A model for dense granular flows down bumpy inclines

Transcription

1 A moel for ense granular flows own bumpy inclines Michel Y. Louge Sibley School of Mechanical an Aerospace Engineering, Cornell University, Ithaca, NY (Date: February 9, 23) We consier ense flows of spherical grains own an incline plane on which spherical bumps have been affixe. We propose a theory that moels stresses as the superposition of a rate-epenent contribution arising from collisional interactions an a rate-inepenent part relate to enuring frictional contacts among the grains. We show that ense flows consist of three regions. The first is a thin basal layer where grains progressively gain fluctuation energy with increasing istance from the bottom bounary. The secon is a core region where the soli volume fraction is constant an the prouction an issipation of fluctuation energy are nearly balance. The last is a thin collisional surface layer where the volume fraction abruptly vanishes at the free surface. We also istinguish basal flows with the smallest possible height, in which the core an surface layers have isappeare. We erive simple closures of the governing equations for the three regions with insight from the numerical simulations of Silbert, et al. [PRE 64, (21)] an the physical experiments of Pouliquen [Phys Fluis 11, 542 (1999)]. The theory captures the range of inclination angles at which steay, fully-evelope flows are observe, the corresponing shape of the mean an fluctuation velocity profiles, the epenence of the flow rate on inclination, flow height, interparticle friction an normal restitution coefficient, an the epenence of the height of basal flows on inclination. PACS numbers: 45.7.Mg,45.7.Vn,45.5.-j,45.7.Ht,45.5.Tn,81.4.Pq, 5.2.D,83.7.Fn I. INTRODUCTION The flows of grains own rough incline planes have serve as a moel for geophysical phenomena like rock slies, unes an avalanches, in which the base of the flow is irregular on the small scale. Pouliquen an Chevoir [1] wrote a review of past an current research on the subject. Two stuies have she recent insight on the phenomenon. In the first, Pouliquen [2] conucte a series of experiments with monoisperse glass spheres in a wie chute roughene by gluing similar beas on the base. In the secon, Silbert, et al. [3] ran numerical simulations, in which they recore profiles of soli volume fraction an of the mean an fluctuation velocity of the grains for ifferent angles of inclination of a bumpy incline surface. From these stuies, it is clear that steay, fullyevelope (SFD) flows own a rough plane are generally ense. The roughness of the base frustrates the motion of the grains, thus leaing to a vanishing granular velocity there. The movement inuce by the gravitational acceleration then shears the entire flow, leaing to granular agitation through the whole epth, except at the rough base, where granular agitation is issipate. Pouliquen [2] mae two principal observations. First, SFD flows only exist within a range of angles of inclination α between the base an the horizontal, α min < α α max. These flows have a minimum height normal to the base h h stop (α), which ecreases with increasing α. They stop if h < h stop (α) or if α α min. They ac- Michel.Louge@cornell.eu; microgravity/ celerate a infinitum with α > α max. Secon, Pouliquen showe that the epth-average grain velocity u scales as h 3/2. This result is in contrast with flows own a flat, frictional incline, where u scales as h 1/2 [4]. Silbert, et al. [3] presente etaile profiles, from which one can istinguish three regions in the epth of the flow. Near the base, the fluctuation energy of the grains increases to reach a maximum within a few grain iameters from the rough bottom surface. We call this region the basal layer. Above this layer, the core of the flow is subject to a shear stress that graually ecreases towar the free surface as the weight of the granular overburen iminishes. In this core region, granular agitation is prouce from the working of the mean shear through the graient of the mean velocity. In turn, the agitation enows the grains with a rate-epenent shear stress that is riven by collisional interactions. However, because of the high packing ensity of the grains, the latter o not only interact through impulsive collisions. They also experience enuring frictional contacts leaing to a rate-inepenent component of the stresses. Remarkably, Silbert, et al. [3] observe that the core region possesses a volume fraction inepenent of epth. The thir region is locate near the free surface. It is energize by the agitation conucte from the core an by the shearing. Its volume fraction abruptly reaches zero at the free surface. Its thickness is only a few grain iameters. We call it the surface layer. In this paper, we present a moel that captures the observations of Pouliquen [2] an Silbert, et al. [3]. Our principal hypothesis follows Savage [5] an others in assuming that the stresses have two components. The first is rate-epenent. It is riven by collisional interactions an is given by the ense kinetic theory of Jenkins [6]

