Nonlocal rheological properties of granular flows near a jamming limit

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1 PHYSICAL REVIEW E 78, Nonlocal rheological properties o granular lows near a jamming limit Igor S. Aranson, 1 Lev S. Tsimring, 2 Florent Malloggi, 3 and Eric Clément 3 1 Materials Science Division, Argonne National Laboratory, Argonne, Illinois 6439, USA 2 Institute or Nonlinear Science, University o Caliornia, 95 Gilman Drive, San Diego, La Jolla, Caliornia 9293, USA 3 PMMH, UMR7636, CNRS-ESPCI-University, P6-P7, 1 Rue Vauquelin, 755 Paris, France Received 18 January 28; revised manuscript received 19 June 28; published 8 September 28 We study the rheology o sheared granular lows close to a jamming transition. We use the approach o partially luidized theory PFT with a ull set o equations extending the thin layer approximation derived previously or the description o the granular avalanches phenomenology. This theory provides a picture compatible with a local rheology at large shear rates G. D. R. Midi, Eur. Phys. J. E 14, and it works in the vicinity o the jamming transition, where a description in terms o a simple local rheology comes short. We investigate two situations displaying important deviations rom local rheology. The irst one is based on a set o numerical simulations o sheared sot two-dimensional circular grains. The next case describes previous experimental results obtained on avalanches o sandy material lowing down an incline. Both cases display, close to jamming, signiicant deviations rom the now standard Pouliquen s low rule O. Pouliquen, Phys. Fluids 11, ; 11, This discrepancy is the hallmark o a strongly nonlocal rheology and in both cases, we relate the empirical results and the outcomes o PFT. The numerical simulations show a characteristic constitutive structure or the luid part o the stress involving the conining pressure and the material stiness that appear in the orm o an additional dimensionless parameter. This constitutive relation is then used to describe the case o sandy lows. We show a quantitative agreement as ar as the eective low rules are concerned. A undamental eature is identiied in PFT as the existence o a jammed layer developing in the vicinity o the low arrest that corroborates the experimental indings. Finally, we study the case o solitary erosive granular avalanches and relate the outcome with the PFT analysis. DOI: 1.113/PhysRevE PACS number s : 45.7.Ht, 68.8.Bc I. INTRODUCTION Gravity-driven particulate lows are central to a variety o chemical technologies, the pharmaceutical, ood, metallurgical and construction industries, as well as agriculture. These industries ace serious challenges in the handling o granular materials, including their storage, transport, and processing. Furthermore, the transport properties o various natural particle-laden lows, such as dune migration, erosiondeposition processes, land slides, underwater gravity currents and coastal geomorphology, are staggered with many unresolved questions around the mechanics o the dense solid raction motion. The dynamics o dense slowly sheared granular media 1,2 are strongly dierent rom dilute rapid granular gases 3. One o the distinct eatures o dense granular assemblies is the luidization transition and the onset low above a certain critical value o the shear stress. A comprehensive survey o granular lows in a variety o experimental settings and conditions perormed by various groups around the world was conducted by a French research network Groupement De Recherche Milieux Divisés MiDi 1. The salient eature o this work was to propose in the region o dense low, not too close to jamming and o course, not in the domain o dilute lows where kinetic theory would prevail, a constitutive equation based on the existence o an eective riction coeicient. The constitutive relation solely depends on a single parameter, the inertia number I 1,4, which received a simple interpretation as the ratio o a microscopic reorganization time scale to a macroscopic time o shear here, the elementary macroscopic volume is typically o one grain size. This approach was proven to be an essential step as it provides the correct phenomenology when tested against kinetic theories developed previously or the dilute lows. It was also successully extended to take into account lows conined between rigid walls 5. Nevertheless, a ull picture o the rheology o granular lows is still lacking a conceptual clarity. Namely, the transition rom a dense low regime to a jammed phase remains a most debated issue 6 8. In particular, systematic low experiments with smooth glass beads on an inclined plane have established an empirical low rule also known as Pouliquen s low rule which has the remarkable property o rescaling all available data remaining ully compatible with a local riction coeicient picture. Nevertheless, one obtains a paradoxical situation where the associated eective riction coeicient involves a length scale describing how the granular lows jams so-called stop height h stop and this is quite ar rom the described lowing regime. Furthermore, recent experiments 9,1 and simulations 11 on the same systems have explicitly shown, in the close vicinity o jamming, deviations rom the Pouliquen s low rule, hence proving the nonlocality o the granular low close to arrest. In addition, such a low rule, relevant or model glass spheres, exhibits substantial deviations rom experiment when other types o particles as rough sandy grains are used 1,12,13. Since or dense, sheared granular assemblies, no ull derivation o the constitutive relations is so ar available, one needs to appeal to phenomenology. Several suggestions were made to justiy the actual constitutive properties 14 17, but no consensus has been reached because the jamming o granular lows is still an area o active research. Some years ago, a continuum approach, the partial luidization theory PFT, was proposed in order to incorporate explicitly a de /28/78 3 / The American Physical Society

