Partially fluidized shear granular flows: Continuum theory and molecular dynamics simulations

Transcription

1 Partially luidized shear granular lows: Continuum theory and molecular dynamics simulations Dmitri Volson, 1 Lev S. Tsimring, 1 and Igor S. Aranson 2 1 Institute or Nonlinear Science, University o Caliornia, San Diego, La Jolla, Caliornia , USA 2 Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA Received 17 February 2003; published 5 August 2003 The continuum theory o partially luidized shear granular lows is tested and calibrated using twodimensional sot particle molecular dynamics simulations. The theory is based on the relaxational dynamics o the order parameter that describes the transition between static and lowing regimes o granular material. We deine the order parameter as a raction o static contacts among all contacts between particles. We also propose and veriy by direct simulations the constitutive relation based on the splitting o the shear stress tensor into a luid part proportional to the strain rate tensor, and a remaining solid part. The ratio o these two parts is a unction o the order parameter. The rheology o the luid component agrees well with the kinetic theory o granular luids even in the dense regime. Based on the hysteretic biurcation diagram or a thin shear granular layer obtained in simulations, we construct the ree energy or the order parameter. The theory calibrated using numerical experiments with the thin granular layer is applied to the surace-driven stationary two-dimensional granular lows in a thick granular layer under gravity. DOI: /PhysRevE PACS number s : Cc, d, y I. INTRODUCTION In the last ew years there have been many experimental 1 7 and theoretical 8 14 studies that explored a broad range o granular low conditions rom rapid dilute lows to slow dense lows, as well as the details o the shear-driven luidization transition. While dilute granular lows can be well described by the kinetic theory o dissipative granular gases 15, dense granular lows still present signiicant diiculty in ormulation o a continuous theory. In Re. 16, Savage proposed a continuum theory or slow dense granular lows based on the so-called associated low rule that relates the strain rate and the shear stress in plastic rictional systems. Averaging strain rate luctuations yields a Binghamlike constitutive relation in which the shear stress has viscous as well as strain-rate independent parts. According to this theory, the stress and strain rate tensors are always coaxial and, urthermore, it also postulates that the viscosity diverges as the density approaches close packing limit. Losert et al. 3 see also Re. 17 proposed a similar hydrodynamic model based on a Newtonian stress-strain constitutive relation with density dependent viscosity without strain-rateindependent component. As observed in Re. 3, the ratio o the ull shear stress to the strain rate diverges at the luidization threshold. This was also interpreted in Re. 3 as a divergence o the viscosity coeicient when the volume raction approaches the randomly packed limit. This description works only in a luidized state and cannot properly account or hysteretic phenomena in which static and luidized states coexist under the same external load such as stick-slip oscillations 2, avalanching 6, or shear band ormation. In many granular lows o interest static and dynamic regions coexist under the same external load conditions. Examples o such hysteretic phenomena include stick-slip oscillations 2, avalanching 6, or shear band ormation. This calls or a uniied theory which would be applicable both in the lowing regime and in the static regime. In our recent papers 18,19 we proposed a dierent approach based on the order parameter description o the granular matter. The value o the order parameter speciies the ratio between static and luid parts o the stress tensor. The order parameter was assumed to obey dissipative dynamics governed by a ree energy unctional with two local minima. This description based on the separation o static and luid components o the shear stress, calls or an alternative deinition o viscosity as a ratio o the luid part o the shear stress to the strain rate. Since the luid shear stress vanishes together with the strain rate, the viscosity coeicient in our theory is expected to remain inite at the luidization threshold. We assumed the simplest Newtonian riction law, so the viscosity coeicient is a constant. This model yielded a good qualitative description o many phenomena occurring in granular lows, such as hysteretic transition to chute low, stick-slip regime o a driven near-surace low, structure o avalanches in shallow chute lows, etc. However, several important issues have not been addressed: mesoscopic deinition o the order parameter, quantitative speciication o the order parameter dynamics and the constitutive relation. In this paper we set out to perorm twodimensional 2D molecular dynamics MD simulations, which should provide us with the way to achieve these goals. We classiy all contacts as either luidlike or solidlike and deine the order parameter as a mesoscopic space-time average raction o solidlike contacts. Using this order parameter we obtain the constitutive relation and the relaxational dynamics o the order parameter directly rom simulations o granular low in a thin Couette geometry at zero gravity. Preliminary account o our results is presented in Re. 20. The paper is organized as ollows. Section II outlines the standard granular hydrodynamics theory based on the kinetic theory o dissipative granular gases. Sec. III introduces the continuum description o partially luidized lows based on the relaxational dynamics o the order parameter. In Sec. IV we describe our 2D molecular dynamics simulations and deine measurement protocol or the order parameter and luid X/2003/68 2 / /$ The American Physical Society

