Granular Rheology and Phase Transition: DEM Simulations and Order-Parameter Based Constitutive Model

Chemical Engineering Science 72 (2012) 20–34

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Chemical Engineering Science

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Granular rheology and phase transition: DEM simulations and order-parameter based constitutive model

V. Vidyapati, S. Subramaniam n

Department of Mechanical Engineering, Center for Computational Thermal-Fluids Research, Iowa State University, Ames, IA 50011, USA article info abstract

Article history: DEM (discrete element method) simulations are used to characterize granular rheology and granular Received 31 August 2011 phase transition by studying order parameter (OP) dynamics. These DEM simulations reveal the Received in revised form existence of a third stable granular phase that is neither completely fluid-like nor completely solid-like. 7 December 2011 Hence, a modified form of the free energy density function is proposed to account for this third stable Accepted 22 December 2011 granular phase observed in DEM simulations. Further, a constitutive model for granular flows is Available online 8 January 2012 developed based on an objective version [Gao et al., Phys. Rev. E 71(021302), 2005] of the original OP Keywords: concept proposed by Volfson et al. [Phys. Rev. E 68(021301), 2003], with the intention of capturing the Granular flow transitional behavior in a continuum description of granular flows. This OP-based model is refined by Order parameter extracting new model coefficients from 3D DEM simulations of homogeneous shear flow. The proposed Discrete element method linear version of the objective OP model has the advantage that the total granular stress is a linear Constitutive model Transitional regime combination of the solid-like and fluid-like stresses, and it is denoted as the refined order parameter (ROP) model. The performance of this ROP model along with other existing constitutive models is assessed in homogeneous shear flow, and the results are explained by analyzing granular stress data from DEM simulations. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction interactions between the particles are important, still lacks a predictive constitutive model and has motivated many studies A quantitative description of the large-scale behavior of over the past decade (Jop et al., 2006; MiDi, 2004; Vidyapati et al., granular flow in industrial devices – such as hopper discharge, 2011). Nevertheless, these theories were unable to capture the chute flow, and dense-phase pneumatic conveying – rely on a transition between solid-like and fluid-like behavior of the continuum description of granular flows (Sundaresan, 2001). The granular material. difficulty in modeling granular rheology is that granular matter Most constitutive models (Savage, 1998; Johnson and Jackson, can exhibit constitutive behavior like a solid (in a sand pile), like a 1987; Srivastava and Sundaresan, 2003) that are used to predict liquid (when poured from a hopper or silo), or like a gas (when it the behavior of granular flows are based on an additive decom- is strongly agitated) (Jaeger and Nagel, 1996). These different position of the total granular stress as a weighted sum of kinetic kin fric constitutive behaviors depend on both the microscale properties and frictional contributions ðsij ¼ sij þsij Þ, with the weight (e.g., particle friction and coefficient of restitution) as well as on factor specified solely as a function of the solid volume fraction. macroscale properties (e.g., solid volume fraction and shear rate). A continuum theory for slow dense granular flows based on the Further, these different behaviors pose significant challenges in so-called associated flow rule is proposed by Savage (1998). This formulating a comprehensive constitutive theory that can theory relates the shear stress and the strain rate in a plastic describe all the regimes of granular rheology. For the two extreme frictional system. Averaging strain-rate fluctuations yields a regimes – rapid and quasi-static – constitutive equations have Bingham-like constitutive relation in which the shear stress has been proposed based on the kinetic theory for rapid flows two contributions: a viscous part, and a strain-rate independent (Goldhirsch, 2003), and soil mechanics for quasi-static flow part. According to this theory the stress and strain rate tensors are (Nedderman, 1992; Schaeffer, 1987). However, the transitional always coaxial. Furthermore, the theory also postulates that the (intermediate) regime, where both collisional and frictional viscosity diverges as the density approaches the close packing limit. A similar hydrodynamic model based on a Newtonian stress–strain constitutive relation with density-dependent visc- n Corresponding author. osity is proposed by Losert et al. (2000). In this model also the E-mail address: [email protected] (S. Subramaniam). viscosity diverges when the density approaches the random close

0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.12.037 V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 21 packing density of grains. Jop et al. (2006) proposed a constitutive which OP dynamics are extracted. The description and refinement relation for dense granular flows inspired by the analogy between of the OP-based continuum model is discussed in Section 4. granular flows and visco-plastic fluids such as Bingham fluids. Section 5 describes the specification of the proposed ROP model. In their work (Jop et al., 2006), granular flow is described as an The performance of the ROP model in different regimes is incompressible fluid with the stress tensor given as a function of presented in Section 6.InSection 7 we assess the performance 0:5 the inertia number, I ¼ g_d0=ðP=rsÞ . of different constitutive models in the intermediate regime, and Experiments in a 2D granular shear cell (GSC) (McCarthy et al., the results are explained by analyzing granular stress data from 2010; Jasti and Higgs, 2008) as well as DEM simulations (Volfson DEM in Section 8. Finally conclusions are drawn in Section 9. et al., 2003b) reveal that grain contacts in the transitional regime are characterized by a mix of enduring solid-like and fluid-like contacts that is indicative of a granular phase transition. In 2. Order parameter description of granular ‘phase’ transition particular, these grain interactions are not determined by the solid volume fraction alone, but are dependent on particle proper- In a homogeneous granular flow, the OP is defined (Volfson ties (such as particle friction coefficient and inelasticity) as well as et al., 2003b) as the ratio of the number of space–time averaged flow properties (such as the shear rate). Consequently, simple ‘‘enduring’’ (solidlike) contacts /ZsS to all contacts /ZS within a additive models are not able to capture the complex constitutive sampling volume behavior in the transitional regime. Since most constitutive models /Z S in use are phenomenological, this observation motivates the r s , ð1Þ /ZS development of a constitutive model for the transitional regime that reflects the phase transition based on microscale physical where /zS and z stand for averaging of z in space and time, interaction between the grains. respectively. The OP is useful in characterizing two limiting cases: Volfson et al. (2003b) proposed a different approach based on (i) a solidlike state when the granulate is in a state of ‘‘enduring’’ the order parameter (OP) description of granular matter. The OP is contacts, and (ii) a fluidlike state when it is strongly agitated, i.e., defined as the ratio of number of solid-like (enduring) contacts to completely fluidlike. In the solidlike state all contacts are endur- all contacts in a given sampling volume. The OP attains its ing and hence r ¼ 1. In the fluid limit /ZsS is zero and /ZS is maximum value of 1 when the granular matter is in a ‘solid’ state small but finite, and therefore r ¼ 0. Since the OP distinguishes and takes its minimum value of 0 in the completely ‘fluid’ limit. between ‘‘solidlike’’ contacts and ‘‘fluidlike’’ contacts in the They decomposed the total granular stress sij into the sum of a granular material, its computation requires a precise definition s f ‘‘solidlike’’ stress sij and a ‘‘fluidlike’’ stress sij. The relative of these two types of contacts. A contact is considered enduring magnitude of the solidlike and fluidlike contributions is a function (solidlike), if it is in stuck state ðFt omtFnÞ and its duration is of the OP. Models are then proposed for the ‘‘solidlike’’ and longer than a typical time of collision tn, which is generally taken