2 2 h b z core α surface layer basal layer FIG. 1: Sketch of a SFD flow own a bumpy incline showing notation use in the text an the three regions consiere in the moel. in terms of the granular agitation an the shear rate. The secon is rate-inepenent an erives from longlasting frictional contacts of the grains. In these SFD flows, we further assume that the corresponing enuring shear stress is proportional to the enuring normal stress through a constant friction coefficient µ E. We begin by writing governing equations for this flow that are consistent with the above hypothesis. We then exploit insight from the numerical simulations of Silbert, et al. [3] to inicate how these equations can provie close solutions. Finally, we compare the preictions with the ata of Pouliquen [2] an Silbert, et al. [3]. II. GOVERNING EQUATIONS We consier flows of monoisperse spherical grains of material ensity ρ s. The local state of flow is characterize by the soli volume fraction ν, an by the granular mean an fluctuation velocities, which are mae imensionless with the square root of the grain iameter an the gravitational acceleration g. The square of the fluctuation velocity is the granular temperature T (1/3) < u i u i >, where u i ũ i u i, ũ i is the instantaneous velocity, u i is its average over time an the inex i = x, y, z enotes three orthogonal Cartesian irections pointing, respectively, ownwar along the flow, across its with an up perpenicularly from the bottom surface (Fig. 1). The granular temperature is a measure of the agitation of the grains. In SFD flows, the mean velocity is parallel to the base, u y = u z = an u x u. Simple force balances on an infinitesimal slice of thickness z yiel ifferential equations for the shear stress S an normal stress N on surfaces parallel to the base, S z = ρ sνg sin α, (1) s x an N z = ρ sνg cos α. (2) In a ense flow, these equations can be integrate to yiel, approximately, an S ρ s νg sin α(h z) (3) N ρ s νg cos α(h z), (4) in which the epth-average volume fraction ν was substitute for its local value. The ratio of shear to normal stress represents the effective friction exerte by the grains on ajacent layers parallel to the base. In SFD flows, it is constant an equal to the tangent of the angle of inclination, S N = tan α. (5) Our approach is to istinguish two components of the stresses, an S = S I + S E (6) N = N I + N E, (7) where the subscript I refers to impulsive interactions leaing to rate-epenent stresses an the subscript E enotes enuring contacts associate with rateinepenent stresses. Recent simulations by Campbell [7] inicate that rate-epenent an rate-inepenent stresses generally coexist. We moel the latter by postulating the existence of an internal friction µ E such that S E N E = µ E. (8) For convenience, we efine the fraction η of the total shear stress that is rate-inepenent, η S E S. (9) Thus, a purely collisional flow like that stuie by Jenkins [6] has a stress fraction η =. A static granular heap that is not flowing has η = 1. In SFD flows, we then combine Eqs. (5) to (9) an express the ratio of any two stresses among S, N, S I, N I, S E an N E in terms of η, tan α an µ E. For example, the effective collisional friction is Ξ S I (1 η) tan α = N I (1 η tan α/µ E ). (1) To moel the rate-epenent stresses, we invoke the theory of Jenkins [6], who assume that velocity fluctuations

3 3 an normal stresses are isotropic, an who provie expressions in the limit where ν is large, S I = a 1 ρ s ν 2 g 12 (ν) T u z, (11) where g 12 is the sphere pair istribution function at contact an a 1 = 8(1 + π/12)/5 π. For such ense flows, we aopt the correction of Torquato [8] to the pair istribution function of Carnahan an Starling [9], an g 12 (ν) = 2 ν 2(1 ν) 3 for ν ν f (12) g 12 (ν) = (2 ν f ) 2(1 ν f ) 3 (ν c ν f ) (ν c ν) for ν f < ν < ν c, (13) where ν f =.49 an ν c =.64 is the ranom close packing fraction. In this ense limit, the normal collisional stress is N I = 4ρ s ν 2 g 12 (ν)t. (14) From Eqs. (11) an (14), it is clear that the rateepenent stresses are governe by the magnitue of the fluctuation energy. To etermine the latter, we write a SFD fluctuation energy balance in an infinitesimal slice of thickness z, q z + (S I + S E ) u γ =. (15) z The balance involves a flux of fluctuation energy provie by Jenkins [6] in the ense limit, q = a 2 ρ s ν 2 g 12 (ν) T T z, (16) with a 2 = 4(1 + 9π/32)/ π, an a volumetric rate of energy issipation where γ = a 3 ρ s ν 2 g 12 (ν)t 3/2 /, (17) a 3 = 24(1 e eff )/ π (18) an e eff is an effective coefficient of restitution combining the collisional energy issipation associate with inelastic an frictional impacts [1]. To erive Eqs. (11), (14), (16) an (17), Jenkins [6] assume that the collisional fluctuation energy issipation is small or, equivalently, that e eff is nearly unity. In Eq. (15), we follow Louge an Keast [4] in allowing both the enuring an impulsive stresses to prouce fluctuation energy by their working through graients of the mean velocity. For flows own a flat, frictional surface, Louge an Keast [4] showe that ignoring the prouction associate with enuring stresses woul lea to restrictions prohibiting the existence of SFD flows for most situations of practical interest. Jenkins [6] note that Eq. (15) is a linear ODE in the epenent fluctuation velocity variable w T. He also wrote that Eq. in terms of the imensionless istance s (h z)/ from the free surface (Fig. 1). Thus, combining Eqs. (3) through (18), we obtain three Eqs. governing SFD flows own an incline plane, { (1 η tan α } )s w s µ E s { a3 (1 η tan α ) 2a 2 µ E 8 tan2 α 1 η } ( ) sw =, (19) a 1 a 2 1 η tan α/µ E an u s = 4 { w 1 η } tan α, (2) a 1 1 η tan α/µ E ν 2 g 12 (ν) = ν cos α 4w 2 η tan α (1 )s. (21) µ E In these Eqs., the asterisks enote velocities mae imensionless with g. Unfortunately, these three Eqs. are not sufficient to etermine the four epenent variables u, w, ν an η. In orer to close this problem, we will exploit insight provie by the numerical simulations of Silbert, et al. [3] in the three regions of the flow. We begin with the core. III. THE CORE REGION In the core, Silbert, et al. [3] observe that the soli volume fraction is remarkably constant. Because the core spans most of the flow epth, we write that its soli volume fraction is roughly equal to the average throughout the flow, ν core ν. (22) We then etermine η using the energy equation, in which we neglect the flux graient term. Later, once the mean an fluctuation velocities are known through the epth, we will justify this assumption by evaluating the relative magnitue of this term. With this simplification, Eq. (19) reuces to a quaratic equation in η, (1 η) tan 2 α (1 η tan α/µ E ) 2 = a 1a (23) To possess real solutions, this equation requires tan α tan α min = µ E 1 + 4µ 2 E /(a 1a 3 ). (24) The inequality etermines the minimum inclination at which SFD flows exist. If it is satisfie, Eq. (23) has two solutions. The larger is typically near unity an graually increases with angle of inclination. We ismiss it

4 4 because it is unphysical for the flow to become less collisional with increasing inclination. Then, in the core, we fin { 1 η = η core = µ E tan α 8µ } E a 1 a 3 4µ E 1 µ E a1 a 3 tan α + 4µ2 E. (25) a 1 a 3 η core 1..5 ν = η In turn, because η < 1, this expression yiels another necessary conition for SFD flows, tan α tan α max = a 1 a 3 /4. (26). Pouliquen We then calculate the fluctuation velocity profile from Eqs. (21) an (25), w Ξ core sin α h z (z) = 2 a 1 a 3 ν core g 12 (ν core ), (27) where Ξ core is given by Eq. (1) in terms of η core, α an µ E. We integrate Eq. (2) to fin the corresponing velocity profile u (z) = u b + 16 Ξ 3/2 sin α core 3a 1 a 1 a 3 ν core g 12 (ν core ) ( h ) 3/2 { (h b h } ) 3/2 ( h z ) 3/2. (28) h We will calculate later the thickness b of the basal layer an the mean velocity u b where it meets the core. Silbert, et al. [3] alreay recognize the general form of the velocity profile in Eq. (28). They erive an expression similar to this equation assuming that the shear stress is proportional to the square of the mean strain rate through a constant to be etermine. One contribution of our moel is to establish the form of that constant. Another is, like Savage [5], to provie angular limits between which SFD flows can exist. It remains to etermine the volume fraction. Saly, we o not know how to o so in the core. Fortunately, the solutions in Eqs. (27) an (28) are relatively insensitive to its exact value. Nonetheless, to close the problem, we note that the simulations of Silbert, et al. [3] suggest the following relation (Fig. 2): ν core η core. (29) The simulations imply that, when the volume fraction is near ranom loose packing, the value of η core tens to zero, an the flow becomes fully collisional, thus losing its SFD character through graual acceleration. Conversely, a linear extrapolation inicates that the flow will lock up when the volume fraction approaches 61% or so. Further insight from numerical simulations is neee to establish whether the correlation in Eq. (29) has any merit beyon the conitions examine by Silbert, et al. [3] ν core.6.65 FIG. 2: Stress fraction versus soli volume fraction in the core. The symbols are volume fraction ata from moel L3 of Silbert, et al. [3], in which we evaluate η core using Eq. (25). The lines show a suggeste fit through the ata. The vertical ashe line inicates Pouliquen s estimate of the mean volume fraction in his experiments an the associate error bar. Because the core solution preicts a vanishing granular agitation at the free surface (Eq. 27), it contraicts evience from the numerical simulations [3]. Thus, the flow near the surface has a ifferent character, which we examine next. IV. THE SURFACE LAYER The core solution, if extene all the way upwar, woul in principle satisfy the free surface bounary conitions for the flux (q = ) an all stresses (S E = S I = N E = N I = ). However, there are several reasons why it is inaccurate near the free surface. First, because the flux graient from Eqs. (16) an (27) iverges as z h, it cannot be balance by collisional issipation in Eq. (15). Secon, the volume fraction is known to ecline from its constant value in the core as the free surface is approache [3]. Thir, the core solution stipulates that enuring an collisional stresses remain proportional (Eq. 25). However, the form of the granular agitation in Eq. (27) an of the energy flux in Eq. (16) imply that there is a net influx of fluctuation energy from the core to the surface layer. Because the surface layer is sheare through its epth, an because it is the scene of a iminishing soli volume fraction, it is likely to involve mainly collisional interactions. Finally, the simulations inicate that the fluctuation energy oes not vanish at the free surface, but instea exhibits an inflexion towar higher values [3]. Thus it is reasonable to assume that the surface layer has istinct physics from the core region. For these reasons, we assume η = in the surface layer. Numerical simulations shoul etermine whether this assumption has any merit.

