Session QN: Granular Flows III

Granular materials: on topology of force chains

One property of granular materials is the lack of spatial scale separation between the one characterizing the particle size and the one characterizing system as a whole. This property requires careful understanding of the features which exist on particle scale, such as force chains, with the hope that this understanding will help us produce appropriate continuum level models. In this talk, we will discuss our initial attempts to characterize force chains. These attempts are based on algebraic topology techniques that will be used to analyze and quantify force chain structures. In particular, we will discuss how these properties differ for the systems exposed to shear versus compression, and correlate the topological measures to the phenomena such as jamming. Furthermore, we will discuss the possibility of using topological techniques to come up with a quantitative way of comparing experiments and simulations. Prelminary results of these comparisons will be shown.   

Energy bursts in shallow granular systems

In a mixture of two species of grains of equal size but differing by their mass, placed in a vertically vibrated shallow box, there is spontaneous segregation. Once the system is at least partly segregated, energy bursts take place: the horizontal kinetic energy of the heavy particles, that normally is small, suddenly increases. An explanation is provided based on the existence of a fixed point for an isolated particle bouncing with only vertical motion between the top and bottom walls. Energy bursts occur when the large energy stored in the vertical motion is partly converted into horizontal energy through a chain reaction of collisions between heavy particles. Depending on the experimental or numerical parameters and initial conditions, the energy bursts can occur either randomly or rather periodically in time.   

3D aspects of mixing and transport in tumbled granular flow

The exploration of the kinematic structures that emerge in 3D flows has only just begun (see, e.g., Focus on Fluids in JFM vol.\ 654, 2010). Tumbled granular flow in a spherical container rotated sequentially about two distinct axes is a convenient physical system in which to investigate these issues. The flow can ``switch'' between 2D motion (dynamics restricted to 2D manifolds) and fully-3D motion depending on the choice of angles of rotation about the axes. We compute explicitly the action- action-angle transformation, period along trajectories and exact location of normally- hyperbolic and elliptic period-one curves from the piecewise-defined nonlinear dynamical system. This provides the basis for a definition of a 3D notion of an ``island.'' These theoretical results also allow for the ``optimization'' of the angles of rotation in the protocol. An extensive numerical investigation of the ``goodness of mixing'' is performed using Danckwerts' intensity of segregation $I$. By fitting the decay rate and asymptotic value of $I$, we can understand the effects of the protocol parameters and how mixing varies across the volume of the tumbler. Finally, we establish the existence of ``adiabatic'' structures (2D manifolds exhibiting chaotic and ergodic dynamics) and study the persistence of these barriers to radial transport as the flow is perturbed into the fully-3D regime.   

Derivation and numerical treatment of the low-Mach number equations for two-phase granular mixtures

In this talk we present a methodology for the derivation of the low-Mach number equations for two-phase flows and apply it to a particular constitutive model for immiscible mixtures of a granular material and a fluid. The proposed methodology is based on non-dimensionalizing the governing equations with respect to a reference thermodynamic state of the phase with the smallest speed of sound. Further, we propose an algorithm for the numerical treatment of these equations. It belongs to the class of fractional-step algorithms and employs a generalized projection method for the momentum equation of each phase. Our discussion concludes with the presentation of some preliminary numerical results for constant density flows.   

Simulations of 2D granular jet impact deadzone formation

Motivated by granular experiments showing the emergence of continuum-like dynamics when a dense jet hits a target, we simulate the impact of a 2D granular jet of frictional, cohesion- less grains upon a fixed target. This is an inertial, dense jet regime where the motion is essentially incompressible. Impact deflects the material in the jet into a hollow conical sheet. The cone angles measured in simulation are consistent with previous experimental studies of the 3D granular jet impact. In addition, experiments have revealed the formation of a ``dead zone,'' a region where the grain motion is negligibly small. The simulation shows that this dead zone can only form when a no- slip boundary condition is enforced at the target. The presence or absence of the dead zone leads to a change in cone angle consistent with the experimentally observed differences in cone angle between the 3D granular flow and the corresponding water bell flow.   

Rheology of simple shear flows of dense granular assemblies in different regimes

Using the discrete element method, simulations of simple shear flow of dense assemblies of frictional particles have been carried out over a range of shear rates and volume fractions in order to characterize the transition from quasistatic or inertial flow to intermediate flow. In agreement with previous results for frictionless spheres [1], the pressure and shear stress in the intermediate regime are found to approach asymptotic power law relations with shear rate; curiously, these asymptotes appear to be common to all intermediate flows regardless of the value of the particle friction coefficient. The scaling relations for stress for the inertial and quasistatic regimes are consistent with a recent extension of kinetic theory to dense inertial flows [2] and a simple model for quasistatic flows [3], respectively. For the case of steady, simple shear flow, the different regimes can be bridged readily: a harmonic weighting function blends the inertial regime to the intermediate asymptote, while a simple additive rule combines the quasistatic and intermediate regimes. \\[4pt] [1] T. Hatano, et al., J. Phys. Soc. Japan 76, 023001 (2007). \\[0pt] [2] J. Jenkins, and D. Berzi, Granular Matter 12, 151 (2010). \\[0pt] [3] J. Sun, and S. Sundaresan, J. Fluid Mech. (under review).   

Rheology of Granular Mixtures Differing in Size and/or Density

Rheology of mono-dispersed granular materials is well understood and it is well known that these materials follow a friction law where the shear stress to pressure ratio is determined by the Inertial number $I$. However, rheology of the general and more common case of granular mixtures of different size and/or different density particles has not received attention of researchers. We study the rheology of binary mixtures flowing over an inclined plane under the influence of gravity by means of DEM simulations. We show that the friction law for single component granular material with appropriate modification in the inertial number expression captures the rheology of the mixtures as well and can predict the viscosity of both, same-size different-density particle mixtures and different-size same-density particle mixtures. For the case of mixture of particles differing in size and density both, we obtain a well-mixed or a segregated state depending upon the over-all composition of the mixture. The modified friction law is able to predict the viscosity for this case as well for both well-mixed and segregated state. Thus we show that friction law with a generalized definition of Inertial number can predict the rheology of granular mixtures differing in size and/or density.   

Rheology of dense granular mixtures: Particle size distributions, boundary conditions, and collisional time scales

We computationally investigate the dependence of the rheology of dense sheared granular mixtures on their particle size distribution. We find that the variation of the rheology with the particle size distribution depends on the boundary conditions. For example, under constant pressure conditions the effective friction coefficient $\mu^*$ (the ratio between shear and pressure stresses at the boundary) increases mildly with the average particle size. On the other hand, under constant volume conditions, $\mu^*$ has a non-monotonic dependence on the average particle size that is related to the proximity of the system solids fraction to the maximum packing fraction. Somewhat surprisingly, then, $\mu^*$ scales with a dimensionless shear rate (a generalized inertial number) in the same way for either boundary condition. We show that, for our system of relatively hard spheres, these relationships are governed largely by the ratio between average collision times and mean free path times, also independent of boundary conditions.