‘‘fluidlike’’ contributions, in terms of the total granular stress as 1.1 times the binary collision time tc (Volfson et al., 2003b). The tensor sij. The postulated form of the free energy density function first requirement eliminates long-lasting sliding contacts and the Fðr,dÞ in Volfson’s (Volfson et al., 2003b) work has only two stable second requirement excludes short-term collisions. When either states for the OP: either zero or unity. In this functional form of of the requirements is not fulfilled, the contact is defined as the free energy density function, r corresponds to the order ‘‘fluidlike’’. parameter and d is the ratio of shear to normal stress. The OP In order to understand granular phase transition through the values obtained from this procedure need to be validated against OP, we extract this quantity from 3D DEM simulations of sheared DEM data in order to verify this postulated form of the free energy granular flow over a range of solid volume fractions, particle density function. The validity of this free energy density function friction coefficients and shear rates. In the following section we is examined in this work using DEM simulations. Also the original describe these 3D DEM simulations of sheared granular flow. OP model (Volfson et al., 2003b; Aranson and Tsimring, 2001) does not satisfy the objectivity requirement (Gao et al., 2005). The original OP model by Volfson et al. (2003b) was general- 3. DEM simulations of sheared granular flow ized to an objective form by Gao et al. (2005), which makes it independent of the coordinate system. The model coefficients of The OP is extracted by performing three-dimensional (3D) the objective OP model specified by Gao et al. (2005) were discrete element method (DEM) simulations of monodisperse, obtained by fitting DEM data (obtained from Volfson et al., non-cohesive spheres of diameter d0 and mass m0 subjected to 2003b) for 2D inhomogeneous Couette flow with wall boundary homogeneous shear over a range of solid volume fractions, conditions. In the present work, new model coefficients for the particle friction coefficients and shear rates. A soft-sphere model objective OP model are extracted from data of 3D DEM simula- is used in which particles interact via contact laws and friction tions of homogeneous shear flows. The objective OP model (Gao only on contact. Since the realistic modeling of particle deforma- et al., 2005) is linearized to allow easy inversion of the total tion is complicated, a simplified contact force model based on a granular stress from fluidlike or solidlike stress relations, and it is linear spring-dashpot combination is used in this work (Silbert found that the simple linear model incurs only 11% more error et al., 2001). Details of the computational model used in the than the full nonlinear model. This linearized OP model with new discrete element simulations are given in Appendix A. coefficients is denoted as the refined order parameter (ROP) These constant-volume DEM simulations of sheared granular model. Following Aranson and Tsimring (2001), it is assumed flow are performed in a cubical domain of side length L ¼ 14d0, for that the fluidlike contribution of the total granular stress can be solid volume fraction ranging from 0.20 to 0.62. The effect of computed using a constitutive relation from the kinetic theory of system size is examined by varying the box length from 7d0 to granular flows (KTGF) (Lun et al., 1984). The performance of this 20d0. It was found that the stress asymptotes once the box length ROP-KT model is assessed by comparing predicted granular stress exceeds 10d0, consistent with the estimates reported by Campbell with DEM data in different regimes of granular flow. (2002). For all the simulations reported, the mass and diameter of The following section describes the order parameter approach the particles are set to 1, so the density of the particles is 6=p. The 5 as applied to granular flow by Volfson et al. (2003b). The next value of normal spring stiffness kn is set to 2 10 (in m0g=d0 section discusses DEM simulations of sheared granular flow from units), which captures the general behavior of intermediate to 22 V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34

high kn systems (Silbert et al., 2001). The value of the coefficient of this algorithm to study homogeneous time-dependent flows. of restitution e is chosen to be 0.7. All these simulations are This difficulty can be greatly alleviated through the use of the performed with zero gravity. The integration time step Dt for all SLLOD algorithm. The SLLOD algorithm originates from ideas in the simulations is selected to be tc=50, where tc is the binary nonequilibrium statistical mechanics (Evans and Morriss, 1990) collision time. This time step is shown to be sufficiently small to where nonequilibrium flows such as shear flow are simulated by ensure temporal convergence (Silbert et al., 2001). Simulations applying a force to the entire system (as opposed to simply are run to a nondimensional time of g_t ¼ 500, which is long moving the boundaries of the system faster or slower, as done enough to attain a statistically stationary solution (Campbell, in Lees–Edwards). Although we do not study time-dependent 2002). After reaching steady state the quantities are time-aver- shear in this article, the SLLOD algorithm (Lois et al., 2005) was aged over a time window corresponding to 200g_ 1. As a first step applied to all the simulations performed in this study to be we verified our OP calculations with previously published results consistent with other work. of Volfson et al. (2003a) for inhomogeneous wall-bounded shear Using data obtained from these homogeneous shear simula- simulation and confirmed that the OP is capable of capturing tions we established a comprehensive regime map by assigning granular phase transition from solidlike to fluidlike behavior. each of these simulations different regimes (inertial, intermediate Results from this study are summarized in Appendix B. and quasi-static) based on the scaling of shear stress with strain rate. In the inertial regime the stress scales as square of the strain 2 3.1. Granular rheology through regime map rate ðspg_ Þ (Bagnold, 1954), whereas in the quasi-static regime stress remains independent of the strain rate ðsaf ðg_ÞÞ (Campbell, In order to characterize the physics of granular phase transi- 2002). In between these two regimes there exists an intermediate tion and to generate benchmark data for model assessment in regime where stress is related to the strain rate in the form of a n different regimes, we performed homogeneous shear simulations power law ðspg_ Þ, where n takes values between 0 and 2 based (where the stress is independent of position) over a wide range of on particle and flow properties (Tardos et al., 2003). solid volume fractions, shear rates and particle friction coeffi- Fig. 1(a) and (b) shows the regime maps obtained from these cients (see Table 1). These homogeneous shear simulations are DEM simulations of homogeneously sheared granular flow in the performed with periodic boundary conditions in all directions space of solid volume fraction n and particle friction coefficient n 3 _ 2 4 ðx,y,zÞ and uniform shear is generated in the domain using mp, for non-dimensional shear rates k ðkn=rsd0g Þ of 2:5 10 and 9 2 the ‘‘SLLOD’’ algorithm (Evans and Morriss, 1990). The SLLOD 10 , respectively. In Fig. 1(a) and (b) the inertial regime (spg_ )is algorithm (Evans and Morriss, 1990) is an improved form of the represented in red, whereas blue indicates the quasi-static regime a Lees–Edwards boundary condition (Lees and Edwards, 1972)to (s f ðg_Þ). In between these regimes, there exists an intermediate n generate simple shear flows. The shearing motion induced by regime (spg_ ,0ono2) which is the green region in Fig. 1(a) and the Lees–Edwards boundary condition takes time to develop. (b). The principal observations concerning granular rheology from Therefore, the flow would not be homogeneous immediately after these regime maps are: a shear rate change, which raises questions about the suitability 1. As particle friction coefficient decreases the intermediate n Table 1 regime expands for both k values shown. This is important Parameters for homogeneous shear simulations. because the friction coefficient for many granular materials (such as glass beads) varies between 0.15 and 0.5, and hence n Solid volume fraction Particle friction coefficient Shear rate (k ) the expansion of the intermediate regime will affect granular flow in practical devices. Studies performed by Campbell (2002) 0.45 0:1,0:5,1:0 2.5 104,105,107,109 0.53 0:1,0:5,1:0 2.5 104,105,107,109 (for monodisperse systems) and Ji and Shen (2008) (for poly- 0.57 0:1,0:5,1:0 2.5 104,105,107,109 disperse systems) also confirm this dependence of regime 0.58 0:1,0:5,1:0 2.5 104,105,107,109 transition on particle friction coefficient in granular media. 4 5 7 9 0.59 0:1,0:5,1:0 2.5 10 ,10 ,10 ,10 2. Fig. 1(b) shows that at higher kn the intermediate regime starts 0.60 0:1,0:5,1:0 2.5 104,105,107,109 at lower values of solid volume fraction, although its extent in 0.62 0:1,0:5,1:0 2.5 104,105,107,109 solid volume fraction remains the same.

