while the rest of the material undergoes some deformation in its solid state (Setion 3); iii) the deformation of YSF interfae with another fluid, in the form of fingers tending to penetrate the material via a loal liquefation proess (Setion 4). Finally these observations show that slow flows of YSF are a kind of extension of plasti flows for very large deformations and without irreversible hanges of the struture. 2. The different rheologial and physial states of yield stress fluids Solid regime The rheologial behavior of a yield stress fluid, apparently intermediate between a solid and a fluid, is something unexpeted from the usual or more sophistiated [Coleman et al. 1966] approahes of material behavior in ontinuum mehanis, i.e. whih distinguished two main lasses of materials, solids and fluids. This is also an original state of matter from the physial point of view [Coussot 2017]. This leads to original flow properties even under homogeneous stress onditions. Indeed let us onsider a series of reep tests with a given yield stress material, onsisting to apply a given stress to a sample and follow the deformation in time, then start again the same test at another stress level on the material plaed in the same initial state. For a stress below a ritial value we get a response typial of a solid [Cottrell 1964] (see Figure 1): the deformation first inreases, then very rapidly tends to saturate and reah a plateau [Coussot et al 2006]. Remarkably, for some YSF, if we maintain the stress appliation for a long time, we may observe an extremely slow flow, in fat a very slight inrease of the apparent deformation [Lidon et al. 2017]. This an nevertheless not be onsidered as refleting a standard visous behavior assoiated with the flow of a given material, sine in that ase the apparent shear rate ontinuously dereases in time (see e.g. pink urve in Figure 1), and the deformation remains very small. This more likely reflets some slight rearrangements of the struture and/or loalization of the deformation. For solids or glasses a similar effet may be observed, whih is alled aging. This effet, whih ours only for some materials, and does not indue signifiant

4 deformation over usual times of observation, will be left apart in the following disussion, and we will essentially be onerned by the approximate deformation plateau. (%) t (s) Figure 1: Deformation vs time for reep tests at different stress values (the numbers in front of eah urve orrespond to the stress values in Pasals) for a diret onentrated emulsion (82%). The liquid regime orresponds to urves tending to a slope 1 (blue urves), the solid regime orresponds to urves tending to a slope 0 (red urves). For the n 1 pink line we have at a given time ( t ): ln( ) nln( t) with n 1, whih implies nt and thus 0 when t. In order to further appreiate the behavior of the material in this solid regime it is useful to plot the stress vs the level of the deformation plateau, as is ommon for a solid material. In that ase, in some range of (small) deformations, the slope of the orresponding urve is onstant (see Figure 2a). Moreover, in this range, if we suddenly release the stress, the sample goes bak to its initial position (deformation equal to zero). Thus we are in a linear elasti regime. Then, for larger stresses, the slope of the stress vs deformation urve dereases progressively. In this regime, when the stress is released, the deformation does not go bak to zero, but to a finite value, whih inreases with the previously applied stress. This means that the material is elastoplasti, it undergoes a part of irreversible deformations. In the example presented in Figure 2a it is remarkable that in this regime the elasti omponent of the behavior still remains linear with the same slope as in the purely elasti regime. Note that here we only onsidered final states of the material under different stress values, but sine the transformations are not instantaneous, some visous omponent of the behavior might also be onsidered. Suh a behavior is naturally assoiated with a jammed struture of the material [Liu and Nagel 1998]. What does mean jamming here? This is the fat that the omponents of the material are elements of a struture spanning throughout the sample, and in whih neighboring elements interat signifiantly even at rest, so that thermal agitation alone annot break the struture. Under suh onditions some fore applied to the system will indue a deformation of the struture, assoiated with slight relative displaements of neighboring elements. For suffiiently small stress, the struture will be fully