5 5 With η =, the energy equation becomes in which we efine ) (s w k 2 sw = (3) s s k 2 a 3 2a 2 8 tan2 α a 1 a 2. (31) Conition (26) guarantees that k 2 >. In general, solutions to Eq. (3) are moifie zeroth-orer Bessel functions of the first an secon kin, respectively I (ks) an K (ks). Because the energy flux vanishes at the free surface, only the Bessel function of the first kin has physical significance. We solve Eq. (3) by matching the magnitue an slope of w at the interface between the core an the surface layer. These two bounary conitions yiel the magnitue of the fluctuation velocity profile w (s) = 2 I (λ ) an the epth of the surface layer, Ξ core λ sin α ν core g 12 (ν core )a 1 a 3 k I (ks), (32) In these Eqs., λ 1.7 is a solution to l = λ /k, (33) z/ (T/g) 1/ eg u/g FIG. 3: Typical profiles of imensionless mean an fluctuation velocities for Pouliquen s system 1 at the inclination shown. The horizontal ashe lines mark the interfaces between surface layer, the core an the basal layer. 15 I (λ ) = 2λ I 1 (λ ), (34) where I 1 is the first-orer moifie Bessel function of the first kin. We then etermine the mean velocity profile by matching its value at the interface between the core an the surface layer, an by integrating Eq. (2) numerically with η =. Finally, we evaluate the profile of soli volume fraction by substituting the fluctuation velocity (32) in the equation of state (21) with η = an by solving the resulting equation in ν. Figures 3 an 4 show typical profiles through the epth, incluing the basal layer iscusse in the next section. Because in this simple approach there is a iscontinuity of η at the interface between the core an the surface layer, neither the slope of the mean velocity profile nor the volume fraction are continuous there (Fig. 4). This efect is small an without much consequence when evaluating integrals leaing to the epth-average velocity or the overall mass flow rate. For cosmetic reasons, one can artificially eliminate the iscontinuity by substituting η core for η = in Eq. (21) before calculating the volume fraction profile. Note that, because in this moel the volume fraction vanishes abruptly at the free surface (Fig. 4), it is legitimate to assume a form for the governing equations that is appropriate for ense flows, rather than invoking more complicate expressions that woul span the entire range of soli volume fractions. Finally, we can now evaluate the error involve in neglecting the flux graient term in our etermination of η z/ eg..2.4 ν.6.8 FIG. 4: Profile of soli volume fraction for the conitions of Fig. 3. The ashe curves enote the volume fractions calculate from Eq. (21) with the value of η preicte by the moel. The soli lines represent curves in which iscontinuities have been remove by artificially aopting η = η core in Eq. (21) for the surface an basal layers. in the core. Using the profiles of mean an fluctuation velocities in Eqs. (27) an (28), we calculate the relative magnitue of the flux term an one of the other two balance terms in Eq. (15), q/z Su/z = a ( 2 ) 2. (35) 2a 3 h z Because the graient of fluctuation velocity becomes

6 6 steeper as the free surface is approache, the relative magnitue of the flux term is greatest at the interface between the core an the surface layer. Thus, the relative error is strictly less than (a 2 /2a 3 )(/l) 2. Because the thickness of the surface layer ecreases with angle of inclination, the worst error occurs when the angle is small, an it rops with increasing inclination. For example, at the smallest angle of 22 at which Pouliquen [2] observe a eep flow in system 1, the relative magnitue of the flux term was less than 13%. At the interface with the basal layer, the error at 22 was own to 6% for h/ = 7 an.1% for h/ = 24. At 28, the largest error was only 7%. Thus, it is legitimate to neglect the flux graient term in Eq. (19) to calculate η in the core (Eq. 23). V. BASAL FLOW AND MINIMUM HEIGHT The simulations of Silbert, et al. [3] clearly reveal the presence of a region above the base consisting of a few layers of grains whose agitation graually rises from zero at the bumpy bounary to a peak value at the interface with the core. We call this region the basal layer. Before eriving its governing equations in the next section, it is instructive to consier first its behavior in the limit where the flow height is minimum. If the height of a eep flow is progressively ecrease, as Pouliquen [2] i, then the entire flow eventually reuces to a passive basal layer that possesses no core or surface layer overhea. We call this iminutive flow a basal flow. Its height is what Pouliquen [2] calle h stop (α). In basal flows, grains tumble an roll over one another. As Louge an Keast [4] pointe out for thin shear layers near a flat, frictional surface, grains acquire angular momentum from frictional interactions with ajacent granular layers above an below. Thus, the shear stress that a horizontal layer exerts over grains above an below prouces angular momentum at a rate P proportional to the total shear stress an inversely proportional to the moment of inertia I an the grain number ensity n, P S/nI S/ρ s ν 2. (36) Conversely, grains lose angular momentum in collisions with other grains in the same horizontal layer. Because the impact protagonists roll at roughly the same angular velocity, their collisions prouce impulses resulting in the frustration of both of their rotation rates. As Louge an Keast [4] showe, the corresponing rate of loss of angular momentum D is proportional to the collision frequency νg 12 (ν) T / an to the impulsive reuction in granular spin T /. Then, D νg 12 (ν)t/ 2. (37) At steay-state, P = D or, using Eqs. (14) an (5) through (9), S tan α = = C, (38) N I 1 η tan α/µ E where C is a constant that we will etermine later. At this stage, we will merely assume that C is constant through the epth. Extracting η from this relation, the energy equation becomes where s (s w s ) Ksw =, (39) K = a 3 8C { ( C 1 µ ) } E + µ E. (4) 2a 2 a 1 a 2 tan α The form of this ifferential equation implies that its solutions are Bessel functions. However, because the fluctuation energy flux an, consequently, the slope of w must vanish at the free surface, the secon kin of Bessel or moifie Bessel functions is exclue. Further, because there is no relative velocity between the base an the flowing grains, the flux of fluctuation energy at the bottom bounary is negative an reuces to q = D (41) where D is a rate of fluctuation energy issipation per unit surface of the base. Consequently, because the bounary can only issipate fluctuation energy with q w/z w /s <, the solutions are zero-th orer Bessel functions of the first kin J, with K <. Defining m 2 K, these solutions have the form w = w J (ms). (42) Using Eq. (4), the conition K < implies that there is a minimum angle of inclination for basal flows, tan α > µ E 1 a 1 a 3 /[16C 2 ] + µ E /C, (43) where the strict inequality inicates that the flow stops altogether as α tens to its lower limit. The simulations of Silbert, et al. [3] an the experiments of Pouliquen [2] clearly reveal that basal flows an their eeper counterparts share the same minimum angle of inclination. This is evient by inspecting iagrams showing the heights at which these authors observe SFD flows versus angle of inclination. In these iagrams, the borer between the presence an absence of flow is a vertical asymptote at a single minimum angle of inclination. A borer having any other shape, such as an oblique asymptote, woul have betraye limiting angles epening on flow epth. Therefore, the minimum angles in Eqs. (24) an (43) are matche. This observation fixes the magnitue of C. From Eq. (38), we then extract the stress fraction in basal flows, η = η b = µ E tan α 8µ2 E a 1 a 3, (44)

7 7 an, from Eq. (4), the value of K m 2, m 2 = a 3 2a 2 { 1 a 1a 3 4µ E ( 1 tan α 1 µ E ) }. (45) The bottom bounary conition etermines the height h stop of basal flows. The simulations of Silbert, et al. [3] suggest that the fluctuation velocity vanishes at the base, w = at z =. From Eq. (42), this conition implies h stop (α) = j 1 /m, (46) where j 1 is the first root satisfying J (j 1 ) =. Finally, the conition η b < 1 must be satisfie for basal flows to exist. We fin that any angle α > α min guarantees η b < 1. However, to satisfy η b, Eq. (44) requires tan α tan α max = a 1a 3 8µ E. (47) Curiously, the moel thus preicts that the maximum angle for basal flows is larger than the maximum angle for eeper flows, α max > α max. Simulations of Silbert, et al. [3] will later confirm this peculiar observation. VI. THE BASAL LAYER If the height of a basal flow is progressively increase from h stop, then the grain assembly evelops a core an a surface layer. Our assumption is that the grains near the base will continue to experience the same stress fraction as in a basal flow (Eq. 44), an thus to be governe by Eq. (39) with K = m 2 from Eq. (45). The solution of the energy equation is then w = w 1J (ms) + w 2Y (ms), (48) where Y is the zero-th orer Bessel function of the secon kin. We calculate w1 an w2 by matching the fluctuation velocity at the interface between the core an the basal layer, an by making it maximum there, w /s =. The thickness b of the basal layer is then set by the bottom bounary conition, which we write, once again, w = at z =. We fin w1 Ξ core sin α h b = 2 a 1 a 3 ν core g 12 (ν core ) Y ( mh )J Y (mh/) ( m h b ) J ( mh )Y w2 = w1 J (mh/) Y (mh/) an an equation that can be solve for b, ( ), (49) m h b (5) J ( mh )Y 1(m h b ) = Y ( mh )J 1(m h b ), (51) TABLE I: Parameters of Pouliquen s [2] experiments system (mm) b (mm) α min ( ) α max ( ) e eff µ E where J 1 an Y 1 are the first-orer Bessel functions of the first an secon kins, respectively. We then evaluate the profile of mean velocity by integrating Eq. (2) numerically with η = η b subject to u = at z =, an fin the value of u b = u (z = b) require in Eq.(28). We note that, because w increases from zero at the base, the concavity of the mean velocity profile is opposite to its counterpart in the core (Fig. 3). Finally, we erive the profile of soli volume fraction from that of the fluctuation velocity using Eq. (21) with η = η b. As with the core/surface layer interface, there are minor iscontinuities at z = b for the soli volume fraction an the velocity graients, which are the result of the jump of η from η b to η core (Fig. 4). Once again, these iscontinuities are of little consequence for evaluating the overall mass flow rate. VII. PREDICTIONS In this section, we compare the preictions of our moel with experimental ata from Pouliquen [2] an with numerical results from Silbert, et al. [3]. We begin with the experimental ata. Pouliquen [2] i not measure the frictional or collisional properties of his glass spheres. Instea, he reporte the range of inclination angles at which he observe SFD flows. From his minimum an maximum angles, we infer the coefficients of internal friction an normal restitution using Eqs. (24) an (26), an µ E = 2 tan α [ max tan α max tan α min ] tan 2 α max tan 2 α min (52) e eff = 1 2 π 3a 1 tan 2 α max, (53) The results are summarize in Table I, in which we use a 1 = 8(1 + π/12)/5 π. The relatively high effective restitution justifies our use of Jenkins expressions [6] for nearly elastic spheres. The internal friction has a magnitue that is typical of granular materials, see for example, Savage [11]. As Fig. 5 illustrates, our expression for h stop (α) in Eq. (46) captures Pouliquen s measurements for basal flows