1 1

Quasi-static Quasi-static Inertial Inertial p p

μ 0.5 μ 0.5

Intermediate Intermediate

0 0 0.45 0.5 0.55 0.6 0.65 0.45 0.5 0.55 0.6 0.65 ν ν

n 3 2 4 Fig. 1. Regime map for granular flows, constructed from data of 3D DEM simulations of homogenously sheared granular flow (a) k ¼ kn=rsd0g_ ¼ 2:5 10 and n 3 2 9 (b) k ¼ kn=rsd0g_ ¼ 10 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 23

3. For sufficiently low values of kn (high shear rates) and particle δ = 0.30 friction coefficient m , the quasi-static regime can completely p δ = 0.35 disappear as seen in Fig. 1(a). 0.1 δ = 0.50 δ There have been other attempts to represent granular rheology of = 0.70 monodisperse systems using similar regime maps. Tardos et al. Solid like (2003) presented a schematic of different regimes in powder flow ρ=δ based on the results obtained from their Couette cell experiments. 0.05 )

However, the effect of particle friction coefﬁcient was not incor- δ ,

ρ Liquid like porated in that map. Campbell (Campbell, 2002) proposed a F( regime map for different values of friction coefﬁcients, however the intermediate regime was not discussed in that work. 0 3.2. OP dynamics from homogeneous shear simulations

As noted earlier, the OP gives one characterization of the phase or state of the granular material. Aranson and Tsimring (2001) in their original work related the OP to the free energy density -0.05 function Fðr,dÞ, that they specified as -0.5 0 0.5 1 1.5 Z r ρ Fðr,dÞ¼ rðr1ÞðrdÞ dr, ð2Þ Fig. 2. Typical profile of the free energy density function Fðr,dÞ postulated by Aranson and Tsimring (2001). through the Ginzburg–Landau equation Dr @Fðr,dÞ ¼ D r2r : ð3Þ 0.8 Dt c @r Further they postulated that this free energy density function Fðr,dÞ (Eq. (2)) has two local minima at r ¼ 1 (completely solid- 0.7 like) and r ¼ 0 (completely fluidlike) to account for bistability near the solid–fluid transition. The relative stability of the two phases is controlled by the parameter d, which is the ratio of shear to normal stress. For small d the solidlike state is more 0.6 favorable, and vice versa. A typical profile of the free energy

density function postulated by Aranson and Tsimring (2001),is ρ shown in Fig. 2, for different values of d. While the OP cannot take any values lower than zero and higher than one, a scale which 0.5 goes below zero and beyond one has been used for the OP in Fig. 2 ν=0.60, Hookean model to clearly show that r ¼ 0 and r ¼ 1 are the two stable states of ν=0.60, Hertzian model the OP for this free energy density function. With the formulation ν=0.59, Hookean model 0.4 of the free energy density function Fðr,dÞ in Eq. (2), the solution of ν=0.59, Hertzian model the Ginzburg–Landau equation (Eq. (3)) always results in a steady ν=0.58, Hookean model state value of the OP which is either zero (completely fluidlike) or ν=0.58, Hertzian model one (completely solidlike), depending on the value of the para- 0.3 meter d. 100 200 300 400 500 . In order to understand the OP dynamics from DEM simula- γt tions, we plot the time evolution of the OP in Fig. 3 for three different values of the solid volume fraction and with two Fig. 3. Time evolution of the OP obtained from DEM simulations for mp ¼ 0:5, n k ¼ k = d3 _ 2 ¼ 105 and e¼0.7. different contact force models, i.e. Hookean and Hertzian n rs 0g (Silbert et al., 2001). Fig. 3 shows that the OP evolves with time and attains a steady state value that is neither zero (completely simulation for n ¼ 0:59, mp ¼ 0:5, and e¼0.7, whereas the circles fluidlike) nor one (completely solidlike), irrespective of the two correspond to unsteady evolution of the OP obtained from DEM contact force models used. This result reveals that there should be simulations for n ¼ 0:64, mp ¼ 1:0, and e¼0.7. These data points one more intermediate local minimum in the free energy density also reveal that the OP evolves with time and attains a steady function postulated by Aranson and Tsimring (2001) (see Eq. (2) state value that is not necessarily zero or one. We further study the behavior of the steady value of the OP for and Fig. 2)atr ¼ r3, which results in the OP attaining a steady value that is neither zero (completely fluidlike) nor one (com- different solid volume fractions, particle friction coefficients and pletely solidlike). To account for this third granular phase at shear rates. We note that for all these DEM simulation conditions the steady value of the OP is neither zero not unity, but r ¼ r3, the following modification to the form of the free energy density function is proposed: corresponds to the third stable phase r3 that is dependent on Z r solid volume fraction, particle friction coefficient and shear rate. n n F ðr,dÞ¼ rðr1Þðrr3ÞðrdÞðrd Þ dr: ð4Þ In Fig. 5(a) these steady values of the OP are plotted with solid volume fraction for three different values of the particle friction Fig. 4 shows typical profiles of the proposed free energy density coefficient. Fig. 5(a) shows that the OP is indeed a strong function n n n function F ðr,dÞ for d ¼ 0:28, r3 ¼ 0:60, d ¼ 0:85, where r3 and d of the particle friction coefficient. An increase of about 300% in the are obtained from DEM simulations. The triangles in Fig. 4 value of the OP is seen when the particle–particle coefficient correspond to unsteady evolution of the OP obtained from DEM of friction increases from 0.1 to 1.0 at the same solid volume 24 V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 fraction. This result is expected because at higher interparticle the OP with solid volume fraction as described in Fig. 5(a), but for friction the particles are prevented from sliding over each other, different values of shear rates. These plots (Fig. 5(a–d)) show that resulting in a greater fraction of solidlike contacts. A sudden jump the OP is most sensitive to the particle–particle friction coeffi- in the OP is seen near the maximum packing limit. This sudden cient, whereas shear rate has the least impact on OP values. increase in the OP is ascribed to the presence of strong force Fig. 5(a–d) reveals that the sudden jump in the OP value (near chains near the packing limit. Fig. 5(b–d) shows similar plots of the maximum packing limit) becomes increasingly sharp as the n 3 2 non-dimensional shear rate k ðkn=rsd0g_ Þ increases (going from Fig. 5(a) to (d)). Also as the particle friction coefficient increases 0.003 this sudden jump in the OP occurs at progressively lower values of δ δ ρ solid volume fraction. This finding is consistent with the results of =0.28, *=0.85, 3=0.60 ν μ =0.59, p=0.5, e=0.7, DEM Song et al. (2008), who studied the jamming of packed spheres ν μ =0.64, p=1.0, e=0.7, DEM through a phase diagram and showed that the minimum solid volume fraction required for jamming decreases with increase in the particle friction coefficient. It is also noteworthy that at n 0.002 the highest value of k (which corresponds to lowest shear rate, see Fig. 5(d)), the OP attains its limiting value of unity near the

) maximum packing limit, whereas the OP approaches its other δ ,

ρ limiting value of zero, when both solid volume fraction and ρ=δ

F*( friction coefficient tend towards zero. Solid like Neglecting the small variation of with shear rate kn,we ρ=δ* r3 0.001 propose the following fit for r3, the third stable value of the OP, as Liquid like ρ ρ = 3 a function of solid volume fraction n and friction coefficient mp using data shown in Fig. 5 b