2 VOLFSON, TSIMRING, AND ARANSON and solid components o the stress tensor. In Sec. V we study the Couette low in a thin granular layer to obtain the ree energy controlling the relaxational order parameter dynamics and to extract the constitutive relations. In Sec. VI the obtained set o equations is used to calculate the stress and velocity distributions in a dierent system, a thick granular layer under nonzero gravity driven by a moving heavy upper plate. II. GRANULAR HYDRODYNAMICS In this section we outline the standard continuum description o granular lows based on continuity equations or mass, momentum, and luctuation kinetic energy or granular temperature. This description is usually applied to dilute granular gases where it can be rigorously derived rom the kinetic theory 15, although slightly modiied hydrodynamics based on kinetic theory oten works reasonably well or relatively dense lows, even though the kinetic theory itsel is not applicable to these conditions. The mass, momentum, and energy conservation equations have the usual orm D Dt u, Du Dt g, DT Dt : q, where is the density, u is the velocity ield, T ( u 2 u 2 )/2 is the granular temperature, D/Dt t (u ) is the material derivative, g is the gravity acceleration, is the stress tensor, q is the energy lux vector, u u is the strain rate tensor, and is the energy dissipation rate. These three equations have to be supplemented by the constitutive relations or the stress tensor, energy lux q, and the energy dissipation rate. For dilute systems, a linear relations between stress and strain rate is obtained, p Tr, q T. In the kinetic theory o granular gases 15, these equations are closed with the ollowing equation o state p 4 T d e G, and the expressions or the shear and bulk viscosities T 1/2 2 1/2 dg 1 2G 1 8 G 2, the thermal conductivity 2 T1/2 8 G T1/2, 8 3/2 d 1/2 dg 1 3G G 2, and the energy dissipation rate 9 16 G T3/2 3/2 d 3 1 e Here 0 e 1 is restitution coeicient and d is particle diameter. The unction G( ) which enters these relations is the spatial particle-particle correlation unction, and or a dilute 2D gas o elastic hard disks was derived by Carnahan and Starling 21, G CS 1 7 / This ormula is expected to work or densities roughly below 0.7. For high density granular gases, this unction has been calculated using ree volume theory 22, 1 G FV 1 e c / 1/2 1, 12 where c 0.82 is the density o the random close packing limit. Luding 24 proposed a global it G L G CS 1 exp 0 /m 0 1 G FV G CS 13 with empirically itted parameters and m However, even with this extension, the continuum theory comprising o Eqs cannot describe the orce chains that transmit stress via persistent contacts remaining in the dense granular lows, as well as the transition rom solid to static regimes and coexisting solid and luid phases. III. ORDER PARAMETER DESCRIPTION OF PARTIALLY FLUIDIZED GRANULAR FLOWS In this section we review briely our continuum theory 18,19 that provides an alternative approach to the ormulation o the constitutive relations in partially luidized granular lows. In the dense low regime, the granular matter can be considered incompressible, so Eq. 1 can be replaced by u 0 and the density c. This also allows us to drop the energy equation 3 and the equation o state 6, asor c, G( ), and T 0, so that G( )T const. This, o course, leads to the amiliar divergence o the viscosity coeicient G( )T 1/2. Next, we separate the stress tensor into two parts, a static contact part s, and a luid part. The latter is assumed to take a purely Newtonian orm

3 PARTIALLY FLUIDIZED SHEAR GRANULAR FLOWS:... p, 14 where p is the partial luid pressure, is the viscosity coeicient associated with the luid stress tensor, which is dierent rom introduced or the ull stress tensor. As we shall see in the ollowing, unlike, does not diverge as c. In our original model 18,19 we simply assumed const. We postulated 25 that the luid part o the o-diagonal components o the stress tensor is proportional to the odiagonal components o the ull stress tensor with the proportionality coeicient being a unction o the order parameter, yx q yx. 15 This assumption stipulates the equation or the static part o the o-diagonal stress components, s yx 1 q yx. 16 Both luid and solid parts o the stress tensor are assumed symmetric,,s yx,s xy. This assumption is conirmed by our numerical simulations see below. We choose a ixed range or the order parameter such that it is 0 in a completely luidized state and 1 in a completely static regime. Thus, the unction q( ) has the property q(0) 1, q(1) 0. In Res. 18,19 or simplicity we took q( ) 1. A similar relationship can be postulated or the diagonal terms o the stress tensor, xx q x xx, yy q y yy, s xx 1 q x xx, s yy 1 q y yy, where the scaling unctions q x,y ( ) can dier rom q( ). Combining Eqs , we obtain the constitutive relation in the closed orm, p /q /q, 19 where, x,y. The order parameter itsel was not related to any microscopic properties o granular assemblies in Res. 18,19. We simply assumed that because o strong dissipation in dense granular lows it has purely relaxational dynamics controlled by the Ginzburg-Landau equation, D Dt D 2 F. 20 Here D is the diusion coeicient and F( ) is the ree energy density, which was assumed to have a quartic polynomial orm to account or the bistability near the solid-luid transition: F 1 d. 21 The control parameter is determined by the stress tensor, which in Re. 18,19 was taken to be a linear unction o max mn / nn, where the maximum is sought over all possible orthogonal directions m and n. It is easy to see that in the interval 0 1, Eq. 20 has two stable uniorm solutions 0,1 corresponding to luid and solid states and one unstable solution. Momentum conservation equation 2 together with Eqs represent a closed set o continuous equations, which, ater being augmented by appropriate boundary conditions, can describe a variety o interesting granular lows such as avalanches in thin chute lows, drum lows, stick-slip oscillation in surace-driven lows, 18,19. Since a rigorous derivation o the continuum model o dense partially luidized lows rom irst principles does not seem easible, the assumptions made ad hoc in the ormulation o model 2 Eqs have to be checked against available experimental and/or numerical data. Unortunately, it appears to be extremely diicult to directly measure the order parameter in the bulk o granular material, as it is determined by subtle changes in the contact abric. In this paper we attempt to extract the properties o the order parameter rom molecular dynamics simulations and on this basis make a quantitative it o the order parameter model. IV. MOLECULAR DYNAMICS SIMULATIONS To model the interaction o individual grains we use the so-called sot-contact approach. 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 spring-dashpot model 9. 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 26. 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 its interactions with all the neighboring grains and walls o the container. Consider a grain i o radius R i located at r i moving with translational velocity v i and angular velocity i. This grain is in contact with grain j whenever overlap n R i R j r i r i 0. The relative velocity at the contact point and its normal and tangent components are given by v n v i v j n ij, v ij v i v j R i i R j j t ij, 22 v t v i v j t ij R i i R j j, 23 where n ij (r i r i )/r ij (n x,n y ) is the inward normal to the surace o i at the contact point with j, and the direction o the tangent t ij (n y, n x ) is chosen so that t ij n ij is colin