5 reovered when releasing the stress. For larger stress (but still below the ritial one) there might be loally some breakage of the link between two (initially) neighboring elements, whih will not be reovered after stress release. These loal link breakage are also named plasti events in physis of jammed systems, and might be at the origin of plastiity in the solid regime of yield stress fluids [Hébraud et al. 1997]. (Pa) Liquid 40 Elastoplasti solid Elasti solid 10 (a) 0 Elasti reovery Plasti deformation (%) Figure 2: Stress vs final deformation from reep tests for (a) a typial YSF (here an emulsion, data from Maimouni et al 2016) and (b) typial solids (brittle or dutile). Liquid regime The above sheme is oneptually analogous to that developed for a solid [see for example Cottrell 1964, Tabor 1991]. For example, the loal link breakages orrespond to disloations for pure solids. For suh a standard solid the next step, i.e. for a suffiiently large stress, is breakage or plasti flow and loalization, depending on whether we are dealing with a brittle or dutile material respetively (see Figure 1b). A fundamental harateristis of this behavior for solids is that the transformation of the material is irreversible beyond some point (I in Figure 2b): in general it will not be possible to get bak the material with exatly the same rheologial behavior, beause the struture has been modified and annot reover its initial state by itself. That in partiular means that, even if in some ases we an observe some (plasti) flow the apparent behavior of the material ontinuously evolves, and this orresponds to an irreversible alteration of the material. The situation is fundamentally different for what we all a yield stress fluid: for a stress larger than a ritial value, i.e. the yield stress ( ), the deformation does not reah a plateau, it ontinuously inreases and its rate of inrease finally reahes a steady value. Thus we an onsider that we have a steady state flow at a onstant shear rate. This is what we will all here the liquid regime. The remarkable point, as ompared to standard solids, is that despite the very large deformations possibly undergone by the material in this regime, the material transformations are reversible: if we release the stress, and prepare the material in the same way as previously, we will get the same evolution of the deformation in the solid and liquid regime, and finally the same apparent visosity in steady state flow. That means that (i) the struture an support thermal agitation without breaking, (ii) it may be fully broken so as to get a flow, and (iii) it fully reovers its initial harateristis after full disintegration of the struture. Note that, sine the struture is not fully broken in the solid regime (despite some possible loal plasti events), we an presume that in the solid regime a slight

6 heterogeneity of the stress distribution will not have signifiant impat and we will get a roughly homogeneous marosopi deformation of the sample. On the ontrary, in the liquid regime, sine the struture is broken we an antiipate that some slight heterogeneity an easily lead to some signifiant heterogeneity of the deformation field (see below). How an we explain suh differenes between a plasti solid and a YSF? This is likely due to the nature of the elements and their mutual interations, and two essential ingredients might be neessary: (a) the elements develop soft elasti interations, in the sense that their mutual fores vary relatively slowly with the distane between elements, so that loal relaxation is fast; (b) the elements are in a disordered onfiguration, i.e. there is no partiular ordered struture taking plae over a large number of elements, and this disorder is suh that two suh onfigurations annot be distinguished. With suh harateristis we start to understand the YSF properties: below a ritial stress it is possible to somewhat deform the material without breaking the struture; beyond this ritial stress the struture is broken but the new (still disordered) struture obtained after this breakage does not signifiantly differ from the initial one, and eventually at rest the material rapidly relaxes towards its initial struture assoiated to the solid state. Finally we are faed with a unique kind of material: a self-healing solid. The situation that we here alled liquid regime differs from the standard definition of a liquid in physis, i.e. assoiated with densely paked moleules undergoing thermal agitation. Here the effets of thermal agitation remain negligible as ompared to the jamming or visous effets, so that by no way the struture resembles that of a liquid as above desribed. And sine as soon as the stress is released, the flow stops and the material reovers an apparent solid state, we an finally onsider that the liquid regime just orresponds to a large, ontinuously inreasing, deformation, of a selfhealing solid material. Suh materials are the objet of physial studies whih onsider an extension of the onept of plasti events to the liquid regime [Sollih et al (1997), Maloney and Lemaître (2006)]. Referring to the standard piture of a plasti material slowly flowing under a onstant ritial stress, it follows that the behavior of a YSF under slow flow onditions would express as 0. In fat, suh an expression annot be onsidered as a proper onstitutive equation sine it does not allow to define the flow rate, and measurements fortunately show that the stress inreases with the flow rate (see Figure 1). It is now well reognized that under simple shear the flow urve (i.e. shear stress vs shear rate in steady state) of simple YSF (i.e. exhibiting negligible thixotropi harateristis) is well represented, over several deades of shear rates, by a Hershel-Bulkley model, whih expresses as: n 0 k (1) in whih is the shear stress amplitude, the shear rate amplitude, and k and n are material parameters (generally n is situated between 0.3 and 0.5). Suh an expression is the point at whih we start to depart from the pure plasti behavior and we have to onsider that visous effets take plae in the liquid regime. This inrease of the stress with the flow rate means that the struture does not have enough time to fully relax so as to reah exatly the same state as at rest, between two steps of deformation, so that the deformation at the next step requires a somewhat larger stress than that needed for the struture initially at rest. 3. Simple shear flows