8 h/ 8 4 (u/g)/(h/) 3/ a 1 = 8(1+π/12)/5π α(eg) FIG. 5: Dimensionless height of SFD flows versus angle of inclination for Pouliquen s system 1. The circles represent experimental values of h stop/. The thick soli line is the preiction of Eq. (46). The small squares show the heights of all relatively eep SFD flows observe. The vertical ashe lines show the limiting angles of the core region α min an α max from inequalities (24) an (26). The ashe curve inicates the sum of the heights of the basal an surface layers l(α) + b(α). Because our steay, fully-evelope moel assumes the existence of three istinct regions, it ceases to be vali when either the surface layer or the basal layer engulfs the entire flow epth i.e., when h > l(α) + b(α). in system 1. Figure 5 also reveals that, because the heights of nearly all of his eeper SFD flows excee l + b, these flows possess a basal layer, a core an a surface layer. The theory also confirms the scaling of the epthaverage velocity that Pouliquen [2] an Silbert, et al. [3] observe for relatively eep flows. Because the mass flow rates of such flows are ominate by the core region, the principal benchmark for gauging the moel s accuracy is to plot the ratio ū/ g/(h/) 3/2 versus angle of inclination. As Fig. 6 shows, a moel that aopts the constants of Jenkins theory [6] agrees well with experimental observations. Because these three constants a 1 = 8(1 + π/12)/5 π, a 2 = 4(1 + 9π/32)/ π, an a 3 = 24(1 e eff )/ π were erive without consiering long-lasting contacts, it is remarkable how well they fare in flows where both rate-epenent an rate-inepenent stresses coexist. Nearly perfect agreement is obtaine using a slightly ifferent value for a 1 = 1.5. In Fig. 6, the proximity of the right-most ata point to the theoretical ashe curve also hints that a moel with the theoretical a 1 works better when the angle of inclination is larger, as expecte from a flow with the greatest collisional contribution. Figure 7 shows the corresponing epenence of the epth-average velocity on height an angle of inclina α(eg) 3 35 FIG. 6: Ratio ū/ g/(h/) 3/2 versus angle of inclination. The symbols are measurements with Pouliquen s system 1. The ashe an soli lines are preictions of the moel with a 1 = 8(1 + π/12)/5 π an a 1 = 1.5, respectively. Note that, as α α min, the core an basal layer isappear, an the remaining basal flow stops. u/g eg 26 eg 24 eg 22 eg h/ FIG. 7: Dimensionless epth-average velocity versus imensionless flow height for Pouliquen s system 1. The squares, circles, triangles an crosses represents inclinations of 22, 24, 26 an 28, respectively. The lines are preictions of the moel with a 1 = 1.5. tion. Note that Pouliquen s system 1 is the only ata set complete enough to permit these comparisons. Figure 8 shows Pouliquen s measurements of h stop (α) for systems 2, 3 an 4. The moel captures ata in systems 3 an 4 reasonably well. However, it clearly overpreicts h stop for system 2. Pouliquen also encountere unexplaine ifficulties with this system, notably in scaling the epth-average velocity. It is possible that for shallow basal flows, the bottom bounary conition may

9 9 h/ h/ h/ α(eg) system 2 system 3 system 4 FIG. 8: Dimensionless height of SFD flows versus angle of inclination for Pouliquen s systems 2, 3 an 4. The symbols are experimental values of h stop. The thick an thin soli lines represent h stop(α) from Eq. (46) an h stop(α) from Eq. (56), respectively. not always be w = at z =. If the flow was more agitate at the base, then Eq. (41) coul be more appropriate. For fully collisional flows, Jenkins an Richman [12] calculate 2 D = 2 π (1 e w) 1 cos θ sin 2 θ N I T (54) where sin θ ( b + )/( + b ) is a measure of the bumpiness of the bounary with spherical bumps of iameter b, is the mean separation between the centers of two ajacent bumps, an e w is the coefficient of normal restitution in the impact of a flow sphere with a bump. Pouliquen s values of an b are foun in Table I. We assume that = b. If we aopte Eqs. (41) an (54) instea of w = at z =, then the bottom bounary conition woul become w s = 4 a π (1 e w) 1 cos θ sin 2 θ w, (55) an the new flow epth h stop woul satisfy 4 2 π (1 e w) 1 cos θ sin 2 θ J (m h stop ) = a 2 mj 1 (m h stop ). (56) Because the solutions of Eq. (39) monotonically ecrease with s, h stop is strictly smaller than h stop in Eq. (46). It so happens that the height of system 2 is more closely preicte by h stop (thin line, Fig. 8) than it is by h stop. Although basal flows are too shallow to have much importance in practical applications, it woul be useful to investigate their bottom bounary conition further. An important preiction of our moel is that basal flows an the core region of eep flows o not share the same physics. The former is compose of grains tumbling at a minimum height h stop, below which no flow can exist. The latter is a region, generally eep, where the volume fraction is constant an the prouction an issipation of fluctuation energy are nearly balance. Therefore, in this view, it is a coincience that the mean velocity of eep flows, which is ominate by the core region, shoul be roughly in inverse proportion to the epth h stop of basal flows. Pouliquen propose such empirical relation after analyzing his experimental ata, u/ gh β(h/h stop ). Although we regar this relation an its constant β as coinciental, our