2 ARANSON et al. scription o a luid-solid transition via a Landau-Ginzburg phenomenological theory o phase transitions. This approach allows or a nontrivial interpretation both o simulation data and o experiments 18,19. It naturally incorporates the constitutive relations in terms o a solid stress tensor combined with a luid stress tensor which is valid in a wide range o shear strain rates. The momentum transport equations, valid on both sides o the jamming transition, allow us to describe a mixture o jammed and lowing phases. It nevertheless depends sensibly on the structure o the eective relations describing the phase transition and how the luid and the solid stress components are mixed together. This question is at the ocus o the present paper. The PFT is extended to match quantitatively a set o numerical and experimental data. Several adjustments to the early ormulation o the model parameters were made by comparisons with numerical and experimental results. For the sake o simplicity, a reduced set o equations have been derived in earlier works 18 ; the momentum transport equations o the ull PFT theory were approximated to obtain a more handy thin layer limit. The model also used a simpliied constitutive approach to render the luid phase i.e., a viscouslike relation, independent o the conining pressure. Remarkably, even in this simpliied orm, all o the qualitative eatures ound so ar or erosion-deposition phenomena involving granular lows were reproduced 12,2,21, and some quite quantitatively. Note that recently this approach has also been used to describe real geophysical lows 22. Important questions still remain such as how to connect the PFT more quantitatively to extended numerical results and to experimental data. Furthermore, one need to understand how dierent various low conditions can be modeled as ar as grain stiness, shape or other granular eatures are concerned. The structure o the paper is as ollows. In Sec. II we discuss various models or sheared dense granular lows, including PFT, and their relation to the most recent experimental results. In Sec. III, we present the results o sot-particle numerical simulations. First we describe the simulation technique and then discuss the existence o an eective local rheology or an extensive set o simulations eaturing a twodimensional granular packing under shear. We also discuss the limitation o the previous approaches based on a single dimensionless parameter inertia number I. We give an interpretation o the results in the ramework o the partial luidization theory, and show how it can be extended such as to capture the existence o particle sotness and the dependence o the liquid phase rheology with the conining pressure. This extended version o the PFT theory will be used to conront in the ollowing sections, several experimental results. In Sec. IV, lows o rough sandy grains down an incline will be considered. We will address speciically the question o low rules and we will ocus on the existence o a jammed layer at the bottom o the low. The relation between the low proiles derived rom PFT and the experimental indings will also be addressed. Section V is dedicated to erosiondeposition waves. We will ocus on the dynamics o a solitary erosion deposition wave and we will compared the PFT results to the outcome o experimental indings or rough sandy grains. Discussion and conclusion are in Sec. VI. PHYSICAL REVIEW E 78, II. BRIEF REVIEW OF THEORETICAL MODELS AND RELATION TO EXPERIMENTS A. Bagnold scaling From heuristic dimensional arguments Bagnold proposed a simple local relation between the strain rate and shear stress shear 23, shear d This relation ollows rom the assumption that the only relevant time scale in a dense quasistatic granular low is the deormation time scale T 1/. This hypothesis was tested and improved in many subsequent presentations 4,14,24,25. While the Bagnold relation 1 is likely satisied in the bulk o a dilute granular low, dynamic inhomogeneities in slow and dense granular lows result in substantial deviations rom this simple scaling law. In addition, deviations rom Bagnold scaling can stem rom grain compressibility and cohesion 11,26. B. Pouliquen s low rule and local eective riction A more general relation between shear stress and the strain rate has been recently proposed in a series o works 1,4,5,27. In dense lows, the normal stress pressure is not directly related to the strain rate as in dilute regime, and must be included as an independent parameter. Pressure p, shear stress xy, strain rate, grain size d, and density deine two independent dimensionless parameters, the eective riction coeicient e 28, e = xy 2 p and the dimensionless shear rate I, I = d. 3 p/ The parameter I can be interpreted as the ratio o two dierent time scales, the deormation scale T =1/, and the microscopic or coninement time scale T p =d / p. The volume raction o granular lows or, correspondingly, density is another quantity controlling granular lows. One reasonable assumption is that the volume raction is a unction o the parameter I. In this situation one can ind a unique local relation between the eective riction e and the rescaled shear rate I. Experiments and numerical simulations indeed support the existence o a local relation between the eective riction e and the scaled strain rate I or a variety o lows. In the most o the experiments the eective riction e is a monotonously increasing unction o I limited by the static riction value 1 or I and the dynamic riction value 2 or high values o I. On the basis o extended experimental results, Pouliquen et al. 4,5,27 proposed the ollowing relation between e and I which its most o the experimental data or glass beads: e I = xy p = I /I