7 Slow flows We an now define more preisely what we all slow flows, on whih we fous in this paper. They orrespond to situations for whih the yielding harateristis play a major role. For a simple shear this is obtained when the first (onstant) term of the stress expression (1) is muh larger than the n n seond term. This orresponds to k 1. On the ontrary, if k 1 the material essentially behaves as a power-law fluid with flow harateristis approahing those of simple liquids. For 3D flow the situation is a priori more omplex. The standard desription (but not fully validated yet) of the onstitutive equation in 3D relies on an extrapolation of the simple shear behavior assuming that the stress tensor depends essentially on the seond invariant ( D ) of the strain rate tensor ( D ) [Oldroyd 1947]. It assumes that the stress tensor ( Σ ) is a sum of one term related (and likely proportional) to the yield stress in simple shear, and one related to the additional stress term related (with a fator proportional to k ) to the seond term of (1), whih gives n (1 n ) 2 Σ pi D D 2 k ( D ) D, where p is the pressure and I the unit tensor. Then II II we an ompare the invariants of the different omponents of this stress tensor expression (leaving apart the pressure term): the invariant of the seond term is, the invariant of the third term is n k 2 D II, whih we an roughly estimate as k V l n, in V is a harateristi flow veloity and l a harateristi flow length. Finally a generalization of our riterion of slow flow is the following approximate riterion: II Bi k V l n 1 (2) in whih Bi is the Bingham number. Loalization in simple shear rheometrial tests It remains that the stress inrease at low shear rate is rather slow, whih partiularly appears when the flow urve is represented in a logarithmi sale: the stress tends to a plateau assoiated with the yield stress when 0. A onsequene of this behavior is that the exat determination of the behavior (in terms of shear rate assoiated with a given value of shear stress) is diffiult as it requires to impose a perfetly homogeneous stress throughout the sample, otherwise the effetive shear rate an differ from the apparent one. As a orollary, a loalization of the deformation is highly probable at the approah of the yield stress. To illustrate that point let us onsider the flow in a one and plate geometry with a very small angle. The applied torque M is transmitted through onial surfaes with dereasing angles towards the lower plate, so that the shear stress in eah of these surfaes 3 writes 3M 2 R. Now, if the urvature of the periphery of the sample is not ideal, i.e. not spherial, R varies from one layer to another. This is a typial phenomenon observed with one and plate or parallel disk geometry, the peripheral free surfae tends to urve inwards, leading to a smaller radius in the middle of the sample. Under these onditions the shear stress may easily vary by several perent or tens of perent over the gap. For example we an frequently observe an inward urvature leading to a smaller radius by about 0.5 mm; for a typial radius of 3 m along the walls this leads to a stress at the neking point larger by 5%. For rheologial parameters with typial values suh that k 3 and n 0. 5, we an have an apparent flow for an apparent stress, whereas the effetive stress is in some entral region, so that the shear rate varies from 0 along the walls to s in the entral region. Thus the effetive shear rate is strongly heterogeneous in the

8 gap, and the apparent shear stress does not provide the exat stress value assoiated with the mean shear rate. At slightly lower stress value we an even expet a omplete loalization of the flow in a thin entral region, so that the rheometrial measurements just provide an estimation of the yield stress, but are not at all relevant onerning the effetive shear rate. This implies that it is rather diffiult to get relevant data onerning the effetive shear rate value in the sample below some -1 value, of the order of 0.1 s in the present example. Another diffiulty is that, in order to reah steady state we have to impose a flow during a time suffiient for the deformation to overome the ritial one assoiated with the liquid regime. Otherwise we do not observe steady state flow but a stress value assoiated with some transient behavior in the solid regime. To sum up, we an hardly onsider that under flow onditions a YSF is in a true liquid state. Instead we may see it as a (self-healing) solid material able to widely deform without losing its basi jammed struture, and inreasing the flow rate just slightly drifts the struture state from its basi state assoiated with rest. The speial ase of thixotropi yield stress fluids For the sake of simpliity, in all the disussion so far we were onsidering YSF exhibiting negligible thixotropy effets. Suh effets take the form of time evolution of the apparent visosity in the liquid regime, due to some progressive destruturing. This means that now, in ontrast with our assumptions above, the struture an signifiantly evolve when the material is deformed. Under these onditions, if the struture evolutions are irreversible we are not dealing with a YSF. Suh a material belongs to the lass of YSF only if its struture evolutions are reversible. This implies that a restruturing proess is expeted under other flow onditions than those whih led to destruturing, suh as, typially, some period at rest. Thus, even if we an observe trends with some similarity with those above desribed, the preise haraterization of suh materials in the solid and in the liquid regime beomes more diffiult than for non-thixotropi YSF, sine it is now neessary to take into aount the impat of the whole flow history on the urrent behavior. On the basis of observations and reasonings it was suggested (Coussot et al. 2009, Moller et al. 2009, Coussot and Ovarlez 2010), that this situation is essentially enountered with olloidal suspensions of aggregated partiles, in whih the partiles are somewhat dispersed and poorly interat when the material breaks; whereas simple YSF are obtained when the struture is made by onentrating relatively large deformable elements in a limited volume, suh as for foams, physial gels, emulsions, et., whih thus rapidly relax and reover a similar struture after breakage. In the present paper we onsider only simple YSF (Ovarlez et al. 2013), i.e. with negligible thixotropi effets. Note that the flow of thixotropi materials are extremely prone to develop shear-banding (see e.g. Ovarlez et al. (2009)). This is so beause the destruturing proess being signifiant, under some apparent shear imposed by boundary onditions, it may soon beome easier for the material to strongly destruture in some speifi region, so that its visosity drops to rather low value, while remaining in its solid state elsewhere. Wall slip A trend whih an play an important role in YSF flows is wall slip, a situation in whih the bulk moves as a rigid blok along a smooth solid surfae, the flow being onentrated in an extremely thin layer of material with a struture, and thus a behavior, differing from that of the bulk (Cloitre and