4 VOLFSON, TSIMRING, AND ARANSON ear with the angular velocity. Then the grain i is subjected to the contact orce due to the interaction with j with normal and tangential components F n k n n 2 n m e v n, 24 t F t sign t min k t t, t F n, t t v t d, t0 25 where k n,t are the corresponding spring constants, n is the normal damping coeicient, m e m i m j /(m i m j ) is the reduced mass, t is coeicient o riction, and t is tangential displacement since the moment t 0 o the initial contact between i and j. When the static yield criterion Eq. 25 is satisied, the magnitude o t is adjusted to an instantaneous equilibrium value providing F t t F n. According to the analytical solution o the linear spring-dashpot model, the coeicient o restitution and the duration o heads-on collision are e exp( n t c ), t c /(k n /m e n 2 ) 1/2. The advantages and limitations o the employed contact orce model were thoroughly studied by a number o authors 9,8,10. In act, this is the simplest model that allows us to account or both static and dynamic riction. When all orces acting on grain i rom other grains, boundaries, and perhaps external ields are computed, the problem is reduced to the integration o Newton s equations or translational and rotational degrees o reedom, m i d2 r i dt 2 mi g c F ic, 26 below deal with steady quantities, the procedure consists o two steps: space and then time averaging. The space averaging o a ield x is perormed over horizontal bins along the low direction o size V b L x 1 and is denoted as x in the ollowing. Contributions o those particles that only partially belong to a certain bin are weighted by the raction o their area. Ater a simulation has reached a steady state, instantaneous proiles are averaged over a suitable number o time snapshots. We shall denote the time averaging with x. For example, steady solid raction proile is given by y y,t, y,t V b 1 i Vb w i y V i, 28 where the summation runs over grains at least partially in a bin V b centered at y, V i is the area o grain i, and w i (y) isa corresponding raction o a grain s area within V b. The coarse-grained velocity ield is u y u y,t, u y,t y,t 1 v i y,t y,t 1 V b 1 i Vb w i y V i v i. 29 For simplicity, we extend the space-time averaging technique described above or other quantities, such as the stress tensor y y,t, I i d i dt Ri c F t ic, 27 y,t 1 r ic 2 F ic mi ṽ i ṽ i, c i 30 where the mass o particle i is denoted by m i and its moment o inertia is I i 1/2m i R i2, g stands or an external gravity ield, and the sums in Eqs. 26 and 27 run over all contacts o particle i. The results o the simulations reported here are presented in dimensionless orm. All quantities are normalized by an appropriate combination o the average particle diameter d, mass m, and gravity g. Equations 26 and 27 were integrated using the ithorder predictor-corrector 27 with a constant time step t. 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 e 0.92, t 10 4, t c 50 t, k n , n 16.7, k t 1/3k n. The computational domain spans L x L y area, and is periodic in horizontal direction x. Unless otherwise mentioned, the granulate is slightly polydisperse to avoid crystallization eects 8. We assume that the grain diameters are uniormly distributed around mean with relative width r. To provide a link between micromechanical quantities obtained through simulations and continuous ields, we deine the ollowing coarse-graining procedure. Since all experiments described where, x,y, r ic r ic e, F ic F ic e, ṽ i v i v i. The stress tensor in Eq. 30 has two distinct components. The irst one virial or contact describes pairwise interactions o grains. The second one kinetic or Reynolds is due to velocity luctuations. A. Order parameter or granular luidization: Static contacts vs luid contacts At any moment o time all contacts are classiied as either luid or solid. A contact is considered solid i it is in a stuck state (F t t F n ) and its duration is longer than a typical time o collision t. The irst requirement eliminates * long-lasting sliding contacts, and the second requirement excludes short-term collisions pertinent or completely luidized regimes. We choose a typical collision to last t* 1.1t c. When either o the requirements is not ulilled, the contact is assumed luid. We deine the order parameter as the ratio between spacetime averaged numbers o solid contacts Z s and all contacts Z within a sampling area 23, y Z s i / Z i