moel oes inicate that it is approximately vali. In fact, we preict that β, espite its relatively weak variations with α, h, µ E an e eff, is nearly constant except as it approaches the limits α α min or h h stop. For example, for eep flows of Pouliquen s system 1, we fin β =.18 ±.3 for α {21, 22,, 32 }, whereas Pouliquen reports β.136. In the limit of basal flows where h h stop, our moel preicts β =.63 ±.9 for α {21, 22,, 32 }. We now turn to the numerical simulations of Silbert, et al. [3]. These authors consiere two kins of threeimensional systems that iffere in their granular contact laws. The first, labelle L3, moels contacts using a linear spring-ashpot system tune to yiel a constant coefficient of normal restitution e =.88. The secon, labelle H3, assumes a Hertzian contact law with viscoelastic amping that prouces a velocity-epenent normal restitution. Both moels assume Coulomb interparticle friction with a coefficient µ =.5 inepenent of relative contact velocity. We o not know how to evaluate the corresponing internal friction coefficient µ E in general. However, as the following two-imensional calculation suggests, we expect a simple relation between µ E an µ when µ is small. Consier two contacting isks having a line of centers making an angle ξ with the y- irection. A force F y < irecte along the negative y-irection is exerte on the center of mass of the top isk. If µ is small enough for the contact to remain frictional, then the bottom isk opposes a force with projection F x = µf y cos 2 ξ F y sin ξ cos ξ along the x- irection. The resulting contribution to internal friction is µ = F x /F y. If all angles ξ have equal probability in the range π/2 ξ +π/2, then the mean value of µ is µ/2. Thus, for small µ, we expect µ E µ with

10 a 1 = 8(1+π/12)/5π 3 h/ 2 1 (u/g)/(h/) 3/ α(eg) FIG. 9: Dimensionless height of SFD flows versus angle of inclination for the simulations of Silbert, et al. [3] with moel L3. For symbols, lines an comments, see Fig. 5. The soli circles represent basal flows at an angle of inclination that excees α max α(eg) FIG. 1: Ratio ū/ g/(h/) 3/2 versus angle of inclination. The triangles an circles represent the simulations of Silbert, et al. [3] for moels H3 an L3, respectively. We believe that the squares correspon to moel H3, although Silbert, et al. [3] labelle them as L3. The ashe, otte an soli lines are preictions of the moel with a 1 = 8(1+π/12)/5 π, a 1 = 1.5, an a 1 = 2.1, respectively. 28 µ E < µ. In the other limit where µ, we expect that the angle of internal friction shoul reach an asymptote that is inepenent of µ. To evaluate the effective coefficient of restitution from the parameters of the simulations, we invoke the collisional theory of Jenkins an Zhang [1], who provie an expression for e eff = e eff (e, µ). With e =.88 an µ =.5, they preict e eff =.58. In general, e eff < e. Figure 9 shows the heights of SFD flows that Silbert, et al. [3] observe with moel L3. From their limiting angles α min = 19.5 an α max = 28.7, we infer µ E =.4 an e eff =.83, which, as expecte, are respectively smaller than µ an e. In aition, we fin that the moel agrees well with the ata for h stop (α). Finally, the simulations also confirm that steay basal flows can exist with angles of inclination exceeing α max (Eq. 47), ark circles in Fig. 9. Figure 1 examines the scaling of the epth-average velocity with height an angle of inclination. Unlike the physical experiments of Pouliquen [2], we fin that the simulation ata epart significantly from the preictions of a moel that assumes the constant a 1 = 8(1 + π/12)/5 π erive from the collisional theory. Instea, we fin that a 1 = 2.1 captures the ata better. However, we note that the magnitues of the epthaverage velocities recore in the simulations of Silbert, et al. [3] can vary by as much as a factor of 1.8 epening on the form of the contact moel employe (moels L3 an H3). We also suspect that Silbert, et al. [3] may have mislabelle some results as L3 rather than H3 in instances mentione in Figs. 1 an 11. The principal benefit of the simulations resies with their insight on granular behavior through the flow epth. We fin that our moel closely captures the profiles of mean an fluctuation velocity reporte by Silbert, et al. [3]. Figure 11 shows mean velocity profiles for several flow heights. Unfortunately, because Silbert, et al. [3] only reporte relative profiles of fluctuation velocity, we coul not evaluate the egree of anisotropy of their velocity fluctuations or compare these with absolute preictions of our moel. Nonetheless, as Fig. 12 illustrates, the moel reprouces relative profiles of fluctuation velocity well. Finally, Silbert, et al. [3] examine the epenence of the mean flow rate on interparticle friction µ an normal restitution e at α = 22. To compare our preictions with their results espite the absence of ata for µ E an e eff, we aopt the theory of Jenkins an Zhang [1] for the effective restitution, an we assume that the ratio µ E /µ remains constant an equal to the value.4/.5 that we etermine earlier. As Fig. (13) shows, this approach captures well the epenence of the mean flow rate on friction when µ is small. At higher values, the mean flow rate becomes inepenent of µ as µ E reaches a constant asymptotic value. Further, the moel clearly agrees with Silbert, et al. [3] that the flow rate is much more sensitive to interparticle friction than it is to normal restitution (Fig. 13). Because purely collisional theories (η =, ν) preict instea that ū/ g/(h/) 3/2 epens strongly on e an weakly on µ, their relevance to ense SFD flows own a bumpy incline is questionable.