( n a sinðbpnmpÞ 0onrn , r3 ¼ 2 2 n ð5Þ A logðBn mpÞþC expðn mpÞ n ononmax, 0 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 with a¼0.804, b¼0.678, A¼0.555, B¼6.769 and C¼0.685. In ρ n Eq. (5), n ¼ 0:53 and nmax is the solid volume fraction correspond- Fig. 4. Typical proﬁle of proposed free energy density function Fnðr,dÞ with third ing to the close-packed limit. In order to verify the new speciﬁca- intermediate local minima at r ¼ r3. tion of the free energy density function in Eq. (4), we solve the

1 1 μ μ p = 0.1 p = 0.1 μ μ p = 0.5 p = 0.5 μ μ 0.8 p = 1.0 0.8 p = 1.0

0.6 0.6 ρ ρ 0.4 0.4

0.2 0.2

0 0 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6 ν ν

1 1 μ μ p = 0.1 p = 0.1 μ μ p = 0.5 p = 0.5 0.8 μ 0.8 μ p = 1.0 p = 1.0

0.6 0.6 ρ ρ 0.4 0.4

0.2 0.2

0 0 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6 ν ν

n n n Fig. 5. The OP (at steady state) plotted with solid volume fraction (a) for k ¼ 2:5 104 and e¼0.7, (b) for k ¼ 105 and e¼0.7, (c) for k ¼ 107 and e¼0.7 and (d) for n k ¼ 109 and e¼0.7. V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 25

Ginzburg–Landau equation 3.5 1 N Dr @Fnðr,dÞ CN ¼ D r2r , ð6Þ R Dt c @r 3 xz 0.8 for a homogeneously sheared granular assembly with the speci- 2.5 fication of r3 in Eq. (4) given by the fit in Eq. (5). Fig. 6 shows that the steady solution of the Ginzburg–Landau equation (Eq. (6)) 0.6 with the new formulation of free energy density function Fnðr,dÞ 2 xz matches the steady OP values obtained from DEM within 5%, CN R N whereas the original form of the free energy density results in a 1.5 0.4 stable value of unity. This result validates the form of the new free n energy density function F . 1 Further we attempt to quantify this third stable phase of the granular material by investigating structural quantities such as 0.2 0.5 the average coordination number NCN, the fabric tensor R and pair correlation function g(r). These structural quantities are chosen because they are relevant to constitutive modeling of granular 0 0 rheology (Sun and Sundaresan, 2010). The average coordination 0 0.2 0.4 0.6 0.8 1 ρ number NCN that is defined as the average number of contacts per particle is a measure that is sensitive to the local particle Fig. 7. Variation of the average coordination number and fabric tensor with the configuration. It has been used to characterize the equilibrium order parameter for a homogeneously sheared granular flow. state in static packings (Silbert et al., 2002). The fabric tensor R describes the anisotropy of the contact distribution in granular media (Bathurst and Rothenburg, 1990; Cowin, 2004). Compo- ρ ≈ 1.0 nents of this tensor can be calculated on the basis of particle 4 contact information using 3 X ρ 1 =0.64 g(r) 2 R ¼ n n , ð7Þ 4 3 ij N i j c c A V 1 0 where Rij is a symmetric second-order fabric tensor, Nc is the 3 01234 number of contacts, ni and nj are the unit vectors corresponding to r/d0 the contact vector from particle center to point of contact. Fig. 7 g(r) 2 shows the variation of average coordination number NCN and 4 ρ ≈ 0.0 fabric tensor R (the xz component) with the order parameter r. 1 3 While the average coordination number is sensitive to change in

the OP, there is no appreciable change in the fabric tensor for the g(r) 2 same change in the OP values. This result indicates that the 0 1 average coordination number is more sensitive to this phase 01234 r/d 0 change as indicated by stable OP values than the fabric tensor. 0 01234 Another quantity that gives insight into the microstructure of r/d granular media is the pair correlation function or the radial 0

Fig. 8. (a) The pair correlation function corresponding to the third stable phase 1 (r ¼ 0:64), inset shows the corresponding snapshot of internal structure for the ν μ δ δ 3 =0.60, p=0.5, *=0.85, =0.28 solidlike contacts at this third stable phase and (b) the pair correlation function corresponding to r 1:0 (completely solidlike phase) and r 0:0 (completely 0.9 ﬂuidlike phase), respectively. Inset shows the corresponding snapshots of internal structure for the solidlike contacts for these two limiting phases. 0.8 distribution function g(r)(Silbert et al., 2002; Donev et al., 0.7 2005). Fig. 8(a) shows the pair correlation function corresponding to the third stable granular phase (r3 ¼ 0:64). The ﬁrst peak (at r ¼ d0) corresponds to the high probability of having a neighbor in

ρ 0.6 contact. We also observed a secondary peak at r ¼ 2d0, and this secondary peak in g(r) diminishes with increase in the particle

0.5 friction coefficient mp (result not shown here). This behavior of g(r) with particle friction coefficient has been previously observed 0.4 by Silbert et al. (2002) in their numerical simulations. Inset of Old G-L equation (Eq.2) Fig. 8(a) shows an instantaneous realization of the internal New G-L equation (Eq.5) structure (shown in a cube of 4d , note that the pair correlation 0.3 0 DEM function has reached its uniform value of one by r ¼ 4d0) for the solidlike contacts corresponding to this third stable granular 0.2 phase. Fig. 8(b) shows the pair correlation function corresponding 0 100 200. 300 400 to the OP value of 1.0 (completely solidlike phase) and 0.0 γt (completely fluidlike phase), respectively. Inset shows an instan- Fig. 6. Solution of Ginzburg–Landau equation (Eq. (6)) with new formulation of taneous realization of the corresponding internal structure of the free energy density function Fn (Eq. (4)). solidlike contacts for these two limiting phases. The lower panel 26 V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 of Fig. 8(b) shows that in the completely fluidlike state (r 0), the 1 ss ¼ s ð1aÞd þð1bÞb g½ðb2Þ ðb2Þ d , ð9Þ third peak in the pair correlation function disappears, whereas it ij 0 ij ij ij 3 ll ij is seen in the other two states. The instantaneous structures where s ¼ s =3 is the scale of stress (summation is implied over confirm the behavior of the average coordination number with 0 ii repeated indices). In Eqs. (8) and (9), b is the normalized, the OP: namely, the average coordination number is clearly a ij symmetric, anisotropy tensor defined as strong function of the steady OP value as evidenced by the sij difference in number of contacting particles in the realization. bij ¼ dij: ð10Þ In a previous study, Jaeger and Nagel (1996) described different s0 regimes of granular flow (i.e. solid, liquid and gas) where they The components of the second-order tensor b2 are defined as point out the lack of scale separation in granular liquids, the tight ðb2Þ ¼ b b , ð11Þ coupling between scales in granular solids, and the importance of ij ik kj 2 inelasticity and energy loss in granular gases. They relate the and ðb Þll is a scalar that is defined as different regimes to the interaction between grains and force 2 networks. The characterization of granular flow in the current ðb Þll ¼ blkbkl: ð12Þ work using the OP and other structural quantities follows the same The significance of the model coefficients, a, b and g is as follows. ideas by exploring the coupling between mesoscale structure and The coefficient g represents the degree of nonlinearity in the macroscale rheology. We have shown that the connection between model, so g ¼ 0 results in a linear model. If we consider a linear structure and grain interaction is provided by the OP dependence model, then a and b represent the respective weight factors that on the particle properties such as interparticle friction. Forterre and multiply the isotropic and deviatoric parts of the total granular Pouliquen (2008) characterized regimes of granular flow using the stress, to obtain the fluidlike stress. If a ¼ b in the linear model, contact time between grains. This idea is closely related to the then the fluidlike stress is coaxial with the total granular stress current work, where the contact time between grains forms an (and then so is the solidlike part). These model coefficients a and integral part of the definition of the OP. b are functions of the order parameter r, which are specified in It should also be noted that the third steady state observed in Gao et al. (2005) as the current study is a combination of ‘fluidlike’ and ‘solidlike’ a ¼ð1rÞ1:8, ð13Þ states. It is a stable state between completely fluidlike and completely solidlike states. Any value of the OP between zero b ¼ð1rÞ2:5: ð14Þ and one represent a state where some of the contacts are ’solidlike’ and the remaining are ‘fluidlike’. However, in the original These model coefficients were obtained using DEM data from 2D definition of the OP by Volfson et al. (2003b), only r ¼ 0 qualifies inhomogeneous Couette flow with wall boundary conditions. as completely fluidlike and r ¼ 1 qualifies as completely solidlike. It should be noted that in 2D the term in g is redundant, and The pair correlation function corresponding to the third stable there are only two coefficients a and b, because the characteristic phase (shown in Fig. 8(a)) suggests that the new granular phase is equation for the stress tensor is a quadratic (instead of a cubic for indeed ‘liquidlike’. With this better understanding of the OP the 3D case), and there are only two invariants: the sum and dynamics in hand, the next step is to explore and improve the product of the two principal values of the stress tensor. objective OP model (Gao et al., 2005) in order to make it tractable. A complete specification of the objective OP model requires data from 3D DEM simulations in order to calculate all three coefficients (a, b , and g), and in this case the nonlinear term may not be 4. Order parameter model description and refinement zero. We obtained model coefficients from 3D DEM data of homogeneously sheared granular flow to specify a complete set 4.1. OP model description of coefficients for the objective OP model.