5 PARTIALLY FLUIDIZED SHEAR GRANULAR FLOWS:... This deinition endures at least two limiting cases: when granulate is in a static state and when it is strongly agitated, i.e., completely luidized. In act, in the ormer quiescent state all contacts are stuck and 1. In the luidized case Z s is 0 and Z is small but inite, thereore 0. Let us note that the order parameter just introduced is expected to be as sensitive to the degree o luidization as the stress tensor. A small rearrangement o the orce network may result in strong luctuations o either ield, while such quantities as the solid raction or granular temperature will remain virtually constant. It is known that granular aggregates exhibit rigidity phase transition near the critical volume raction c 11. When the volume raction is near or above c, the granular material has elastoplastic rheology and below this critical value it behaves as a luid. Near and above the critical volume raction or the same boundary conditions the granular aggregate may exhibit either slow creeping low, solid-like behavior, or both. The latter case, known as stickslip phenomenon, is observed both experimentally and numerically 2,11,28. In an experiment under constant volume boundary conditions, a series o stick-slip events occur without a signiicant change in the solid raction, which is insensitive to global rearrangements relieving the accumulated stress. We expect the order parameter to be able to relect such rearrangements and ultimately describe the corresponding phase transition. B. Stress tensor The ull stress tensor 30 consists o a contact virial part and the Reynolds part. In turn, the contact stress tensor can be split into the solid contact component s and the luid contact component in the same ashion as was done with contacts themselves. Combining the luid contact component with the Reynolds stress, we obtain the ull stress tensor as a sum o two parts s, 32 FIG. 1. Color online Sketch o the granular shear low model. V. TESTBED SYSTEM: COUETTE FLOW IN A THIN GRANULAR LAYER A. Biurcation diagram We studied the luidization transition in a thin (50 10) granular layer between two rough plates under ixed pressure P and zero gravity conditions Fig. 1. The parameters o simulations were t 0.5, e 0.92, r 0.2, k n , k t /k n 1/3, n We used the same material parameters o grains throughout this section. The rough plates were simulated by two straight chains o large grains twice as large as an average particle diameter glued together. The layer was chosen narrow enough, so the properties o the granular layer shear rate, shear stress components, order parameter, etc. in the absence o gravity were approximately constant across the layer within 10% accuracy. 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 weak orces not suicient to initiate a shear low and slowly ramped them up in small increments well above the critical yield orce at which the granular low started. Ater that we ramped the shear orces down until the granular layer was jammed again. At every stop we measured all stress components, strain rate, and the order parameter, and averaged the data over the whole layer and over time o each step. Figure 2 shows the strain rate we drop subscripts yx s 1 2 r c c F c, c R 1 2 c r c F c m i ṽ ṽ, 33 where summation in and is restricted to solid and luid contacts, respectively. 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 rigid 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. This picture is reminiscent o the concept o bimodal character o stress transmission in static contact network introduced by Radjai et al. 29,30. FIG. 2. Color online The strain rate a and the order parameter b vs shear stress in a thin Couette geometry o Fig. 1 with 500 particles (10 50) at P

6 VOLFSON, TSIMRING, AND ARANSON biurcation diagram, which merges with the stable branch at 0.26, we can make a simple analytic it o this curve as F, * * 2 exp A 2 * FIG. 3. Color online The order parameter as a unction o the normalized shear stress in a thin Couette geometry. at the strain rate and the order parameter as unctions o the shear stress yx, which is approximated by the applied orce F normalized by the layer length L x, or P 40. As it is to be expected, the strain rate remains 0, and the order parameter is one until the shear stress reaches a certain critical value This value diers slightly or dierent runs because o the inite system size and absence o selaveraging in the static regime. Above the yield stress, the strain rate abruptly jumps to a inite value 0.35, and the order parameter drops to At larger yx, the strain rate increases aster than it does linearly, and the order parameter rapidly approaches 0. The return curve corresponding to the diminishing o the shear stress ollows roughly the same path, and then continues to another smaller value o the shear stress ( 2 9.4). At this value the layer jams, the strain rate returns to 0, and the order parameter jumps back to 1. The most striking eature o this igure is the hysteretic behavior o both the strain rate and the order parameter as a unction o the shear stress. This hysteresis was anticipated in our order-parameter model 18,19 ; however, now we are in a position to it the model equations quantitatively. We repeated these simulations at several dierent values o the compressing pressure P. Data or dierent pressure values in the low regime all onto the same universal curve i one normalizes the shear stress by the pressure see Fig. 3. Assuming that there is an unobserved unstable branch o the with 0.6,A 25, 0.26 see Fig. 3, line and use it in * * the polynomial expansion o the ree energy density which enters the order parameter equation 20 : F F, d. 35 We also measured the density and the granular temperature o the grains as we decreased the shear orce. These measurements could only be perormed in the range 0.55 since or larger the partially luidized state is unstable. Note that the density o grains stays almost constant in a wide range o the order parameter 0.1 see Fig. 4 a. The granular temperature which is deined as T ũ i ũ i /2) normalized by the applied pressure P appears to be a unique unction o the order parameter Fig. 4 b. B. Relaxation dynamics o the order parameter We probed the relaxation dynamics o the order parameter by perorming the ollowing numerical experiment. The granular layer was prepared as in the preceding section. Lateral shear orces were increased adiabatically until the granular system reached a metastable solid jammed state within a hysteretic region. Then the layer was perturbed by applying random orces to a small randomly selected raction o particles. The dynamics o the order parameter varies depending on the magnitude o the perturbation. Figure 5 shows an example o the evolution rom the same jammed state or two dierent magnitudes o initial perturbation. Interestingly, the relaxation back to the jammed state is very ast, whereas the relaxation toward stable shear low is much slower. We believe that it has to do with inertia o grains and the upper plate, so the intrinsic time scale o the order parameter relaxation is rather small, O(1). Unortunately, our thin Couette low system does not allow us to probe the local coupling o the order parameter FIG. 4. Color online The density a and the normalized granular temperature b vs the order parameter in a thin Couette geometry or three dierent values o pressure P