9 Bonneaze 2017). The struture of YSF, generally made of elements suspended in a liquid phase, is partiularly well fitted to the development of this effet. Indeed a simple extration of a very small amount of the interstitial liquid from the bulk, or the natural steri depletion of the density of suspended elements at the approah of the wall, are suffiient to form, along the wall, offering a resistane to shear flow smaller than the yield stress (in that aim one just needs to derease the displaement veloity at a suffiiently low level). As a onsequene the YSF bulk an move along the wall thanks to the shearing of this layer, even if the applied stress is smaller than the yield stress. This indues a ritial hange of the apparent behavior of the material whih, although it behaves as a solid unable to flow below a ritial stress with rough surfaes, turns to a apparently simple liquid in the same range of stresses with smooth surfaes. An illustration of this effet is shown in Figure 3: a heap of onentrated emulsion lying on a smooth solid surfae an flow (as a rigid blok) when this surfae is inlined, and without leaving traks of fluid behind, whih would be the hallmark of bulk shearing along the solid. There might instead remain a very thin layer of interstitial liquid, soon evaporating. Note that there are nevertheless small traks remaining along the lateral edges of the sample, likely due to some adherene of the line of ontat (see Zhang et al (2017)). Figure 3: Suessive views of a heap of emulsion (yield stress: 40 Pa) put on a smooth surfae, at times: (a) 0, (b) 15.6 s, () 26 s. The sample length is about 3 m. [Courtesy Xiao Zhang] 3. Loalization in omplex flows The original rheologial behavior of YSF, inluding a solid and a liquid regime, obviously leads to partiular flow properties. A first trend results from the fat that these two different states depend on the stress value as ompared to the yield stress value. For rheometrial flows in simple shear and when the stress field is approximately homogeneous the material will mainly be, as a whole, in a given regime, depending on the applied stress, exept at low shear rates due to some residual slight heterogeneity of the stress distribution (see Setion 3). As soon as the stress field is signifiantly inhomogeneous the situation is muh more omplex: the fluid an flow in its liquid regime in some regions while remaining at rest (while possibly deformed) in other regions; the spatial distribution of these different regions is expeted to evolve as the global flow rate is inreased, in partiular the volume of the liquid region should inrease. However the exat distribution of stress, whih will determine the distribution of arrested or flowing regions is in general a priori unknown, it results from the interplay between the boundary onditions and the rheologial behavior of the material. We start by onsidering a simple ase (flow in a onduit) for whih this stress distribution is known and then move to other more omplex geometries. Conduit flow

10 The typial situation of a YSF flow through a straight ylindrial onduit (of radius R ) illustrates these harateristis: for slow flows the liquid regime is onfined in a thin layer along the wall while the rest of the material in the middle of the onduit advanes as a plug; as the flow rate inreases the thikness of the sheared layer inreases and the plug size dereases. This strongly ontrasts with the laminar flow of a Newtonian fluid: in that ase the loal flow intensity simply inreases proportionally to the global flow rate so that the qualitative flow harateristis are independent of the flow rate. In partiular, for YSF, the thikness ( e ) of the sheared layer is expeted theoretially to tend to zero when the veloity tends to zero, or equivalently when the Bingham number tends to infinity. More preisely, for a HB fluid, we have e R 1/( m 1) m /1 m ( m 1) Bi (with m 1 n and Bi n n R kv ) (Chevalier et al (2013)). This poses a serious problem when this thikness falls below the lengthsale of the representative elementary volume of material. In that ase the apparent flow properties of the material through the onduit rely on the shearing of a layer of material whose behavior may differ from that of the bulk, so that the marosopi flow properties may differ from those expeted from theory and onstitutive equation determined from homogeneous flow onditions. Note that the wall roughness may also play an important role in suh a situation, as it an interat with the sheared thikness. It may also lead to a speifi situation if it is lose to zero (with regards to the size of the elements of the jammed struture), this is wall slip. Flow through a larger volume Let us onsider the simple ase of a YSF through a straight onduit with a sudden enlargement of the onduit diameter over a limited length before a sudden restrition into the initial diameter. In the main onduit, at large Bingham number, we have essentially a plug flow with a thin sheared layer along the wall. Surprisingly these flow harateristis are weakly modified during the motion through the larger onduit, the entering fluid essentially keeps a plug state, while most of the rest of the fluid in the larger onduit is undeformed (see Figure 4). Finally, only a thin layer of fluid, situated between these two solid regions, is sheared, ensuring the veloity ontinuity. Moreover the shear rate in this layer is approximately onstant. A similar situation, with similar flow harateristis (after resaling by the plug veloity) and similar thikness of the sheared layer is observed for different values of the Bingham number as long as it remains larger than 1 (Chevalier et al, 2013), but it seems diffiult to predit the size of the sheared layer from simple analytial arguments. Figure 4: Flow of a yield stress fluid (onentrated emulsion with =74 Pa) through a sudden enlargement (diameter 7 m) then ontration (bak to 3.5 m diameter). Veloity map in a longitudinal ross-setion for a mean veloity in the small onduit of 0.16 mm/s. The veloity amplitude (in mirons per seond) in eah voxel is represented