The original order parameter model was developed by 4.2. Refinement of the OP model Aranson, Tsimring and Volfson in a series of papers (Aranson and Tsimring, 2001; Volfson et al., 2003a,b). The fundamental The model coefficients a, b and g for the objective OP model f premise of this model is that one can define an OP in granular (Gao et al., 2005) that best fit the fluidlike stress tensor sij relation flows similar to that used in the Landau theory of phase transi- given by Eq. (8) are computed using 3D DEM data for the total tions (Landau and Lifshitz, 1980). This original OP model (Volfson granular stress sij (from which s0 and bij are computed) and f et al., 2003a) decomposes the total granular stress tensor into fluidlike stress sij from homogeneous shear flow simulations. In a ‘‘solidlike’’ and ‘‘fluidlike’’ contributions based on the OP. The OP 3D granular flow there are six independent non-zero components model gives expressions for the ‘‘solidlike’’ and ‘‘fluidlike’’ stress for the fluidlike stress tensor (assuming the stress tensor is tensors that are functions of the order parameter r and the total symmetric). As there are three unknown model coefficients, a, b and g and six equations for the fluidlike stress, one can only solve granular stress tensor sij. In the original Aranson and Tsimring (2001) OP theory, the fluidlike stress was modeled using a the system of equations using a least-squares method. We solve constitutive relation from the KTGF (kinetic theory of granular this set of equations over a range of flow conditions (simulation flows), and the total granular stress and solidlike contribution parameters for these cases are summarized in Table 1) for which were obtained through relations that are coordinate-system we performed DEM simulations, and the corresponding OP values dependent. for each of these flow conditions correspond to the abscissas of This original OP model by Volfson et al. (2003a) is generalized the data points in Fig. 9(a). The ordinate of the data points in to an objective form by Gao et al. (2005). The objective expres- Fig. 9(a) corresponds to the least-squares solutions for the model f s coefficients obtained using this method. Fig. 9(a) shows the sions for sij, the ‘‘fluidlike’’ contribution, and sij, the ‘‘solidlike’’ contribution to the total granular stress, which are coordinate variation of model coefficients a, b and g with the OP, and the system independent, are (Gao et al., 2005) lines are a polynomial fit to the data. The coefficients a and b are very nearly equal, indicating that the fluidlike stress is nearly f 2 1 2 coaxial with the total granular stress, although not exactly so. s ¼ s0 adij þbbij þg½ðb Þij ðb Þlldij , ð8Þ ij 3 Note that the magnitude of the third model coefficient g remains V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 27

1 1 α α β β 0.8 γ 0.8

0.6 0.6 γ , β β , , 0.4 α α 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ρ ρ

Fig. 9. The objective OP model coefﬁcients as a function of the order parameter (a) for a nonlinear objective model and (b) for a linear objective model.

0.15 objective models is less than 11%. The percentage error incurred ∧ε Non Linear in the linear model is approximately same as the error incurred in ∧ε Linear the nonlinear model. Therefore, a linear version of the objective OP model with new model coefficients extracted from 3D DEM 0.1 data of homogeneous shear flow is now proposed. This linear version of the objective OP model is referred to as the refined order parameter (ROP) model. In the following section we present < ε the complete specification of the proposed ROP model. 0.05

5. Speciﬁcation of the ROP model

0 The model equations for the proposed linear ROP model are f 0 0.2 0.4 0.6 0.8 1 sij ¼ s0fadij þbbijg, ð17Þ ρ s sij ¼ s0 ð1aÞdij þð1bÞbij : ð18Þ Fig. 10. Error in the total granular stress objective models as a function of the OP for both linear and nonlinear models. The model coefficients (a and b) of the linear ROP model are specified as polynomial fits to the data in Fig. 9(b) with a ¼ aþbrþcr2 þdr3, ð19Þ close to zero for the complete range of OP values. Since the model where a¼1.0, b ¼1:23, c ¼0:31 and d¼0.54, and coefficient g determines the magnitude of the nonlinear terms in b ¼ AþBrþCr2 þDr3, ð20Þ Eq. (8), this indicates the possibility of forming a linear model. with A¼1.0, B ¼1:69, C¼0.76 and D ¼0:07. Specification for The model coefficients a and b corresponding to a linear the order parameter is taken from its steady values obtained objective model are computed by dropping the term containing r from DEM simulations (as shown in Fig. 5(a–d)). The advantage of g in Eq. (8), and performing the least-squares solution of Eq. (15) the linear ROP model is that now the total granular stress can be f sij ¼ s0 adij þbbij : ð15Þ inverted from the solidlike and fluidlike stress relations, as follows: Fig. 9(b) shows model coefficients a and b with the OP for the "# f proposed linear objective model. At r ¼ 0, the model coefficients s0 sij sij ¼ þdijðbaÞ , ð21Þ a and b are equal to unity, which indicates that the total granular b s0 stress is solely due to the fluidlike contribution. At r ¼ 1, the f model coefficients a and b are zero, which indicates that the total where s0 ¼ sii=ð3aÞ. One should note that Eq. (21) diverges as the granular stress is due to only the solidlike contribution. The order parameter goes to unity (its solidlike limit), reflecting the error incurred in both (nonlinear and linear) objective models fact that the ROP-KT model for the total stress does not contain is quantified by the vector norm of the relative error in the any information about the solidlike stress. least-squares solution Previously Aranson and Tsimring (2001) showed that a constitutive relation from the kinetic theory for the fluidlike stress gave a JKxyJ E^ ¼ 2 , ð16Þ good match for the kinematic variables in dense chute flow. There- JyJ 2 fore, we follow Aranson and Tsimring (2001) and model the ‘‘fluid- where x is the solution vector for the model coefficients, Kx is the like’’ stress using a constitutive relation from the kinetic theory of total granular stress components given by the OP model and y is granular flows (KTGF) even in the dense regime. Once the fluidlike the total granular stress from DEM. contribution of the total granular stress is known, the total granular