7 PARTIALLY FLUIDIZED SHEAR GRANULAR FLOWS:... FIG. 5. Color online Relaxation o the order parameter towards shear low solid line and jammed state dashed line or two dierent initial perturbations in a thin Couette system o 500 particles with P 40, yx 12. since the order parameter is uniormly distributed throughout the system. In the absence o this data, this coupling was modeled by the linear diusion term in Eq. 20 with the constant diusion coeicient. As we will see in the ollowing section, this approximation indeed provides a good description or spatially nonuniorm near-surace low; however, the value o the diusion coeicient appears to be a unction o local stress. C. Fitting the constitutive relation The next step is to it the constitutive relation rom MD simulations. To this end, we use the same Couette low simulations, but now we analyze the luid stress and the s static stress separately during our ramp-down simulations at three dierent values o P. Figure 6 shows a sample o these data or P 30. At large F/(L x P), when the order parameter is low, the total stress is dominated by the luid component, but as the low stops and approaches unity, the luid stress turns to 0, and the total stress is equal to the static stress. Plotting yx / yx as a unction o the order parameter or dierent P Figure 7 a, we observe that all data collapse onto a single curve, which is well itted by q( ) (1 ) 2.5. The lines in Fig. 6 show the it o the luid and static stress tensors using Eqs. 15 and 16 with q( ) (1 ) 2.5. FIG. 6. Color online Fluid and static components o the shear stress in a thin granular layer (10 50) at external pressure P 30 as a unction o normalized external shear stress F/(L x P): direct calculation points, obtained rom total stress using relations Eqs. 15 and 16 with q( ) (1 ) 2.5 lines. The luid as well as solid parts o the stress tensor are nearly symmetric,,s yx,s xy, so the ratio xy / xy is described by the same scaling unction q( ). On the other hand, the same procedure or the diagonal elements o the stress tensor yields a noticeably dierent scaling see Fig. 7 b. Furthermore, a small but noticeable dierence is evident between xx / xx and yy / yy. More detailed analysis shows that, in act, luid parts o the diagonal components o the stress tensor xx and yy are nearly identical, and the dierence is due to the solid part o the normal stresses see Fig. 8. This observation is consistent with the act that the diagonal terms o the static stress tensor are determined by the details o the external loading. On the other hand, in a completely luidized state the diagonal terms are all equal to the hydrodynamic pressure p 31. In a partially luidized regime, the diagonal terms o the shear stress can be expressed as xx p /q x, yy p /q y. 36 FIG. 7. Color online Ratios o the luid stress components to the corresponding ull stress components / vs or three dierent pressures P: a shear stress components, closed symbols yx, open symbols xy, line is a it q( ) (1 ) 2.5 ; b normal stress components, closed symbols xx, open symbols yy, lines are the its q x ( ) (1 ) 1.9, q y ( ) (1 1.2 )

8 VOLFSON, TSIMRING, AND ARANSON Both unctions q x,y ( ) should approach 1 as 0 the normal stresses should be equal in the luid state, but they may have dierent unctional orm to relect the anisotropy o the static stress tensor. In our simple Couette low, the diagonal stress tensor components can be well itted by q x ( ) (1 ) 1.9 and q y ( ) (1 1.2 ) 1.9 see Fig. 7 b. Soweobserve that even in a partially luidized regime, the luid phase component indeed behaves as a real luid with a wellbehaved partial pressure p which is zero in a solid state at 1 and is becoming the ull pressure in a completely luidized state 0. Plotting the luid shear stress versus the strain rate, we can test the validity o the Newtonian model or the stress-strain relation 14. Figure 9 shows yx vs or three dierent pressures P 20,30,40. At small all three lines are close to the same straight line yx 12, which indicates that the Newtonian scaling or luid shear stress holds reasonably well. The deviations at large are evidently caused by variations o temperature and density in the dilute regime. Note that in contrast, the ull shear stress does not goes to 0 as 0 see Fig. 9 b, so a viscosity coeicient conventionally deined as the ratio o the ull shear stress to the strain rate diverges at the luidization threshold as observed in Re. 3. Combining the Newtonian law or the luid stress-strain dependence with the order parameter scaling o the luid stress tensor, we arrive at the relationship between the ull stress tensor and the strain rate tensor 19 with 12, q x ( ) (1 ) 1.9, q y ( ) (1 1.2 ) 1.9, q( ) (1 ) 2.5. FIG. 8. Color online Diagonal stress components luid, solid, and total in a thin granular layer (10 50) at external pressure P 30 as a unction o external shear stress F/L x normalized by the external pressure P. D. Toward dilute granular lows granular temperature revisited While the above ittings have been made or the regime o a slow dense low with c, it is tempting to generalize the theoretical model so that it smoothly crosses over to the standard kinetic continuum theory 1 10 or 0. This goal can be achieved by including the equation o state 6 and the equation or the granular temperature 3 back into the theory. The important dierence with respect to the standard kinetic theory is that the pressure, which we calculate with Eq. 6, is not the total pressure, but the partial pressure associated with the luid part o the stress tensor. O course, as 0, the static part o the stress tensor disappears, and the partial pressure becomes the total pressure. We can test the relevance o this combined approach by calculating the spatial correlation unction G( ) (1 e) 1 d 2 p /4 T 1 with the values o luid pressure p, temperature T and density calculated in our testbed Couette low at dierent external pressures P and comparing it with the theoretical unctions G CS ( ),G CS ( ),G L ( ) Eqs Figure 10 shows that the Carnahan-Sterling ormula works very well in the dilute range 0.67 as expected. In the high density regime G( ) approaches the ree volume result 12, and the overall dependence is in agreement with the global interpolation G L ( ) by Luding 24. FIG. 9. Color online Stress-strain rate relation or a thin granular Couette low at three dierent external pressures: a luid shear stress vs strain rate, the straight line is a constant viscosity it 12 ; inset: scaled luid shear stress 1 yx as a unction o ; b ull shear stress vs strain rate