3 NONLOCAL RHEOLOGICAL PROPERTIES OF GRANULAR Here I.279 is a constant, and parameters 1, 2 depend on the materials properties o grains. C. Flow rule The low rule relates the mean downhill velocity o a chute low U = h v x z dz/h with its thickness h and with the chute angle. For the most studied experimental system, corresponding to monodisperse spherical glass beads lowing on rough substrates o glued beads, an empirical relation was obtained. It was shown that this relation allows us to collapse within a precision o experiment all experimental data onto a single curve or the dimensionless velocity U / gh or Froude number Fr 1,2,5, Fr = U h * 5 gh h stop with constant *.13; here the characteristic height h stop corresponds to the minimum thickness o the granular layer sustaining steady-state low. Note that this relation is consistent with a local rheology e I. It yields a Bagnold velocity proile characterized by a ratio r o the mean velocity U to the surace velocity V s, r= U V s =3/5. More recent complimentary measurements perormed in the vicinity o the jamming limit h h stop have extended this low rule 9,1 to a more general orm Fr = U gh * h h stop 1 with *.22. Thereore, close to jamming, the rheology locality is now under question as a simple dependence o the eective riction e against a single parameter I cannot work anymore. Moreover, experimental measurements o the velocity ratio yield r=u /V s 1/2, i.e., a value corresponding to a linear shear low. Systems involving irregular grains such as rough sand oten show substantial deviations rom the standard Pouliquen s low rule 1,13 as well. In particular, the experiments revealed the existence o a static layer o thickness z stat below the continuous low and signiicant deviations o the ratio r=u /V s 3/5 ound or the lows o glass beads. We will discuss these issues below. D. Partial luidization theory A popular phenomenological approach to dense gravitydriven granular lows is based on the two-phase description o granular matter; one phase corresponding to rolling grains and the other phase corresponding to static ones. This approach, the so-called Bouchaud, Cates, Ravi Prakash, Edwards BCRE model, was suggested in Re. 29 and was urther elaborated on by many other research groups, see, e.g., Re. 3. However, in real granular lows there is no sharp distinction between a rolling phase and a static phase: The transition rom a static to lowing regime is continuous. An attempt to develop a more general theory describing the nonstationary behavior o dense partially luidized granular 6 lows was made in Res. 18,19. This theory is based on the intuitive idea that in dense granular lows a signiicant part o the stress is transmitted through quasistatic contacts between particles in addition to the luid stresses transmitted through particle collisions and sliding riction. Thus, the stress tensor is represented as a sum o luid and static components, = s +. 7 The key idea o PFT is that the separation o the stress tensor into static and luid parts is controlled by the so-called order parameter,. For the shear component o the stress, this separation is deined by a separation unction q, xy PHYSICAL REVIEW E 78, s = q xy, xy = 1 q xy. 8 Without loss o generality, the order parameter is scaled in such a way so that in granular solid =1 and in a well developed low granular liquid. On the microscopic level the order parameter can be deined as a raction o the number o static or persistent contacts o the particles Z s to the total number o the contacts Z, = Z s /Z within a mesoscopic volume which is large with respect to the particle size but small compared with the characteristic size o the low. Deined in such a orm the order parameter can be relatively easily extracted rom the molecular dynamics simulations o sot particles 19. The separation unction q must conorm to two boundary conditions q =1 =, q = =1. For the sake o simplicity, a linear dependence q=1 was assumed in the irst presentations 18. More recently, we extracted this unction rom a two-dimensional sot particles molecular dynamics simulation, yielding a more speciic dependence 31 q = 1 u 9 with u 2.7. Note that the unction q, which ixes the mixing ratios between the static and the luid parts o the stress might be material dependent. In the ollowing, we will examine in more detail its inluence on the jamming properties o the low. Due to a strong dissipation in dense granular lows the order parameter is assumed to obey purely relaxational dynamics controlled by the Ginzburg-Landau-type equation or the generic irst-order phase transition, t + v = l2 2 F,. 1 Here is the characteristic time scale o the problem, and l is the order parameter correlation length which is assumed to be o the order o the grain size. These parameters can be scaled away by the renormalization o the length, time, and velocity t new = t/, r new = r/l, v new = v /l. 11 For the sake o brieness below we drop the subscript new. F, is a ree energy density which is postulated to have two local minima at =1 solid phase and = luid phase to account or the bistability near the solid-luid transition. The relative stability o the two phases is controlled by the

4 ARANSON et al. parameter which in turn is determined by the stress tensor. The simplest assumption consistent with the Mohr-Coloumb yield criterion is to take it as a unction o eective slope =max mn / nn, where the maximum is sought over all possible orthogonal directions m and n we consider here only two-dimensional ormulation o the model. For planar two-dimensional shear low we deine the control parameter = = xy / yy. In the context o the low on an inclined plane the parameter is simply the slope o the plane, =tan, where is the inclination angle 19. III. SOFT PARTICLES UNDER SHEAR AND CONSTITUTIVE RELATIONS A. Numerical techniques To model the interaction o individual grains we use the so-called sot particles approach described in detail in Res. 19,24. The grains are assumed to be noncohesive, dry, inelastic disklike particles. Two grains interact via normal and shear orces whenever they overlap. For the normal impact we employ the spring-dashpot model 32. This model accounts or repulsion and dissipation; the repulsive component is proportional to the degree o the overlap, and the velocity-dependent damping component simulates the dissipation. The model or shear orce is based upon the technique developed by Cundall and Strack 33. It incorporates tangential elasticity and Coulomb laws o riction. The elastic restoring orce is proportional to the integrated tangential displacement during the contact and limited by the product o the riction coeicient and the instantaneous normal orce. The grains possess two translational and one rotational degrees o reedom. The motion o a grain is obtained by integrating Newton s equations with the orces and torques produced by the grains interactions with all o the neighboring grains and walls o the container. The advantages and limitations o the employed contact orce model were thoroughly studied by a number o authors 24,32. In act this is the simplest model accounting or both static and