Chemical Engineering Science 72 (2012) 20–34 Contents lists available at SciVerse ScienceDirect Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces 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.