9 PARTIALLY FLUIDIZED SHEAR GRANULAR FLOWS:... the presented data strongly suggest that the constitutive relations or the luid part o the stress tensor are well described by the standard granular hydrodynamics. E. Order parameter description o partially luidized granular lows take 2 Let us summarize the equations o the continuum theory as speciied on the basis o the 2D molecular dynamics simulation o the thin Couette low. The mass, momentum, and energy conservation conditions are expressed by Eqs The order parameter equation now has the orm FIG. 10. Color online Particle-particle correlation unction G( ) calculated via equation o state 6 using the values o luid pressure, temperature, and density in a thin granular Couette low at three dierent external pressures. Inset: the same data in a semi-log scale We can carry this analysis one step urther and test the kinetic theory prediction or the shear viscosity. I we scale yx by the shear viscosity calculated using kinetic ormula 7 with globally itted correlation unction G L ( ) and actual temperature and density values rom corresponding runs, all three lines in Fig. 9 a collapse onto the same straight line dependence 1 yx see Fig. 9 a, inset. One more test o the kinetic theory predictions can be perormed by analyzing the granular temperature as a unction o the shear strain rate. According to Re. 16, T 1/2 A or a plane parallel shear low where A is a material constant weakly dependent on volume raction. Our numerical data or three dierent external pressures shown in Fig. 11, are consistent with this scaling law although small deviations can be observed at small. We have not done a similar comparison or the bulk viscosity, thermal conductivity, and the energy loss; however, FIG. 11. Color online Granular temperature as a unction o strain rate in a thin granular Couette low at three dierent external pressures. D Dt D * * 2 exp A 2 * 2 37 with * 0.6, * 0.26, A 25. As mentioned beore, or the lack o simulation data, we assume linear diusion coupling o the order parameter with a constant nondimensional diusion coeicient D. The constitutive relation now reads p /q Tr /q /q 38 with q x ( ) (1 ) 1.9, q y ( ) (1 1.2 ) 1.9, q( ) (1 ) 2.5. The equation o state and expressions or viscosity, thermoconductivity, and the energy dissipation have the same unctional orm as Eqs. 6 10, but they are now written or the luid parameters p,,, and. VI. SURFACE-DRIVEN SHEAR GRANULAR FLOW UNDER GRAVITY In this section we apply the theoretical description that was ormulated in the preceding section on the basis o numerical simulations o a thin Couette low with no gravity to another model problem. We consider shear granular low in a thick granular layer under gravity driven by the upper plate, which is pulled in a horizontal direction, see Fig. 12. A similar system has been studied experimentally by Nasuno et al. 2, as well as by Tsai et al. 7. We simulated up to particles in a rectangular box under a heavy plate, which was moved either with a constant speed V x or a constant orce F x. Periodic boundary conditions were assumed in a horizontal direction. Ater a transient, a quasistationary luidization and shear low established in the near-surace layer, while near the bottom grains remained in a nearly static jammed regime. Here, we show the results o several dierent runs with small large pressure and small large shear. The details o these runs are given in Table I. The vertical proiles o the density, low velocity, and the order parameter are shown in Fig. 13. The density o grains remains nearly constant, very close to the maximum random packing density value, except or a narrow near-surace boundary layer. The most signiicant density variations were observed or the case P10V50 corresponding to a light, ast moving upper plate. The horizontal

10 VOLFSON, TSIMRING, AND ARANSON TABLE I. Parameter values or the simulations o deep Couette lows or dierent geometries and boundary conditions. The irst six runs were perormed at constant velocity o the top plate and the last two runs were or a constant horizontal orce applied to the plate; or all runs t 0.3, e 0.82, r 0.2, k n , k t /k n 1/3, n Run ID N p Lx Ly P V x F x /(PL x ) yx P10V P10V P50V P50V P50V5L P50V50L P20F P20F20XL FIG. 12. Color online The geometry o the MD simulation o a surace-driven shear low. velocity decays roughly exponentially o the plate in agreement with experimental evidence 2,7. The vertical proiles o the order parameter demonstrate a well-deined transition rom luid state near the upper plate to solid state below. The width o the interace grows with the applied pressure P, which indicates the stress dependence o the diusion coeicient or the order parameter. Surprisingly, we ound that the order parameter does not approach 1 at large depths, but instead seems to saturate at some value slightly below 1. We believe that this behavior is inherent to our 2D geometry with periodic boundary conditions on side walls. The moving upper plate oscillates vertically and produces vibrations in the bulk o granular layer. These slowly decaying with depth vibrations break weak contacts between particles that are not strongly pressed against each other e.g., lying under FIG. 13. Color online Density a, horizontal velocity b, temperature c, and the order parameter d proiles in a deep granular layer driven by upper moving plate or our dierent runs rom Table I