dynamic riction. All quantities are normalized by an appropriate combination o the average particle diameter d, mass m, and gravity g while the simulation are perormed or zero gravity, the gravity acceleration g is used only to construct ormal time scale t = d/g. Thus, distances, x,y, time, t, velocity, v, orce, F, spring elastic constant k, and stress, are, respectively, measured in units o d,t,v = dg, F =mg, k =mg/d, =mg/d 2. The equations o motion are integrated using the ithorder predictor-corrector 34 with a constant time step. The spring constant k n and damping coeicient n were chosen to provide the desired value o the restitution coeicient e and guarantee an accurate resolution o an individual collision. Typically, we used a tangential riction coeicient between particles t =.5, we varied the dimensionless stiness o the grains k n in the range , and we adjusted the viscous damping n to obtain a ixed normal restitution coeicient e =.92. Simulations or smaller values o the restitution coeicient e=.87 produced qualitatively similar results. We also used the ixed ratio o the tangential and normal spring constants k t /k n =1/3. PHYSICAL REVIEW E 78, B. Simulation o a two-dimensional packing o sot grains under shear The simulations were perormed in two-dimensional geometry in the absence o gravity. The computational domain spans a 5d 1d area, and is periodic in the horizontal direction x. The granulate is slightly polydisperse to avoid crystallization eects. We assume that the grain diameters are uniormly distributed around mean size d with 2% relative width. To provide a link between micromechanical quantities obtained through simulations and continuous ields, a coarse-graining procedure was applied 19. Since all experiments described below deal with steady quantities, the procedure consists o two steps: space and then time averaging. The top and bottom rough plates were modeled by two straight chains o large grains twice as large as an average particle diameter. The two plates were under ixed pressure p. The layer was chosen narrow enough, so the properties o the granular layer shear rate, shear stress components, order parameter, etc. were constant across the layer. Two opposite orces F 1 = F 2 were applied to the plates along the horizontal x axis to induce shear stress in the bulk. We started with large orces well above the luidization threshold and slowly ramped them down in small increments until we reached the critical yield orce at which the granular low stopped. At every stop we let the system reach the stationary regime and measured all o the stress components, the strain rate, and the order parameter. Finally, we averaged the data over the whole layer and over the duration o each step. At any moment o time all contacts were classiied as either luid or solid. A contact was considered solid i it was in a stuck state F t t F n and its duration was longer than a characteristic time t * which was slightly larger than collision time t c we used t*=1.1t c. The irst requirement eliminates long-lasting sliding contacts, and the second requirement excludes short-term collisions pertinent to completely luidized regimes. When either o the requirements is not ulilled, the contact is assumed luid. We deine the order parameter as the ratio between space-time averaged numbers o solid contacts Z s and all contacts Z within a sampling area 24, y = Z i s / Z i. 12 We split the ull stress tensor into the solid component, s, and the luid component,, in the same ashion as was done with contacts themselves. The luid part o the stress tensor is due to short-term collisional stresses and the Reynolds stresses, whereas the solid part accounts or persistent orce chains. The Reynolds contribution to the stress is negligibly small in the vicinity o the phase transition, but comes into play when the granular aggregate is highly luidized. In the system which is neither completely static nor completely luidized we expect the coexistence in time and space o both phases. A particular grain may have both types o contacts at the same time thus contributing to both and s. C. Rheology and constitutive relations Following the procedure outlined in the preceding section, we now report the rheology or a sheared two-dimensional

5 NONLOCAL RHEOLOGICAL PROPERTIES OF GRANULAR PHYSICAL REVIEW E 78, µ e =σ xy /p a µ /I p=25 ξ=2e-4 p=4 ξ=2e-4 p=6 ξ=2e-4 p=25 ξ=5e-4 p=4 ξ=5e-4 p=6 ξ=5e-4 p=25 ξ=2e-3 p=4 ξ=2e-3 p=6 ξ=2e-3 p=25 ξ=8e-3 p=4 ξ=8e-3 it p=4 ξ=2e-4 it p=4 ξ=5e-4 it p=4 ξ=2e-3 it p=4 ξ=8e b 2 1 µ =σ xy /p µ s =σ s xy /p c p=25, ξ=2e-4 p=4, ξ=2e-4 p=6, ξ=2e-4 p=25, ξ=5e-4 p=4, ξ=5e-4 p=6, ξ=5e-4 p=25, ξ=2e-3 p=4, ξ=2e-3 p=6, ξ=2e-3 p=25, ξ=8e-3 p=4, ξ=8e I=γ. /p 1/2 FIG. 1. Color online Eective riction e = xy / p a, luid riction = xy / p b, and solid riction coeicient s = s xy / p c in planar shear cell experiment vs inertial parameter I= / p or various values o pressure p and values o the grain stiness. Dashed line on the panel a depicts the riction law, Eq. 4, with parameters 1 =.22, 2 =.4, and I =.6. granular packing. We perormed multiple runs with N = 5 particles or several values o the pressure p, shear strain rate, and inverse spring elasticity constant. Figure 1 a shows the ull shear stress xy normalized by the normal pressure p as a unction o the dimensionless internal parameter I = / p here and in the ollowing unless explicitly stated, we set the grain size d=1 and mass m=1. We see that all data collapse rather well on a single curve although there is a systematic drit o the data, and the elasticity o the grains appears to have little eect on the eective riction coeicient riction slightly increases with the stiness constant o grains. Our numerical data qualitatively agrees with the Pouliquen phenomenological law 4 with 1 =.22, 2 =.4, and I =.6 the original Pouliquen law, Eq. 4, gives I.279; smaller value o I in our case is likely due to two-dimensional geometry o the simulation cell. Now, in accord with the partial luidization theory, we can proceed I=γ. /p 1/2 FIG. 2. Color online Scaled data or /I= xy / pi in planar shear cell experiment vs scaled strain rate inertia parameter I = / p. urther and separate the eective riction solid contribution rom its luid counterpart, e = s +, 13 where s = s xy / p is the solid riction coeicient and = xy p is the luid riction. Both luid and static components o the shear stress depend on the dimensionless internal parameter I= see Figs. 1 b and 1 c. Here we can see a p signiicant spread o the data among runs with dierent particle s elasticity parameter =1/k n, and we observe a clear lack o data collapse with I in the limit o low shear rate or alternatively, soter particles. For compressible grains, the sotness, i.e., the inverse spring elasticity constant =1/k n, brings a new time scale e = 1/2 and then one can build an additional dimensionless parameter S when it is compared with the coninement time scale T p =1/ p see also Re. 4 where a similar parameter was introduced, S = 1/2 p 1/2. 