The error incurred in terms of this vector norm is shown in stress tensor sij can be expressed in terms of the ‘‘fluidlike’’ stress and Fig. 10 for the complete range of OP, for both nonlinear and linear the ROP model coefficients (a and b) using Eq. (21). The kinetic theory objective models. As Fig. 10 shows, the error incurred in both the closures are taken from Lun et al. (1984) to compute the fluidlike 28 V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 contribution of the total granular stress tensor, which are Table 2 ^ _ 2 2þa Comparison of granular temperature T ¼ T=ðd0gÞ obtained from Eq. (29) and f k DEM. The last column shows the corresponding OP values from DEM data. sij ¼½rsnð1þ4Zkng0ÞTZkmbr udij 3 n 3 2 5 Simulation parameters: m ¼ 0:5, e¼0.7 and k ¼ kn=r d g_ ¼ 10 . 2m 8 8 6 p s 0 1þ nZkg0 1þ Zk 3Zk2 g0 þ Zkmb Sij, g0Zkð2ZkÞ 5 5 5 Solid volume fraction Algebraic equation DEM OP ð22Þ 0.45 0.529 0.603 0.412 0.53 0.514 0.570 0.416 1 @u @u 1 @u i j i 0.58 0.508 0.553 0.542 Sij ¼ þ , ð23Þ 2 @xj @xi 3 @xi 0.60 0.505 0.550 0.740 0.62 0.503 0.528 0.784 5r d ðpTÞ1=2 m ¼ s 0 , ð24Þ 96

256mn2g Eq. (21). The next step is to assess the performance of the m ¼ 0 , ð25Þ b 5p proposed ROP model, which is presented in the following section.

ð1þeÞ Z ¼ , ð26Þ k 2 6. Assessment of the ROP model for homogeneous shear ﬂows

1 ð Þ¼ ð Þ The ROP model with the constitutive relation for the fluidlike g0 n 1=3 , 27 1ðn=nmax Þ stress contribution obtained from the kinetic theory of granular flows is denoted the ROP-KT model. The ROP-KT model’s predic- a ¼ 1:3, ð28Þ k tions for the total granular stress are compared with those from f where sij is the fluidlike part of the stress tensor, rs is the density of DEM simulations of homogeneously sheared granular flow in the solid particle, n is the solid volume fraction, T is the granular different regimes that are characterized by a regime map in temperature, u is the mean velocity vector, and Sij is the strain rate Fig. 1(a). The validity of the kinetic theory closure for the fluidlike tensor. For inhomogeneous granular flows the granular temperature T f stress is also assessed in different regimes by comparing sij with is obtained as the solution to a transport equation (Lun et al., 1984). the corresponding fluidlike stress tensor obtained from DEM data. However, for homogeneous shear flows the granular temperature can be obtained through a simple algebraic relation. 6.1. Inertial regime (solid volume fraction of 0.45)

5.1. Homogeneous shear case Fig. 11(a) shows a logarithmic plot of the elastic scaling of the shear component of the total granular stress as a function of shear For a steady homogeneously sheared granular flow the gran- rate for a solid volume fraction of 0.45. In this scaling, stress ular temperature results from a balance of production and values in the inertial regime where spg_ 2 corresponds to a line dissipation terms. This balance results in a algebraic equation with slope 1 (in Fig. 11(a) the slope of the line is denoted by m). for the granular temperature T (Syamlal et al., 1993) The shear component of the total granular stress obtained from 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi92 the ROP-KT model is shown by blank diamonds, whereas the < 2 2 2 2 = K1nSii þ K1ðSiiÞ n þ4K4n½K2ðSiiÞ þ2K3ðSijSijÞ filled squares show the data from DEM simulations. The total T ¼ : ; , ð29Þ 2nK4 granular stress predicted using ROP-KT model closely follows the data obtained from the DEM simulations. The ROP-KT model is where constants K1, K2, K3 and K4 are able to predict the total granular stress in the inertial regime within 5%. The total granular stress obtained from both the model K1 ¼ 2ð1þeÞrsg0, ð30Þ and DEM follows the inertial scaling ðspg_ 2Þ of stress with applied pffiffiffiffi 2 shear rate. Fig. 11(b) shows that the fluidlike contribution to the K ¼ 4d r ð1þeÞng =ð3 pÞ K , ð31Þ 2 0 s 0 3 3 total granular stress obtained using kinetic theory closely follows pffiffiffiffi the fluidlike stress contribution obtained from DEM simulations. ð þ Þ d0rs p 8ng0 1ffiffiffiffi e Both the DEM data and predictions obtained from kinetic theory K3 ¼ ½0:5ð3eþ1Þþ0:4ð1þeÞð3e1Þng0þ p , 2 3ð3eÞ 5 p follow inertial scaling ðspg_ 2Þ with the shear rate. This type of ð32Þ scaling of the shear stress with the applied shear rate has been previously reported by Campbell (2002) in the inertial regime. 2 12ð1e Þrsg0 K4 ¼ pffiffiffiffi : ð33Þ d0 p 6.2. Near transitional regime (solid volume fraction of 0.53)

We extract the granular temperature using Eq. (29), and In order to quantify the performance of the ROP-KT model near compare those values to the granular temperature obtained from the transition from inertial to intermediate regime, we considered DEM simulations in Table 2. The interparticle friction coefficient a higher solid volume fraction of 0.53. Fig. 12(a) compares the used for these simulations is 0.5, with a coefficient of restitution total granular stress predicted by the ROP-KT model with data n 3 2 of 0.7. The non-dimensional shear rate k ðkn=rsd0g_ Þ is set to be from DEM simulations. In the near transitional regime (n ¼ 0:53) 105 for this comparison. Table 2 shows that the maximum the ROP-KT model predicts the total granular stress well (see difference in the steady state granular temperature obtained from Fig. 12(a)), with an maximum error of 5% when compared with the algebraic equation (Eq. (29)) and DEM simulation is less than the DEM data. At this volume fraction there are multiparticle 14% over the range of solid volume fractions considered. contacts as indicated by the mean coordination number value of With the specification of the fluidlike contribution to the total 1.6 obtained from DEM simulations (result not shown here). f granular stress sij and the model coefficients (a and b), one can In Fig. 12(b) the variation of the fluidlike contribution of the solve the ROP model to obtain the total granular stress sij using stress obtained from kinetic theory as well as from the DEM data V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 29

Fig. 11. (a) The total granular stress as a function of shear rate kn and (b) the ﬂuidlike stress contribution to the total granular stress as a function of shear rate kn.

Simulation parameters: n ¼ 0:45, mp ¼ 0:5, e¼0.7.

Fig. 12. (a) The total granular stress as a function of shear rate kn and (b) the ﬂuidlike stress contribution to the total granular stress as a function of shear rate kn.