11 PARTIALLY FLUIDIZED SHEAR GRANULAR FLOWS:... FIG. 15. Color online Ratios o luid and ull components o the stress tensor as a unction o the order parameter or our dierent runs at dierent speeds and pressures: a shear stress component yx, b normal stress components xx,yy. Closed symbols correspond to xx, open symbols correspond to yy. Solid lines show the its q( ) (1 ) 2.5 in a, and q 1 ( ) (1 1.2 ) 1.9 in b. FIG. 14. Color online Stationary proiles o the vertical a, horizontal b, and shear c stress components or run P20F10. c Here is the contact part o the luid stress deined in Eq. 33 arches. We believe that in ull 3D simulations with more realistic boundary conditions this eect may be less pronounced. In principle, it may be included in the theoretical description by proper averaging o luctuations o the strain tensor in the spirit o Savage 16. By averaging velocity luctuations and orces acting on individual particles, we calculated vertical proiles o the luid and solid parts o the shear and normal stress components see Sec. IV. These proiles or run P10V50 are shown in Fig. 14. Strong luctuations o the horizontal component o the stress tensor xx are mostly related to the static component o the tensor. According to Eqs. 15 and 16, luid and solid parts o the shear stress components should be related to the shear component o the ull stress tensor which in this geometry is roughly independent o y) via the unction q( ). Figure 15 a depicts this unction as a parametric plot o yx (y)/ yx vs (y) made by using vertical proiles o stresses and the order parameter. As seen rom this igure, the same it q( ) (1 ) 2.5 approximates the data quite well. However, unlike the zero-gravity case o the thin Couette low, the normal stress components seem to be isotropic xx yy and they both are well described by q y ( ) see Fig. 15 b. The dependence o the luid part o the shear stress component on the strain rate Fig. 16 shows the same behavior as or the thin Couette system: at small the viscosity is nearly constant 12, and at larger shear thinning is observed. Interestingly, the dependence o the local Reynolds shear stress on the local strain rate is well described by the Bagnold scaling R yx 2 yx. Eventually, at large, this scal

12 VOLFSON, TSIMRING, AND ARANSON FIG. 16. Color online Fluid shear stress vs local shear strain rate or several runs: a total luid shear stress, b Reynolds part o the shear stress. ing should dominate the ull stress-strain rate relationship. Figure 17 compares the behavior o the viscosity coeicient yx / and yx / calculated along the vertical proiles o and, as a unction o density and order parameter. While the ormer diverges as c and 1, the latter approaches the constant value 12, in agreement with the results o Sec. V. As in the preceding section, we can extract the particleparticle correlation unction by calculating G( ) (1 e) 1 d 2 p /4 T 1 using the vertical proiles o yx,t, and. Again, we obtain a good agreement with theoretical predictions based on the kinetic theory or luid component o the stress tensor Fig. 18. Finally, we can compare the stationary vertical proiles o the order parameter and the horizontal velocity with theoretical predictions. In most o our numerical simulations we speciied the velocity o the upper plate rather than the applied orce. That allowed us to study the regimes o slow dense lows, which would be unstable had we applied a constant shear orce. The shear stress tensor component yx in the stationary regime was indeed constant across the layer see, or example, Fig. 14 c. However, due to slippage near the moving plate the relation between the plate speed and the shear stress is complicated. We do not address the issue o FIG. 17. Color online Full viscosity yx / solid symbols and luid viscosity yx / open symbols coeicients as unctions o density a and the order parameter b or two runs. boundary conditions here as it is a subject o a separate study see or example, Re. 32. Here we simply use the values that are obtained in numerical simulations the last column in Table I, as parameters in our theoretical model. In the stationary regime, the relevant stress tensor components are speciied as ollows: FIG. 18. Color online Particle-particle correlation unction as a unction o density or several runs o the thick Couette low simulations

13 PARTIALLY FLUIDIZED SHEAR GRANULAR FLOWS:... FIG. 19. Color online Proiles o the order parameter and velocity in a thick granular layer driven at the surace by a heavy moving plate or run P10V5 a, P10V50 b, P50V5 c, P50V50 d. Lines show the theoretical results obtained rom continuum models 37 and 38, empty symbols indicate numerical data. Insets show the velocity proiles in the logarithmic scale. yy P H y, yx xy const, where P is the external pressure applied to the upper wall, and H is the thickness o the granular layer note that it is dierent rom L y due to compaction. In the regime o slow dense low, the volume raction o grains is nearly constant, close to the random close packing density, and so the low can be assumed incompressible. We also need to speciy the boundary conditions or the order parameter at the top and bottom plates. This is a serious issue in its own right, which we will address elsewhere. Here we simply impose no-lux boundary conditions both at the top and the bottom plate or the order parameter, y (0) y (H) 0. We limit ourselves with the case o slow dense low regime, when the granular temperature plays a minor role, and the granular low can be considered incompressible. This allows us to use the reduced set o Eqs. 2, 37 and 38 with the ixed viscosity 12. The stationary shear low solution o the continuum equations can be ound numerically as ollows. Since the components o the ull stress tensor are assumed known, we solve the time-dependent order parameter equation 37 using the pseudospectral method until the solution reaches a stationary state. The resulting solution or the order parameter is then used to obtain the velocity proile by integrating the constitutive relation 38 rom the bottom (y 0) up. Since the grains are strongly compressed near the rough bottom plate due to gravity, we assume the no-slip boundary condition or the horizontal velocity at y 0. The momentum conservation equation 2 is satisied automatically. Thus, obtained proiles o velocity and the order parameter were compared with our 2D molecular dynamics simulations. The results o the comparison between the velocity and order parameter proiles obtained in simulation and by using