14 Thereore, while the total eective riction is rather insensitive to the compressibility o grains, the relative contribution o the solid and luid stress components depend signiicantly on the grain compressibility. This can be understood because soter grains can adjust their shape to maintain static contacts and thereore have greater proportion o shear stress carried by static contacts somewhat similar conclusion was obtained in Re. 11. In the ollowing, we ocus on the dynamics o the luid component o the stress. The data suggest that or large I all o the curves appear to approach a certain asymptotic slope. It is more evident rom Fig. 2 showing the ratio /I vs I,

6 ARANSON et al. µ =σ xy /p.2 p=25 ξ=2e-4 p=4 ξ=2e-4 p=6 ξ=2e-4.1 p=25 ξ=5e-4 p=4 ξ=5e-4 p=6 ξ=5e-4 p=25 ξ=2e-3 p=4 ξ=2e-3 p=6 ξ=2e p=25 ξ=8e p=4 ξ=8e-3 γ. /pξ 1/2 FIG. 3. Color online Scaled data o the luid riction xy / p in planar shear cell experiment against I/S= / p/s. Collapse o the data occurs or small strain rates. I Thus, our simulations suggest the ollowing scaling o the luid part o the stress tensor xy vs shear strain rate or suiciently large strain rates : xy = p. 16 While a similar expression was obtained earlier in Re. 4, the result was interpreted as a maniestation o generalized Bagnold rheology with the eective viscosity diverging at the luidization transition as 1/ c 2 where c is the closed packed solid raction. For small strain rate values we ind that the data collapse occurs when is plotted against the parameter I/S and is in act controlled by the spring elasticity, see Fig. 3. For small strain rates the simulations suggest a dierent scaling, xy = 1/2, 17 where 1.5 is another dimensionless constant. The act that this constant is very close to 1 possibly implies that the elastic response is rather local one or ew grain sizes and is related to a single grain deormability. Indeed, by the analogy with Eq. 3, constant 1 can be interpreted as an eective renormalization o the typical length in Eq. 3 this length simply coincides with the grain size d. Correspondingly, or very small shear rates we obtain the ollowing expression or the solid riction coeicient: s = e = e p 1/2 = e I/S, 18 where s e or. To summarize, our simulations suggest two distinct scaling behaviors or small and large values o the strain rate. Thereore, the constitutive relation 4 is only approximate and valid or not too small values o the inertial parameter I determined by the condition I S. In general, the constitutive A,A S=ξ 1/2 p 1/2 relation should be the unction o I and pressure p. We can describe the data in the whole range o p and values by the ollowing empiric law: S exp I/S. 19 I = + PHYSICAL REVIEW E 78, α,β S FIG. 4. Fitting o simulations data to Eq. 19. Each curve in Fig. 2 is itted individually to the ollowing expression: xy / p =I + A exp A 1 I, =1.95. The plot shows coeicients A and coeicient A 1 vs S, solid line shows best it A = /S and dashed line A 1 = /S correspondingly. Inset: Parameters =A S and =A 1 S vs S. Solid line shows =1.5 and dashed line =2.25. The best it or the simulation s data shown in Figs. 1 b and 2 yields the ollowing values o the parameters: 1.95, =1.5, and =2.25. Figure 4 shows the parameters o Eq. 19 in a wide range o the parameter S. As we see rom Fig. 4, the parameters and are almost constant over the whole range o S more than a decade. As it ollows rom Eq. 19, the transition between two scaling regimes occurs when the inertia parameter I becomes o the order o the compressibility parameter S. Thus, the corresponding value o the strain rate is c 1/2 p. For very sti grains the range o the second scaling given by Eq. 17 shrinks, and the rheology is described by Eq. 16 with a single parameter I, in agrement with Re. 4. Moreover, due to the exponential dependence o on I/S, the last term in Eq. 19 is exponentially small or, and, thereore, the single-parameter constitutive relation in the orm I is a rather accurate approximation in the majority o practical situations. The parameter S is relevant in numerical simulations or in real experiments when the elastic moduli o the grains are comparable or much smaller than the conining pressure p. For gravity-driven avalanches it would mean that elastic moduli are o the order gd, which is in practice quite unrealistic. However, or higher conining pressures acting on sot and highly compressible e.g., polymeric grains, this eect may become relevant and would justiy the use o a modiied rheology as described by relation

7 NONLOCAL RHEOLOGICAL PROPERTIES OF GRANULAR PHYSICAL REVIEW E 78, ρ D. Calibration o the transition unction The zeroes o the transition unction, derived rom the derivative o the ree energy density 1 =df/d see Eq. 1, can be obtained rom the same simulations o shear low as in the two-dimensional planar cell described in the preceding section, see Re. 19 and Fig. 5. The overall behavior o the order parameter as a unction o the control parameter, i.e., ratio o shear stress to normal stress, can be summarized as ollows: Ramping up the shear rate results in an abrupt jump o the order parameter rom the static solid value =1 to a small value.2 the onset o low at.3 and urther decrease o towards zero with the increase o. Ramping down the value o rom a well luidized state results in a gradual increase in and consequent jump rom = *.6 to =1 low arrest at.25 hysteresis. A good it to the biurcation diagram is given by the ollowing simple expression: = 2 2 * + * 2 1 or 1, p=4 p=3 p= δ= σ xy /p FIG. 5. Color online Biurcation diagram or planar shear cell showing the