11 by a olor aording to the sale at the bottom right (in mirons/s). [Figure 1 of Chevalier et al (2013)]. Displaement of a plate along one of its axis A similar result is obtained for a plate moving vertially through a yield stress fluid bath in a ontainer. For a plate penetrating the bath or extrated from the bath (dip-oating) a liquid region exists only in a thin layer along the moving plate, while the rest of the fluid is slightly deformed in its solid regime (see Figure 5). One again it appears that the shear rate is almost onstant in the sheared layer and the size the liquid region does not seem to hange when varying the Bingham number in a range above 1, i.e. the flow fields are similar after resaling by the plate veloity. However, prediting the value of the sheared layer from simple analytial arguments remains hallenging. Figure 5: (left) Deformation undergone by a bath of Carbopol gel during the vertial penetration of a plate (6 mm thik) (here seen from its side). The material was initially set up in the form of suessive layers with different olors (but same rheologial behavior). The deformation of the interfaes, observed along the transparent wall, allows to see the material deformation, and we know from PIV measurements (see Boujlel et al 2012) that this well reflets the deformation inside the bath. Two regions an be distinguished: within a distane of the order of the plate thikness the YSF is strongly deformed; beyond this distane it is slightly deformed. [Courtesy Jalila Boujlel]. (right) Sheme of the deformation field around the plate, as dedued from diret internal measurements of the flow field in time (adapted from Boujlel et al 2012). The solid and liquid regions were identified from the evolution of the total deformation undergone by material elements along their trajetories: the deformation does not overome a small value in the solid region, while it ontinuously inreases in time and reahes large values in the liquid region.

12 Blade oating Similar harateristis are observed in blade-oating, although the existing observations are muh less preise (Maillard et al (2016)) or obtained from rough numerial simulations (Maillard (2015)). The priniple of the experiment is the displaement of a vertial or inlined blade along the surfae (and somewhat dipped in) of a uniform YSF layer lying over a horizontal hannel. The proess leads to the aumulation of an inreasing volume of material against the front of the blade, and leaves behind a uniform thikness of material (see Figure 6). The data show that the flow is essentially the result of the displaement of a rigid YSF blok attahed to the blade, relatively to a material layer at rest along the bed, while some thin layer is sheared to ensure the ontat between these two solid regions. Obviously, sine the region stuk to the blade is ontinuously fed by additional material, the above desription is an approximation valid over a short displaement. Finally rough observations and numerial simulations tend to show that the sheared layer thikness does not hange signifiantly as the veloity is inreased. Figure 6: Evolution of the experimental horizontal veloity field during the blade-oating in a hannel filled with a yield stress fluid (Carbopol gel, or yield stress 98 Pa. The blade veloity is V=5 mm/s, the initial (uniform) height of fluid is 17 mm, the depth of penetration of the blade into the fluid is 7 mmm). The pitures orrespond to suessive steps of displaement of 100 mm. In this representation the blade has a veloity equal to zero (dark blue) while the hannel has a veloity V (red). Synthesis From the above desription we an dedue several important points: - Despite omplex boundary onditions, YSF tend to develop a deformation field suh that the liquid region undergoes essentially a simple shear; this result is absolutely not trivial, for example the entrane flow in a die was onsidered as involving mainly an elongational flow involving all the material volume (Benbow and Bridgewater 1993). - The resulting sheared regions exhibit some analogy with shear-bands as observed in plastiity (Nadai 1950): indeed, they support the entire relative motion of the material and the boundaries, their thikness is in general small ompared to the sample size, the shear rate is almost onstant inside the band and their thikness does not vary signifiantly with the veloity for slow flows. - In ontrast with usual solid plasti materials in whih shear-bands an develop, here the large deformation undergone by the YSF in the band do not alter irreversibly its mehanial behavior.