14 VOLFSON, TSIMRING, AND ARANSON the continuum theory are shown in Fig. 19 or our runs P10V5 a, P10V50 b, P50V5 c, P50V50 d. The only itting parameter used was the diusion constant D in the order parameter equation, which has not been determined in our testbed analysis. We used D 1 or runs P10V5 and P10V50, D 5 or P50V5 and D 10 or P50V50. From this we can conclude that the diusion coeicient depends on the local stress tensor; however, more elaborate numerical experiments are needed to pinpoint this dependence more quantitatively. All other parameters were identical or all our cases as speciied in Sec. V E. The vertical proiles o the order parameter and the horizontal velocities are reasonably well described by the theory. However, or low pressure runs P10V5 and P10V50, the horizontal velocity proiles deviate rom the numerical data presumably because the viscosity coeicient is no longer a constant in a dilute region near the top plate. VII. CONCLUSIONS In this paper we perormed a series o numerical simulations o 2D wall-driven granular lows. These simulations were designed with a speciic goal, to quantiy the continuum theory o partially luidized granular lows, which was introduced ad hoc earlier 18,19. We deined the order parameter as a ratio o the number o static contacts to the total coordination number averaged over a small mesoscopic volume. Using simulations o a thin Couette low between two rough plates, we determined the ree energy density or the order parameter. Simulations conirmed that the ratio o the shear to the normal stress in the bulk o the granular low can parametrize the stationary states o the order parameter equation. The same simulations allowed us to determine the detailed structure o the constitutive relation. We split the total stress tensor into the luid and solid components, in which the ormer comprises the Reynolds stresses and the stresses transmitted through short-term collisions, while the latter is ormed by the orce chains through persistent contacts. The ratio o luid and solid stress components is indeed determined by the order parameter through scaling unctions q( ), q x,y ( ). Remarkably, the luid component o the stress tensor is a linear unction o the strain rate in the slow dense low regime. This justiies the Newtonian scaling o the stress-strain relationship adopted in the theory. Using the calibrated theory, we studied the low structure o a thick surace driven Couette granular low under gravity. We ound the theoretical predictions to be in a good quantitative agreement with simulations. The evidence presented here suggests an intriguing interpretation or the order parameter description o dense and slow granular lows. The granular material under shear stress appears to be similar to a multiphase system with the luid phase immersed into the solid phase. The luid phase behaves as a simple Newtonian luid or small shear rates when the density is almost constant, but exhibits shear thinning at larger shear rates when the density begins to drop. We observed that the Reynolds part o the luid shear stress obeys the Bagnold scaling R yx 2. We anticipate that or very large shear rates when the Reynolds stress becomes dominant, the overall stress tensor should exhibit Bagnold scaling locally. While this theory is primarily intended or dense and slow granular lows, we have shown that it can be combined with existing models o rapid granular lows based on the kinetic theory o granular gases. This requires to drop the assumption o incompressibility and include the equation or the granular temperature. Our simulations showed that the kinetic theory works well or the luid part o the stress tensor in the whole range o densities rom dilute regime to the critical random close packing density. Many issues still remain open. The spatially nonuniorm dynamics o the order parameter requires a more detailed study. We ound that the diusion constant postulated in Eq. 37 appears to be a unction o the normal shear stress as well as the local strain rate; however, we do not have suicient numerical data to provide a quantitative description o this dependence. It would be o interest to analyze the propagation o a luidization ront in a granular layer prepared in a meta-stable static regime. Such simulations could provide an insight into the mechanisms o the local coupling o the order parameter. The molecular dynamics algorithm employed is based on a number o approximations. These approximations, however well tested and widely accepted 8 10, directly aect the results o our itting the continuum model. For example, i one replaces the Hookian model o particle interaction with a Hertzian one, an appreciable dierence in the structure o the order parameter may be observed. More numerical work is needed to quantiy the relationships between the microscopic parameters o the system nature o collisions, restitution coeicient, riction, etc. and the parameters o the continuum model. Finally, our simulations were limited by 2D systems, and o course the resulting continuum theory can only be directly applicable to 2D systems. While we anticipate the structure o the model to remain in 3D systems, the speciic orm o the itting unctions should change. This uture work will allow us to perorm a comparison o the 3D model not only with numerical simulations but also with experimental data. ACKNOWLEDGMENTS The authors are indebted to B. Behringer, P. Cvitanović, J. Gollub, T. Halsey, and J. Viñals or useul discussions, and to J. C. Tsai or sharing his unpublished experimental data. This work was supported by the Oice o the Basic Energy Sciences at the U.S. Department o Energy, Grant Nos. W ENG-38 and DE-FG03-95ER Simulation were perormed at the National Energy Research Scientiic Computing Center