order parameter at steady-state regime vs control parameter obtained rom simulations or dierent values o applied pressure p reproduced rom 19. Open symbols correspond to ramping up external driving orce F, and closed symbols corresponds to ramping down. Hysteretic behavior is evident. Thick lines show the it to Eq. 2 ; solid lines indicate stable branch and dashed line unstable branch. = 2 * 1+ or 1, 2 where =A 2 2 with A=15, =.25, * =.6. The rationale or the expression 2 is the ollowing: Equation 2 has the roots 1,2 =, 1,2 = * 1 or 1 and 2 = * 1+, 1 = or 1. Thus, one sees that one o the roots, 2 = * 1+, remains identical or all. The other root, 1 = * 1, becomes identically zero or 1. It prevents unphysical negative values o the order parameter or large values. The parameters A, can be ound on the basis o two-dimensional simulations or extracted rom a comparison with experimental data. E. Stability domain, h stop, h start The parameters,a o the unction depend sensitively on material properties o grains. In order to extract these parameters rom the chute low experiment, we need to ind the vertical proile o the order parameter z or a steadystate low on an inclined plane and calculate independently the value o h stop or each value o the inclination angle. The low velocity can be obtained rom the constitutive relation or the luid part o the stress. The order parameter equation or steady-state shear low is o the orm 18,19 2 z 1, = 21 with the ollowing boundary conditions: at the bottom z = one imposes =1 rough bottom avor solid state, and on ree surace z = h the most natural boundary condition is z h = order parameter value is not ixed. Parameters deining the transition unction can be then determined rom matching the experimental values o h stop and h start, the value o thickness at which a static layer irst becomes unstable at a given slope and corresponding parameter. The value o h start can be obtained rom the linear stability analysis o the time-dependent equation 21. Substituting a solution o the orm =1+ exp t sin z/2h,, we obtain or the growth rate o small perturbations = 1, 2 / 4h 2. The height h start is obtained rom the condition =, which gives h start = 2 1,. 22 Using Eq. 2 one inds that h start or =A 2 2 = 1/ * 1 2. This condition deines the asymptotic static repose angle. For the value o h start decreases as h start 1/. In order to determine h stop we need to ind a nontrivial trajectory z const o Eq satisying the boundary conditions =1, z z=h = and attaining minimal thickness o the layer h. This can be done by minimization o the length o trajectories o Eq. 21. Equation 21 can be integrated by multiplication on z see or detail 18, yielding z 2 2F = c, 23 where the ree energy density F = 1, d and c is the constant o integration. Since z = at z=h, one can set c = 2 1, d, where is the value o at z =h. Thus, h stop is given as 1 d h stop =. 24 min 2F + c This integral can be evaluated numerically. Analytical estimates can be obtained or two cases, h stop and or. The value o h stop diverges at stop, i.e., the dynamic repose angle. The value o stop can be obtained analytically rom the ollowing consideration. The critical value o = stop is deined by the condition that the length o nontrivial trajectory o Eq. 21 diverges, which corresponds to

8 ARANSON et al. PHYSICAL REVIEW E 78, h/d U /(g cos(ϕ)h) 1/ tan ϕ h/h stop FIG. 6. Comparison o experimental data or sand on velvet with expressions or h start and h stop. Symbols are experimental points or h stop and h start 13,2, lines are its according to Eq. 22 dashed line and Eq. 24 solid line with parameters * =.6, A=5.5, =.6. the existence o a separatrix connecting two ixed points, =1 solid and = 1 = * 1 liquid. Other trajectories o ininite length do not satisy the boundary conditions. For the chosen orm o the ree energy F ourth-order polynomial o the existence o such a separatrix ollows rom the conditions that the ixed points o Eq. 21 are equidistant: 1 2 = 2 1 or more general orm o F one should use the Maxwell rule d = d. The condition o equidistant roots yields the ollowing expression stop as unction o parameters A, * : 2 A stop 2 = 1/ * 1 2 /9. 25 Then, the dependence o h stop vs or stop can be obtained rom integral 24 by expanding the denominator or stop and 1, 1. The calculations results in the ollowing behavior: h stop ln stop or stop. In order to match h stop and h start curves, the outcome o PFT theory is compared with experimental data to ind values o A,. As one sees rom Fig. 6 corresponding to the low o sand on velvet 13,2, the best it occurs or A =5.5 and =.6. However, there is some systematic deviation rom experimental data, especially or h start curve. It likely stems rom the act that we use a simpliied unction see Eq. 2 which has been deduced rom twodimensional simulations o a relatively small system 5 grains only. Using a more complicated unction would certainly improve the agreement. However, it would also introduce additional adjustable parameters which we preer to avoid or the sake o simplicity o analysis. IV. FLOWS OF ROUGH SANDY GRAINS DOWN AN INCLINE FIG. 7. Color online Eective low rule U / g cos h plotted against h/h stop. The experimental results or dierent angles are solid black symbols corresponding angle in degrees rom =32.1 to 37 is indicated in the legend to the plot. The results or PFT theory are solid lines. For better agreement with the experiment the value o parameter is set to 2, The dashed line is y=.8 x 1. Experiments on smooth, nearly spherical beads have been widely used to extract the rheology o sheared granular assemblies in the inclined plane geometry 1. As we mentioned earlier, the systems involving irregular grains e.g., rough sand oten show substantial deviations rom the standard Pouliquen s low rule 1,13. The problem o the low rule and the associated constitutive relation was studied earlier by Malloggi et al. 13 using Fontainebleau sand o a rather narrow size distribution d=3 6 m. The shape o those grains was rough and aceted; very dierent rom the regular spherical grains used by Pouliquen 2. Now, we recall here briely the most salient eature needed or a comparison with the PFT. The experimental system is similar to the one used by Daerr and Douady 2, and is represented by a low o Fontainebleau sand over