First normal stress difference and crystallization in a dense sheared granular fluid

Abstract

In the last decade, a lot of research activity took place to unveil the properties of granular materials 1,2 , primarily because of their industrial importance, but also due to their fascinating properties. This has unraveled many interesting and so far unresolved phenomena (for example, clustering, size-segregation, avalanches, the coexistence of gas, liquid and solid, etc.). Under highly excited conditions, granular materials behave as a fluid, with prominent non-Newtonian properties, like the normal stress differences 3 . While the normal stress differences are of infinitesimal magnitudes in a simple fluid (e.g. air and water), they can be of the order of its isotropic pressure in a dilute granular gas 4 . From the modelling viewpoint, the presence of such large normal-stress differences readily calls for higher-order constitutive models 5,6 even at the minimal level.Studying the non-Newtonian behaviour is itself an important issue, since the normal stresses are known to be the progenitors of many interesting and unique flow-features (e.g. rod-climbing or Weissenberg-effect, die-swelling, secondary flows, etc. 7 ) in non-Newtonian fluids. Also, normal stresses can support additional instability modes (for example, in polymeric fluids and suspensions 7−10 , which might, in turn, explain some flow-features of granular fluids. For example, particle-clustering 11−13 has recently been explained from the instability-viewpoint using the standard Newtonian model for the stress tensor 12,14,15 The kinetic theory of Jenkins & Richman 16 first showed that the anisotropy in the second moment of the fluctuation velocities, due to the inelasticity of particle collisions, is responsible for such normal stress behaviour. They predicted that the first normal stress difference (defined as N 1 = (Π xx − Π yy )/p, where Π xx and Π yy are the streamwise and the transverse components of the stress deviator, respectively, and p is the isotropic pressure, see section IIB) is maximum in the dilute limit, decreases in magnitude with density, and eventually approaches zero in the dense limit. Goldhirsch & Sela 4 later showed that the normal stress differences appear only at the Burnett-order-description of the Chapman-Enskog expansion of the Boltzmann equation. Their work has clearly established that the origin of this effect (in the dilute limit) is universal in both atomic and granular fluids, with inelasticity playing the role of a magnifier and thus making it a sizeable effect in granular fluids. While the source of the normal stress differences in the dilute limit has been elucidated both theoretically and by simulation, its dense counterpart has not received similar attention so far. This is an important limit since the onset of dilatancy (volume expansion due to shear 17,18 ), crystallization, etc. occur in the dense regime, which in turn would influence the normal stress differences.Previous hard-sphere simulations 19,3,11 did look at the normal stress differences, but they did not probe the dense limit in a systematic way. The...

Shear-induced crystallization of a dense rapid granular flow: Hydrodynamics beyond the melting point

“…Though crystallization under shearing is well documented in MD simulations [7,8], a recent experiment [23] showed that, under certain conditions, shearing can lead to disorder.…”

“…It was investigated experimentally (usually for slow flows) by many groups [4,5,6]. The crystallization dynamics in inelastic hard sphere fluids has been also extensively studied in MD simulations [7,8]. In this work we will be dealing with an idealized model of inelastic particle collisions characterized by a constant coefficient of normal restitution r, and focus on a plane shear flow.…”

Collision statistics in sheared inelastic hard spheres

“…8) results primarily from collisions between pairs of particles orientated in the x-y plane. This occurs when the change of the streaming velocity over the diameter of a particle becomes significant in comparison to the average relative peculiar velocity [21]. Particles separated in the y-plane then have a significantly increased relative velocity which increases their probability of collision.…”

“…Ring theory [9] is capable of including particle correlations; however, further approximations are required to make the resulting theory tractable. These correlations are present in moderately dense to dense systems of elastic particles, but they are enhanced by the clustering in inelastic systems [21,22]. The failure of Boltzmann and Enskog theories at high densities is therefore expected, even for elastic hard sphere systems.…”

Velocity profiles, stresses, and Bagnold scaling of sheared granular system in zero gravity

“…2,3,4,5,6,7,8,9 In particular, there are very few efforts to model realistically the physical boundaries, and this is the most common discrepancy between the set-up of laboratory experiments and the set-up of numerical simulations. This effect of physical boundaries, such as the walls of the container, is often and rightfully considered responsible for the discrepancy between the results of the experiments and of the simulations, or between the predictions of the existing theories and the results of experiments or simulations.…”