Simulation parameters: n ¼ 0:53, mp ¼ 0:5, e¼0.7. is shown. This plot shows that the kinetic theory closure performs 6.4. Summary of ROP model performance surprisingly well in predicting the fluidlike stress contribution when compared with the DEM data. Based on this assessment study, we conclude that the ROP-KT model has the capability to accurately predict the total granular stress up to a solid volume fraction of 0.53. As the solid volume fraction 6.3. Deep intermediate regime (solid volume fraction of 0.62) exceeds 0.53 the flow transitions to the intermediate regime and the ROP-KT model fails to capture the correct trend of shear stress with To assess the performance of the ROP-KT model in the deep shear rate. The differences in the magnitude of the stress prediction in intermediate regime, we selected a case with solid volume the intermediate regime is attributed to the fact that the ROP-KT fraction of 0.62 and interparticle friction coefficient of 0.1. At this model assumes that the fluidlike stress contribution follows the solid volume fraction the ROP-KT model does not predict either kinetic theory closure even in the dense regime. However, this the magnitude or the scaling of the total granular stress assumption does not hold in the deep intermediate regime where (Fig. 13(a)) or the fluidlike stress (Fig. 13(b)) correctly. The both collision and frictional interactions between the particles are fluidlike contribution obtained from DEM data clearly shows important. Although the ROP model coefficients a and b include a the intermediate scaling of the stress ðspg_ n,n ¼2m ¼ 0:66Þ dependence on shear rate and particle friction coefficient through the with shear rate, whereas the kinetic theory closure necessarily OP, this dependence is not able to accurately predict the stress–strain follows the inertial scaling of the stress ðspg_ 2Þ with applied scaling in the deep intermediate regime of flow. shear rate. Although the ROP-KT model decomposes the total granular stress into solidlike and fluidlike parts, unlike other models (Savage, 1998; Johnson and Jackson, 1987; Srivastava 7. Performance evaluation of different constitutive models in and Sundaresan, 2003) the weighting factors for these contribu- the intermediate regime tions depend on the shear rate and particle friction coefficient through the OP (see Eqs. (19) and (20)). Note that as a conse- The performance of different constitutive models is assessed in the quence the total granular stress predicted by the ROP-KT model intermediate regime of granular flow. In Fig. 14, the shear component actually shows an intermediate scaling (spg_ 2,n ¼2m ¼ 1:48) of the total granular stress is plotted with shear rate for a solid with shear rate, even though the fluidlike stress follows a inertial volume fraction of 0.62 with interparticle friction coefficient of 0.1 scaling. (this combination of solid volume fraction and particle friction 30 V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34

Fig. 13. (a) The total granular stress as a function of shear rate and (b) ﬂuidlike stress contribution to the total granular stress as a function of shear rate. Simulation parameters: n ¼ 0:62, mp ¼ 0:1, e¼0.7. coefﬁcient corresponds to the intermediate regime). The different 10-1 10-1 constitutive models assessed are listed below: 10-2 10-2 1. Losert (2000): Losert et al. (2000) proposed a constitutive model with density-dependent viscosity. The shear stress in 10-3 10-3 this model is given as 10-4 10-4 sxy ¼ Zg_, ð34Þ n n /k /k 0 -5 -5 where viscosity is a function of the density as follows: 0 d 10 10 zx Pd

1:75 σ Z ¼ðnmaxnÞ : ð35Þ 10-6 10-6 Fig. 14 shows that the shear stress predicted using this model DEM fails to capture the correct scaling of shear stress with shear 10-7 Losert (2000) 10-7 rate in the intermediate regime. It should be noted that this ROP-KT -8 ROP-FSM -8 model (Losert et al., 2000) was proposed based on the experi- 10 ROP-DEM 10 mental data obtained from shear flow of granular material in a Π Pressure-DEM Couette geometry. 10-9 10-9 2. ROP-KT: This is the constitutive model proposed in present 104 105 106 107 108 109 1010 work, where the ROP model is coupled with the kinetic theory ρ 3 γ2 k*=kn/ sd0 of granular flows (KTGF) (Lun et al., 1984) for the fluidlike stress contribution of the total granular stress. As discussed Fig. 14. Shear component of the total granular stress (different symbols represent earlier, this model fails to capture the correct trends of shear result obtained with different constitutive models) plotted with shear rate. ¼ ¼ ¼ stress with shear rate in the deep intermediate regime. Simulation parameters: n 0:62, mp 0:1, e 0.7. 3. ROP-FSM: A frictional stress model (FSM) is proposed by Srivastava and Sundaresan (2003) for the frictional part of with the ROP model for the solidlike stress contribution predicts the total granular stress. The FSM model is used to compute stresses that are independent of the shear rate (a behavior s the solidlike stress contribution sij and then the ROP model is characteristic of the quasi-static regime). However, the data solved to obtain"# the total granular stress as follows: obtained from the DEM simulations show a dependency of shear s s0 sij stress on shear rate in this regime. The ROP-FSM results show that sij ¼ þdijðbþaÞ , ð36Þ ð1bÞ s0 it is not simply a matter of modeling the fluidlike or solidlike parts s of the total granular stress. Rather, what is lacking is a funda- where s0 ¼ sii =3ð1aÞ. One should note that Eq. (36) diverges as the order parameter goes to zero (its fluidlike limit), which reflects mental description of the dependence of stress on strain rate in that the ROP-FSM model for the total stress does not contain any theintermediateregime. information about the fluidlike stress. This frictional stress model 4. ROP-DEM: In this constitutive model, the fluidlike contribution is based on the critical state theory of soil mechanics. At the to the total granular stress is supplied from the DEM data and critical state the granular assembly deforms without any volume then the ROP model (see Eq. (21)) is solved to compute the change and the frictional contribution of the stress is given by total granular stress. Fig. 14 shows that the ROP-DEM model is capable of predicting the total granular stress accurately rfric pffiffiffi S ¼ I 2 sin f pffiffiffiffiffiffiffiffiffiffi , ð37Þ (within 5%) even in the intermediate regime, provided the p ðnÞ c S : S fluidlike (or the solidlike) stress contribution is taken from where the form for pcðnÞ (critical state pressure) is taken from the DEM data. This result shows that the limitation is not in Johnson and Jackson (1987) the objective ROP model concept, but in the constitutive model ( r ðnnmin Þ for the fluidlike (or solidlike) stress contributions in the F s if n4nmin, ðnmax nÞ pcðnÞ¼ : ð38Þ intermediate regime. 0ifnrnmin where F, r and s are constants, taken from Srivastava and Fig. 14 also shows the variation of the isotropic pressure (right Sundaresan (2003).AsshowninFig. 14, this model when coupled vertical axis) obtained from the DEM simulations with shear rate V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 31 for a homogeneously sheared granular assembly. This plot shows These contributions to the total granular stress are plotted with that the pressure follows the same scaling with shear rate as the shear rate kn,inFig. 16(a) and (b), respectively. It is noteworthy shear stress in the intermediate regime. Hence, it is important for that both the solidlike and fluidlike stress follow intermediate n a constitutive model to capture the correct behavior of the scaling of the stress (spg_ ,nsolidlike ¼ 0:20,nfluidlike ¼ 0:44), because pressure with strain rate in the intermediate regime. both of them carry portions of contact stress as well; the only difference is whether the contacts are enduring or not.