a velvet cloth. This system displays a rather wide bistability hysteretic domain between curves h stop and h start, see Fig. 6. To analyze the rheology o sand lows, an eective low rule was obtained irst, as in Re. 1, by using the ront velocity V ront o the avalanche spreading over the bare plane see Fig. 7. Note that the identiication o V ront with the mean depthaveraged low velocity U is only valid when the low goes down to the bottom o the plane. To identiy the presence o a jammed static layer o sand below the continuous low o grains, two independent experiments were then conducted. Both experiments were conducted as close as possible to the regime o continuous steady-state low. Correspondingly, the sand lux at the intake o the chute was determined by the opening o a reservoir illed with sand. Since the amount o granular material was limited, the duration o continuous low in our experiments was between 5 and 5 seconds, depending on the lux values. For the irst experiment, a sooted blade was inserted in the low inducing an erosion o the soot limited to the upper granular layers. Ater cessation o the low, the blade was removed and photographed. A sharp transition between the eroded and the noneroded parts o the sooted blade was observed rom the image processing o the soot gray levels. Our experiment shows that or the constant lux condition the transition height between eroded and noneroded soot parts was well deined. Consequently, the width o noneroded part was related to the width o the jammed layer. For the second experiment, a sand deposit was prepared at a constant height close to h stop. Then, a trench spanning

9 NONLOCAL RHEOLOGICAL PROPERTIES OF GRANULAR PHYSICAL REVIEW E 78, v z /d stat h /d stop r=u /V s δ=.63 δ=.68 δ= h/h stop z FIG. 8. Color online Representative vertical velocity proiles, V x z/d or = at dierent driving luxes. Inset: rescaled depth o the static layer z stat /d vs h stop /d solid line. Symbols show that experimental data, squares show measurements using erosion methods and circles showing colored sand method see text or explanations. about 1 cm across and 2 cm along the slope at a downhill distance o about 2/3 o the plane length was dug down to the bottom o the deposit. Then, the trench was illed to the initial deposit height by packing layers by layer colored grains, with each layer corresponding to a dierent color. Once the deposit was reconstructed, the low was initiated at some ixed lux value. Ater cessation o the low, the deposit was excavated layer by layer and the remaining height o immobile grains was determined rom the corresponding grain s color. Both experiments consistently conirmed the presence o a jammed layer see inset o Fig. 8, at least or an experimental duration o continuous low between 5 5 seconds. Note that quasistatic or creeping layers were observed already in many other experiments, see, e.g., 5,7,35, but they likely stem rom lateral boundary riction eects 5,36. In the present case o sandy grain low, the static layer comes rom the interplay between rough bottom boundary and possibly, highly irregular shape o the grains. Indeed, in our experiments the avalanche plane had a width about 12 times the grain size d and the resulting granular lows had a maximal height o about 2d. The jammed layer is around 3 to 8 times the grains size maximum. Thereore, we can neglect the lateral boundary eects as a pertinent mechanism to induce a jammed layer and modiy the low characteristics 5,35,36. Within experimental uncertainties, an empirical relation or the static layer was established as z stat =Ad +Bh stop, with the itting constants A= and B =.7.1. The ratio o the average velocity to the surace velocity, r=u /V s, was measured as well. Note the values o r were ound to dier consistently rom the local rheology value: 3/5 and to increase with the rolling layer thickness R see Fig. 9. The data points were ound around an average value r=.8.1. A. Calibration o the low rule In order to obtain the low velocity we need to calculate the vertical proile o the order parameter rom Eq. 21. It FIG. 9. Color online Values o velocity ratios r vs normalized low depth h/h stop. We have r=u /V s where U is mean low velocity, V s is surace velocity, the symbols display the experimental data or Fontainebleau sand, the solid lines are theoretical results or r obtained or u=2.7 and three dierent values o, and the dashed lines display r the or the parameter values but or u =.1. can be easily done rom the integral 23, which gives implicitly z, 1. Then the velocity is obtained rom the constitutive relation xz = d p = q xz, = const, 26 where unction q is calibrated rom the relation, q = 1 u, see Eq. 9. In the ollowing, we have used the value u = 2.7 obtained rom the early numerical simulations 19. However, we anticipate that in act the value o u depends on material properties, such as tangential riction, grain shape, etc. We actually ound that this parameter u controls the relative width o the jammed layer which increases with increasing o u and shrinks as u. Moreover, with the decrease in the value o the parameter u, the ratios o depth averaged to surace velocities r increases and approaches the 2/3 value corresponding to the Bagnold rheology, see Fig. 9. In the chute low geometry the shear stress is xz =sin h z and the pressure is o the orm p=cos h z. Thus, rom the strain rate = z v x we obtain an equation or the downhill velocity in the rescaled variables z sin v x z = q z h z dz. 27 cos Then, the mean low velocity U =h 1 h v z dz can be obtained by partial integration o Eq. 27, h U = 1 sin v x z dz = q h z h cos h 3/2 dz. 28 Then, or each value o, we determined the value o h stop and mean low velocity U in the range o all possible low thicknesses, h stop h h max, where h max is the maximal depth above which the steady-state low or a given value o the angle either ceases to exist or become unstable. h