13 Suh properties are harateristis of plasti flows, but with the speifiity that the YSF is still able to self-heal. This suggests that a speifi treatment of slow flows of YSF should be developed, inspired from plasti flows or, onversely, slow flows of YSF provide a way to study plasti flows. 4. Deformation of YSF interfaes Let us now examine what happens for less onstrained flows of YSF, i.e. a flow with large interfaes with a simpler fluid (air or a Newtonian liquid). Loalization in (attempted) extensional flows: jets and droplets Let us onsider the formation of a droplet, whih is generally obtained by pushing a fluid through an opened onduit. For a simple fluid this forms a jet whih will separate in droplets after some distane depending on the visosity, the surfae tension and the flow rate. This phenomenon is governed by the Plateau-Rayleigh instability. The instability basially ours beause the surfae energy of a ylinder of fluid is larger than for the same fluid volume separated in spherial droplets of a partiular diameter. During the separation, visous effets an nevertheless slow down the proess so that this separation will our in a harateristi time depending on the balane between apillary and visous effets. A similar situation is expeted for a YSF, but now the visous energy needed to separate the material in droplets is finite, i.e. it is larger than a finite value whatever the rate of the transformation, whih obviously an prelude the development of the instability if surfae tension effets are not suffiiently large. In order to appreiate the tendeny of the fluid to separate a jet of YSF (initially a ylinder of radius R ) in droplets we an rely on rough estimates: the surfae energy 2 reated per droplet is of the order of 2 R (where is the surfae tension of the fluid, see Appendix 1); the visous energy needed to deform the fluid during this separation is of the order of that needed to deform the volume of a strain 1, whih would be (with the droplet volume, 3 whih is of the order of 4 R 3 ). We dedue that YSF droplets should spontaneously form if R. This means that a jet of a few millimeter diameter will easily separate in droplets if the yield stress is smaller than a few Pasals. Otherwise the jet of YSF will tend to keep its ylindrial shape, possibly deformed when reahing a solid surfae and then forming a oil (see Rahmani et al 2011). However, in most ases, gravity will start to play a role beyond some ritial length of the jet, and will lead to the rapid separation in droplets. This effet an easily be understood. Aording to the standard 3D expression for the onstitutive equation (see Setion 2) the ritial normal stress needed to indue an elongational flow of a ylinder of YSF is equal to 3. On the other hand the normal stress ating on a layer of fluid in a jet is equal to the weight of fluid situated below this layer, divided by the jet setion area, i.e. gl. When this normal stress due to fluid weight is larger than 3, an elongational flow starts, whih tends to derease the setion area, so that the differene between the two stresses inreases, leading to faster elongation. Sine we are dealing with strongly shearthinning fluids, this leads to a atastrophi derease of the fluid setion area and finally a separation of the material in that region in a very short time (see Figure 7). A similar effet ours when one attempts to elongate a yield stress fluid volume initially omprised between two parallel plates that are then moved progressively away from eah other. Instead of simple uniaxial elongational flow we observe a separation of the material into two approximately onial parts attahed to the plates: see Figure 8.

14 In fat this experiment provides a deeper insight in the proess of separation of a volume of YSF into two parts: we an see that the whole material is initially deformed but soon the regions lose to the disk surfaes and situated at the periphery stop flowing, sine the shape of these regions remain fixed during the next steps; a similar proess goes on for new fluid volumes (now loser to the entral axis) as the disks are moved farther, so that inreasing volumes stop flowing along the disks, and in the final step there remains only a small entral region whih is deformed and eventually breaks (see Figure 8). Thus this flow may be seen in a different way: it an be onsidered as the penetration, through the YSF, of a radial air finger towards the entral axis. This neking proess is reminisent of the dutile breakage observed for example with metals during a tration test (Nadai 1950). We find again a strong analogy of the properties of slow YSF flows with those of plasti materials. Figure 7: Suessive views of a mayonnaise sample at different times during extrusion. The initial time (0 s) orresponds to the separation of the previous extrudate. [Figure 1 of Coussot and Gaulard 2005]. Figure 8: Suessive views of a YSF shape during a tration test. [Courtesy Jalila Boujlel] Saffman-Taylor instability The Saffman-Taylor instability (STI) is observed when a fluid pushes a more visous fluid in a onfined geometry. The term onfined here means that the distane between the solid walls is muh smaller than the harateristi length in the flow diretion. Suh boundary onditions are typially