11 z/ z/ 23 eg 5 39 h/ = eg 5 1 u/g w/w 3 FIG. 11: Profiles of mean velocity versus epth at an inclination of 24 an for the relative heights shown. The lines are preictions of the moel with a 1 = 2.1. The squares, iamons an circles are for moel L3 of Silbert, et al. [3]. We suspect that the triangles belong instea to moel H3, which woul explain the unexpecte iscrepancy between moel an ata at h/ = 39. VIII. CONCLUSIONS We have outline a moel for granular flows own bumpy inclines in which the stresses have a collisional, rate-epenent part coexisting with a rate-inepenent, frictional contribution [5]. For eep flows, the moel istinguishes three regions: a basal layer where grains graually acquire fluctuation kinetic energy away from the base; a core where the soli volume fraction is constant; an a collisional surface layer, where the volume fraction ecreases rapily near the free surface. The moel also escribes the behavior of basal flows, which possess the smallest sustainable height. With insight from the numerical simulations of Silbert, et al. [3], we have propose simple closures for the governing equations of SFD flows in the three regions. The resulting solutions provie expressions for the limiting angles at which such flows exist, an for the profiles of mean an fluctuation velocities though the epth. They also preicte the scaling of the mean flow rate with angle of inclination an flow height. Although our results compare favorably with the physical experiments of Pouliquen [2] an the numerical simulations of Silbert, et al. [3], several issues warrant further research. Foremost, it remains to explain why the soli volume fraction remains constant in the core, an to preict its value without resorting to an empirical closure. It is possible, for example, that the constant volume fraction may be associate with a phase transition between two states in which collisional an enuring FIG. 12: Relative profiles of fluctuation velocity versus epth at an inclination of 23. The abscissa is w relative to its value w 3 at z/ = 3. The lines are preictions of the moel with a 1 = 2.1. The squares, iamons an circles are the ata of Silbert, et al. [3] for moel L3 in the x, y an z irections, respectively. stresses respectively ominate. In this context, relate work shoul focus on eriving constitutive relations for rate-epenent stresses an collisional bounary conitions that account explicitly for the presence of enuring contacts. In aition, the angle of internal friction shoul be calculate in terms of frictional properties of the contacts, or relate to other physical parameters, such as the angle of repose, so that limiting angles for SFD flows can be preicte from inepenent measurements. Another opportunity for research is to reexamine the closure in the basal layer of eep flows, or in basal flows of a minimum height. If, as we suggest, the angular momentum of the grains plays an important role there, then it may be fruitful to exten the micropolar flui theory of Hayakawa [13] an Mitarai, et al. [14] from purely collisional flows to granular flows experiencing enuring contacts as well. Finally, the numerical simulations shoul be interrogate further to examine whether the moel correctly preicts the absolute level of fluctuation velocity, an to inform the evelopment of a better moel. Acknowlegments This work was supporte by NASA grants NCC3-468 an NAG3-2112, an by the International Fine Particle Research Institute. The author gratefully acknowleges the help of Haitao Xu an the benefit of iscussions with Patrick Richar, Renau Delannay, Alexanre Valance,

12 eg.1 3/2 (u/g)/(h/) µ e FIG. 13: Ratio ū/ g/(h/) 3/2 versus interparticle friction an normal restitution for an inclination of 22 an h/ = 4. The symbols represent the simulations of Silbert, et al. [3] for moels L3. The lines are preictions of the moel with a 1 = 2.1, µ E /µ =.8, an e eff = e eff (µ, e) from Jenkins an Zhang [1]. Luc Oger, James Jenkins, Deniz Ertaş, Leonaro Silbert, Thomas Halsey, James Lanry an Gary Grest. He is also grateful for the financial support of the Université e Rennes for his visits there uring the summers of 1998, 1999, 2 an 22. [1] O. Pouliquen an F. Chevoir, C. R. Physique 3, 163 (22). [2] O. Pouliquen, Phys. Fluis 11, 542 (1999). [3] L. Silbert, D. Ertaş, G. Grest, T. Halsey, D. Levine, an S. Plimpton, Phys. Rev. E 64, 5132 (21). [4] M. Louge an S. Keast, Phys. Fluis 13, 1213 (21). [5] S. Savage, in Mechanics of Granular Materials - New Moels an Constitutive Relations, eite by J. Jenkins an M. Satake (Elsevier, NY, 1983), pp [6] J. Jenkins, Hyraulic theory for a ebris flow supporte on a collisional shear layer (IAHR, Tokyo, 1993), pp [7] C. Campbell, J. Flui Mech. 465, 261 (22). [8] S. Torquato, Phys. Rev. E 51, 317 (1995). [9] N. Carnahan an K. Starling, J. Chem. Phys. 51, 635 (1969). [1] J. Jenkins an C. Zhang, Phys. Fluis 14, 1228 (22). [11] S. Savage, J. Flui Mech. 92, 53 (1979). [12] J. Jenkins an M. Richman, J. Flui Mech. 171, 53 (1986). [13] H. Hayakawa, in Traffic an Granular Flow 21, eite by Y. Sugiyama an D. Wolf (Springer, NY, 21), p. in press. [14] N. Mitarai, H. Hayakawa, an H. Nakanishi, Phys. Rev. Lett. 88, (22).