10 ARANSON et al. Here we must comment that the accessible range o thicknesses h/h stop is somewhat smaller than that in the experiment since we consider steady-state low only. There are several actors limiting the range o existence or the steadystate low. As we showed early in Re. 18, or each value o the slope angle the stationary low regime persists up to some critical thickness. A urther increase in low width h leads to gradual accumulation o the granular matter down the low direction and, consequently, readjustment o the slope angle. Also, or very high low rates the low may become unstable with the transition to accelerating low 2. Alternatively, with the decrease in the low width h, the low may become unstable with respect to long-surace wave instability so-called Kapiza waves 37. As we have shown in Re. 6, the Kapiza waves phenomenon is also captured in the ramework o partial luidization theory. Note also that the Kapiza waves are also observed or sandy grain lows in this range o Froude numbers 37. Overall, 15 curves derived rom the PFT theory are displayed in Fig. 7. They correspond to a broad range o slopes rom =.61 angle =31.4, close to stop in Fig. 6 up to =.93 =42. As one sees, all curves, theoretical and experimental, show the same tendency to cluster near a single line but dierently rom the Pouliquen low rule: The lines do not have intercepts at the origin. We may also note that within the data scatter, a line going through h/h stop =1, Fr= would be a irst order air representation o this raw low rule. Note that this is qualitatively similar to the data o Deboeu et al. 1 in the vicinity o jamming. Unortunately, the presence o various instabilities discussed above prevents us rom going simply to much higher values o h/h stop. Here, we chose to display the Froude number Fr =U / gh cos with the g cos component that represents the actual conining part o the pressure in an avalanche experiment. Nevertheless, this choice made or the sake o consistency with a theoretical description using an eective rheology, has no practical inluence on the results and on the eective low rule, because o the remaining data scatter. Thereore, this choice o parameters provides a quantitative agreement, not only or the stability diagram but also or the eective low rule. Note that we also explored the low rule or a generalized constitutive relation in the generalized orm 19 ound or sot spheres. It produced only a slight improvement, but qualitatively the results remain the same. Thus, we conclude that the part o the constitutive relation associated with the grain compressibility is not maniested in the low rule at typical parameter values. PHYSICAL REVIEW E 78, B. Velocity proile and the onset o low Now we ocus on the low proiles near the onset. Figure 8 shows the vertical velocity proiles obtained rom Eq. 27 or dierent values o h. As one sees, or this choice o parameter, all velocity proiles display a quasistatic layer z stat where the velocity is almost zero. In the lowing layer the velocity changes almost linearly, except or a small boundary layer due to no-lux boundary condition at the ree surace. The velocity proiles are similar to those obtained in simulations or lows over a rough bottom, see 1. Thereore, the quasistatic layer is a new intrinsic eature o partially luidized lows at the onset o low jamming. It corresponds to a region where the velocity is signiicantly smaller than the mean or the surace velocity. This also could be called a creeping region. It is rather challenging to measure the velocity proiles in the static layer. One can only test i the displacement o the grains at a given location in the bulk are noticeable over the duration o the experiment in practice limited by the available mass o grains. In this work, we deine the location o the static layer by the condition v z=z stat =.1v z=h other threshold conditions give similar results. Moreover, our studies show that the static layer thickness z stat does not practically depend on the total low thickness or total grain lux. It allows to parametrize the dependence o z stat on the value o h stop only, as it is shown in the inset o Fig. 8. As it ollows rom Fig. 8, the depth o the static layer shrinks with the decrease o h stop, i.e., with the increase o the slope angle. For comparison, we show the experimental data or z stat normalized by the grain size or the two dierent methods described earlier. It turns out that the numerical data goes right in between the experimental points that can be viewed as higher and lower bounds or the determination o the static layer. To obtain insight into the low proiles and the relationship between theory and experiment, we compared dierent low velocity ratios. Note that or the Bagnold low with a velocity going down to zero at the bottom o the chute one obtains a ratio o the mean low equal to U to the surace low V s with a value r=.6. Our analysis shows that the apparent value o r=u /V s is o the order o.3 or h h stop and gradually increases approaching the value.45 with the increase in low thickness h. Thus, the apparent ratio r is smaller than that o the Bagnold or linear velocity proile, which is obviously due to the presence o a static layer. For comparison, Fig. 9 depicts ratios r=u /V s or both theory and experiment. The experimental and theoretical domains or the corresponding values o h/h stop only weakly overlap. Nevertheless, in both cases, we evidence an increase o r at larger lowing heights. However, it seems that or this choice o theoretical parameters, the experimental values.5 r.55 are slightly higher in the overlapping domain than the theoretical one,.4 r.45. Furthermore, the theoretical analysis also indicates that the value or r depends on the value o the exponent u in Eq. 9 : r increases with the decrease o u, as it is illustrated in Fig. 9. The corresponding low rule is also closer to what was ound by 1 or an assembly o rough glass spheres, where a crossover between a Pouliquen-like low rule and a line intercepting at zero or h/h stop =1 was observed. For larger values h/h stop 4 the parameter r increases and approaches the.8 which can be interpreted as a ormation o a plug low v z z const. However, on the qualitative level the ormation o a plug low proile could be treated as a strong velocity gradient in the jammed region which would act almost as an eective inite velocity slip at the scale o one grain. It would be interesting to see i the PFT model pushed in the limit o thick lows gravity surace waves included would be able to describe this eature. Our study shows that ater a thorough calibration against experiments, we are in a position to provide a clear-cut vi