15 enountered in porous media or between two parallel plates (i.e. Hele-Shaw ell). Under stable onditions the length of the interfae between the two fluids remains minimal, so that it is straight for a flow in a single diretion, or irular for a radial flow. When the STI develops, the interfae evolves in the form of fingers whih, will then evolve in different ways depending on the rheologial behavior of the material (Lindner et al. 2002). The most frequent situation for whih the STI is observed with a YSF is the separation of two plates initially in ontat with a thin layer of paste; as the plates are moved away the layer thikness inreases, whih indues a radial flow towards some entral position; if the distane between the plates is suffiiently small, the radial veloity is muh larger than the axial one, so that the flow approximately orresponds to a radial flow driven by the air entering the gap, whih orresponds to the boundary onditions under whih the STI an be onsidered. This phenomenon an be observed as soon as some thin layer of paint, glue, puree, yoghurt, is squeezed between two solid surfaes (a tool, a spoon, et) whih are then separated: we get a harateristi fingering shape as shown in Figure 9. Note that it is possible to observe suh pitures beause the fluid leaves arrested regions behind the flow front, whih finally give this definitive shape. This ontrasts with simple liquids for whih the fingers soon relax under the ation of wetting effets and a uniform layer rapidly reforms. Figure 9: Fingering aspet of the remaining deposit of yield stress fluid after separation of two plates initially squeezing a thin fluid layer. [Courtesy Quentin Barral] For simply visous fluids the origin of the instability is that any unevenness (loal urvatures) of the interfae allows a subsequent easier advane of the fluids in front of the urvature in the flow diretion, while for a urvature in the opposite diretion the motion is more diffiult, it requires more pressure beause the volume of the more visous fluid to push is larger. It results that the perturbation will tend to develop further, exept if surfae tension, whih on the ontrary works against the deformation of the initial interfae, is suffiient to ounterbalane this effet. We an express this mathematially following the approah of Homsy (1987). We onsider a material pushed by an invisid fluid and flowing in a given diretion x between two parallel plates separated by a small distane 2 b. We assume that the interfae (initially situated in x 0 ) undergoes a small perturbation 0 exp( ikz) in whih k and 0 are two parameters. the perturbation. This indues a urvature of the interfae approximately equal to 2 k is the wavelength of 2 1 '' 1 k, whih leads to an additional pressure differene along the interfae k 2. Let us onsider a small band of fluid along the diretion x. The additional pressure speifially needed to displae this band of the additional loal distane is found from a momentum balane on the small volume displaed: 2bdp 2 w, in whih w is the shear stress along the wall. Finally the total additional

16 k 2 0 pressure required is b dp w at the front. If this term is of the sign of, i.e. if with 2 b w, the additional pressure will tend to push even more the regions whih have already been urved towards positive values of x, the perturbation will be amplified and the instability will develop. We see that for the instability to develop, the visous fores must be suffiiently large ompared to the fores resulting from surfae tension. For a simple fluid this means that the instability will anyway develop beyond some ritial veloity. It is tempting to use suh an approah for a yield stress fluid, remarking that as soon as the material is flowing w is larger than, whih implies that the instability for a YSF would appear at any flow rate if 2 b 2 L. More omplete and preise theoretial or numerial approahes of this instability for yield stress fluids have been developed, still within a fluid mehanis frame, i.e. essentially taking into aount the behavior of the material in its liquid regime. This led to detailed instability riteria and desriptions and analysis of various trends (Coussot 1999, Fontana et al 2013, Ebrahimi et al. 2016). The validation of these onlusions by onfrontation with experimental data remains extremely limited, and when this was done a reasonable agreement for the finger size (Lindner et al. 2000, Derks et al 2003, Maleki-Jirsaraei et al 2005), but a strong disrepany was found onerning the instability riterion (Barral et al 2010), suggesting some important effet is missed. Thus, for a YSF we have an original situation as we an get a hydrodynami instability at vanishing veloity. Another original aspet of this instability for YSF is that at suffiiently low flow rate the fingering proess leaves arrested fluid volumes behind the advaning front. This again results from the speifi solid-liquid behavior of YSF: as soon as some fingering starts, the regions for whih the urvature is in the diretion opposite to flow, require a larger pressure gradient to flow (see above equation for additional pressure); if the (average) flow rate is imposed at a moderate value the pressure to maintain the flow in the other regions will be smaller than the ritial pressure to indue motion in the above regions (situated behind). This differene inreases as the fingers grow so that the regions behind will remain definitively arrested. Atually, these observations suggest that the desription, within the frame of fluid mehanis, of this instability for YSF, might not be appropriate. More preisely we have important differenes with the assumptions at the basis of the standard fluid mehanis approah. The instability develops from an initial state at rest, whih means that the material is in its solid regime when one looks at the possible destabilization; a linear stability analysis in fluid mehanis neglets the initial deformation required to reah the liquid regime, whereas it relies on the analysis of the evolution of a small perturbation of the interfae, whih is assoiated with the very first deformation of the material, still likely in its solid regime in most of its volume. Obviously, in this ontext, the standard fluid mehanis approah assumes that the evolution of this perturbation takes plae while all the material is flowing (possibly at different veloities), and thus does not take into aount the arrested regions. Finally, taking into aount these features, we an propose a different view of the problem, that seems loser to the involved physial proesses. Sine the material regions along the air fingers remain at rest all along the advane of the rest of the material, we may see the proess as the progression of air fingers penetrating the YSF initially at rest. Under suh onditions we expet that, as for an objet penetrating a bath of YSF (see Setion 2) the material will be in its liquid regime only in a small volume along the finger. But the situation is likely even more ritial here as this is not a solid objet whih penetrates the material: sine air indues a negligible stress along the finger sides, we expet a liquid regime to exist at a given time only in the newly deformed region around the finger tip. This would imply that the treatment of this problem has nothing to do with the usual treatment in fluid mehanis, it is essentially a matter of flow loalization in a large solid volume