8. Decomposition of the total granular stress from DEM

In order to better understand the scaling of the granular stress 9. Conclusions in the intermediate regime, the total granular stress obtained from DEM simulations is first decomposed into contact (virial) The characterization of granular phase transition using an order and streaming (dynamic) contributions. The contact contribution parameter (OP) is explored. DEM simulations reveal that indeed the due to particle contacts in a domain of volume V is given by OP is capable of capturing the granular phase transition from solidlike to fluidlike behavior through its dependence on particle 1 XN X 1 rcontact ¼ rðiÞðjÞ fðiÞðjÞ, ð39Þ properties (such as particle friction coefficient) and flow properties V 2 i j,j a i (such as shear rate). Through these DEM simulations we also discovered a third stable granular phase that is neither completely where rðiÞðjÞ is the vector pointing from the center of particle j to fluidlike nor completely solidlike. Hence, a modification to the form the center of particle i, fðiÞðjÞ is the contact force acting on particle i of the free energy density function is proposed to account for this by particle j, and denotes a dyadic product. The streaming third stable granular phase. The refined OP model (ROP model) contribution arises from momentum flux and is given as enables a linear implementation of theobjectiveform(coordinate 1 XN system independent) of the OP model (Gao et al., 2005)througha rstreaming ¼ mðiÞv0ðiÞ v0ðiÞ, ð40Þ V 0 simplification that allows inversion of total granular stress from i solidlike and fluidlike stress relations. 0 where m0 is the mass of a particle, v is the fluctuating velocity A regime map constructed from DEM data of homogeneously and i is a particle index. In Fig. 15(a) the contact contribution to sheared granular flow reveals that the intermediate regime the total granular stress is plotted with shear rate for a range expands as the particle friction coefficient decreases. The of solid volume fractions. Fig. 15(a) shows that the contact ROP-KT model (fluidlike contribution obtained from the kinetic contribution of the total granular stress follows the same scaling theory) model accurately predicts the total granular stress (to ðspg_ n,n ¼2m ¼ 0:28Þ with shear rate as the total granular within 5%) up to a solid volume fraction of 0.53. Beyond a solid stress in the intermediate regime (data points corresponding volume fraction of 0.53 the flow transitions to the intermediate to the intermediate regime are shown with blank squares). regime. Comparison of the total granular stress predicted by However, the streaming contribution (shown in Fig. 15(b)) of different constitutive models with DEM data for homogeneous the total granular stress in the intermediate regime still follows shear flow in the intermediate regime reveals that none of the the inertial scaling of the stress ðspg_ 2Þ with the shear rate. models captures the correct trends of shear stress with shear rate. As expected, at this high value of the solid volume fraction the The assumption in the ROP-KT model that the fluidlike contribu- contact part of the stress contributes more than 95% to the total tion is adequately modeled by KTGF clearly breaks down in the granular stress. Hence it is critical for the performance of any intermediate regime, although it appears to be adequate even up constitutive model in the dense regime to accurately capture the to a solid volume fraction of 0.53 where multiparticle interactions behavior of the contact stress. are present in the flow dynamics. However, modeling the solidlike In order to evaluate the ROP model directly, the total granular stress using the ROP-FSM (Srivastava and Sundaresan, 2003) stress obtained from the DEM simulations is decomposed into formulation does not result in accurate rheological prediction solidlike (stress arise from solidlike contacts) and fluidlike (stress either. The models due to Losert (Losert et al., 2000) also do not arise from fluidlike contacts plus streaming stress) contributions. reproduce intermediate regime behavior.

Fig. 15. (a) Contact (virial) contribution of the total granular stress as a function of shear rate and (b) streaming (dynamic) contribution of the total granular stress as a function of shear rate. Simulation parameters: mp ¼ 0:1, e¼0.7. The data points corresponding to the intermediate regime are shown with blank square symbols. 32 V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34

Fig. 16. (a) Solidlike contribution of the total granular stress as a function of shear rate and (b) ﬂuidlike contribution of the total granular stress as a function of shear rate.

Simulation parameters: mp ¼ 0:1, e¼0.7. The data points corresponding to the intermediate regime are shown with blank square symbols.

Decomposition of the DEM granular stress into contact and local tangent plane of contact. Normal and tangential forces streaming parts reveals that the contact (virial) contribution to acting on particle i are the stress dominates (495%), and follows the same scaling as the Fn ¼ f ðd =dÞðknd n g m vn Þ, ðA:5Þ total granular stress in the intermediate regime. However, the ij ij ij ij n eff ij streaming contribution always follows the inertial scaling even in Ft ¼ f ðd =dÞðktut g m vt Þ, ðA:6Þ the intermediate regime. The decomposition of the granular stress ij ij ij t eff ij obtained from DEM into solidlike and fluidlike contributions where kn,t and gn,t are the spring stiffness and viscoelastic (based on the OP) reveals that both these follow the same scaling constants, respectively, and meff ¼ mimj=ðmi þmjÞ is the reduced as the total granular stress in the intermediate regime. This mass of spheres with masses mi and mj. The corresponding indicates that an entirely new model for the contact stress is contact force on particle j is simply given by Newton’s third needed in the intermediate regime. law, i.e., F ¼F . The function f ðd =dÞ¼1 is for the linear spring- ji ij qffiffiffiffiffiffiffiffiffiffiij

dashpot model, and f ðdij=dÞ¼ dij=d is for Hertzian contacts with Acknowledgments viscoelastic damping between spheres. Static friction is implemented by keeping track of the elastic The authors wish to acknowledge the ﬁnancial support from shear displacement throughout the lifetime of a contact. The US department of Energy National Energy Technology Laboratory static yield criterion, characterized by a local particle friction

(grant DE-FG26-07NT43070), administered under advanced coal coefficient m, is modeled by truncating the magnitude of utij as 9 9 9 9 research at US colleges and universities. The authors would also necessary to satisfy Ftij o mFnij . Thus, the contact surfaces are 9 9 9 9 like to thank Dr. Jin Sun for assistance with the simulation setup treated as ‘‘sticking’’ when Ftij o mFnij , and as ‘‘slipping’’ when and Dr. Sankaran Sundaresan for useful comments on a draft of the yield criterion is satisfied. this paper. The amount of energy lost in collisions is characterized by the value of the coefficient of restitution, which is defined as the negative ratio of the particle normal velocity after collision to the Appendix A. Contact model description velocity before collision. For the linear spring-dashpot model, the coefficient of normal restitution and contact time can be analy- For two contacting particle {i,j}, with radii {ai,aj} at positions tically obtained: {ri, rj}, with velocities {vi, vj} and angular velocities fxi, xjg, the en ¼ expðgntc=2Þ, ðA:7Þ normal compression dij, relative normal velocity vnij , and relative tangential velocity vtij are (Silbert et al., 2001) where the contact time tc is given by dij ¼ d0rij, ðA:1Þ 2 1=2 tc ¼ pðkn=meff gn=4Þ : ðA:8Þ The value of the spring constant should be large enough to avoid vnij ¼ðvij nijÞnij, ðA:2Þ particle interpenetration, yet not so large as to require an unreasonably small simulation time step Dt, since an accurate vtij ¼ vijvnij ðaixi þajxjÞnij, ðA:3Þ simulation typically requires Dt t =50 (Campbell, 2002). After 9 9 c where d0 ¼ ai þaj, rij ¼ rirj, nij ¼ rij=rij, with rij ¼ rij and the contact force is calculated, the equations of motion, which are vij ¼ vivj. Note that there is no sum over repeated indices. The ordinary differential equations, can be numerically integrated to rate of change of the elastic tangential displacement u , set to tij get the particle trajectories. zero at the initiation of a contact is, dut ðut vijÞrij ij ¼ v ij : ðA:4Þ dt tij 2 Appendix B. Verification of the order parameter extraction rij The last term in Eq. (A.4) arises from rigid body rotation In order to verify our OP calculations with previously published around the contact point and ensures that utij always lies in the results of Volfson et al. (2003a), we performed inhomogeneous V. Vidyapati, S. Subramaniam / Chemical Engineering Science 72 (2012) 20–34 33

0.8 transition from solidlike to fluidlike behavior. A contour plot of the OP for an inhomogeneous wall-bounded shear flow is shown in Fig. B2. The OP is minimum near the wall and maximum near 0.6 the centerline. This near-wall behavior is justified because the granular material will behave more like a liquid near the moving walls, than near the centerline.

ν 0.4

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