17 (note that for a tration test this assumes that the aspet ratio is very small, so that residual elongational flow does not allow the material to reah its liquid regime). Rayleigh-Taylor instability (RTI) The Rayleigh-Taylor Instability (RTI) is an instability whih ours when a denser fluid rests on top of a lighter one (Rayleigh 1883, Sharp 1984) as a result of gravity effets. As it develops, the two fluids penetrate one another, in the form of fingers. The instability is driven by density differene and the aeleration to whih the fluids are submitted, while surfae tension provides a stabilizing effet. A example of situation where this instability an be enountered is in oil well ementing operations, in whih yield stress fluids of different densities (drilling muds and ement, e.g), whih behave as solids at rest, may be pumped into the well in an ill-favored density order (Bittleston and Guillot 1991). For Newtonian fluids the instability is well desribed. The interfae starts to deform and give rise to fingers penetrating the initial volume of the other phase. The wavelength of these fingers dereases when the density differene inreases or interfaial tension dereases. The fluid visosities do not play any role in the instability riterion, but have an impat on the dynamis. It is worth noting that stable situations may be obtained in some ases when the above fluid has a density larger than the bottom fluid, if the size of the interfae is suffiiently small. This explains why the liquid may not flow from a suddenly turned over bottle, if its nek is suffiiently small. A typial example of the development of the instability for two simple liquids, just beyond the stable regime, is shown in Figure 10a: a regular sinusoidal interfae forms with a wavelength slightly larger than the bath length, and an amplitude growing in time. Beyond some amplitude the fingers tend to aelerate and finally form jets rapidly penetrating the other phase, with mushroom-like shape after some distane. For a yield stress fluid the instability harateristis differ signifiantly from those above desribed. Let us onsider a simple (denser) liquid plaed above a bath of a YSF. Under some onditions the interfae does not move at all, even after a long time. This orresponds to a perfetly stable situation. Under onditions that an be slightly different (in partiular onerning the density differene, the yield stress) the interfae destabilizes. A very fast motion of the YSF into the Newtonian solution ours in the form of mushroom-like bursts (see Figure 10b). These bursts penetrate the liquid and reah the top of the tank in a time of the order of one seond. At the same time (although not visible in these images) more or less symmetri liquid fingers penetrate the YSF bath.

18 Figure 10: RTI development for (a, left) visous Newtonian fluid (Silione oil, 0.35 Pa.s visosity) initially below a denser olored water-ethanol solution ( 1.3 % ) at times (from top to bottom): 0 (mid-time of gate opening), 6, 12, 18, and 24 s; and (b, right) a onentrated (white) emulsion ( 9.6 Pa ) initially below a (brown) denser salt solution -3 ( 600kg.m ) at times: 0, 1.9, 3, 3.8, and 4.5s. Figure 1 of Maimouni et al Some features are remarkably different from those usually observed with simple liquids: i) even just beyond the stable onditions the instability develops in the form of a atastrophi proess as desribed above; ii) in unstable ases, the size of the unstable bursts (i.e. half the instability wavelength) approximately keeps a onstant value, signifiantly smaller than the interfae length, whatever the experimental onditions (tank size, density differene, yield stress). For simple fluids the linear stability analysis of the problem onsists to assume some small perturbation of the interfae and see how it should evolve. This leads to a riterion for the instability to our whih orresponds to a omparison between apillary effets and gravity effets. Obviously suh a riterion annot work for a YSF, sine the yield stress, like apillary effets, tends to resist to the interfae deformation, whih is naturally assoiated with bulk deformation. In fat, one again, this is not exatly the yield stress, but the strength of the material in its solid regime, whih ats to damp a possible perturbation of the interfae from its initial solid state. Atually, this instability has already been studied for solids, beause it relates to various observations of natural phenomena in whih a liquid apparently opens its way through a layer of denser solid material above it: slowly areting neutron stars (Blaes et al 1990), volani island formation (Marsh 1979), salt dome formation (Zaleski and Julien 1992), and more generally magmati diapirism in the Earth s mantle and ontinental rust (Piriz et al. 2013, Burov and Molnar 2008). In this frame, assuming negligible surfae tension effets, it was remarked that two main origins of destabilization an be onsidered depending on the material behavior.