Session BH: Granular Flows II: Rapid Rlows

Phase Diagram of Vertically Shaken Granular Matter

A shallow, vertically shaken granular bed in a quasi 2-D container is studied experimentally yielding a wider variety of phenomena than in any previous study: (1) bouncing bed, (2) undulations, (3) granular Leidenfrost effect, (4) convection rolls, and (5) granular gas. These phenomena and the transitions between them are characterized by dimensionless control parameters and combined in a full experimental phase diagram.   

Relaxation of number density waves in freely cooling granular systems

In this study, we explore the effect of strong inhomogeneities in the initial number density on the evolution of a system of inelastically colliding hard spheres. The characteristic length scale of the inhomogeneity in number density (viz. n/(grad n) ) is varied compared to a characteristic length scale associated with the initially prescribed pair correlation function of hard sphere positions. Effect of initial translational kinetic energy, volume fraction and restitution coefficient on the system evolution is explored. Implications of this study on scale separation underlying the revised Enskog theory are summarized.   

Gravitational-Inelastic Collapse of a Granular Gas

We experimentally explore the collapse of a granular gas that occurs when the energy input is halted. In a gravitational field, the gas rapidly collapses to a static state at the bottom of the container. In the quasi-2D experiment with glass spheres confined between two glass plates, video particle tracking can be used to measure the granular temperature, mean velocity, and density fields with good resolution in time and space. The collapse process shows fascinating structure including nearly complete collisional cooling during free fall and a compressional heating shock. We will discuss the ability of hydrodynamic models to reproduce the experimental results.   

Granular Flow in a 2D Couette-Taylor Experiment

The dynamics of granular materials pose interesting problems both in gravity and low-gravity environments due to their high dissipation in collisions and their propensity to jam. In addition, it has been shown that even moderate flows can achieve supersonic conditions within a granular medium.\footnote{E. Rericha, C. Bizon, M. D. Shattuck, and H. Swinney, Phys. Rev. Lett., \textbf{88}, 014302 (2002).} A few thousand permeable spheres are motivated to flow in a 2D geometry via a magnetic field and then sheared mechanically at a boundary wall. The novel Couette-Taylor design of the apparatus with one soft boundary allows for shear flow in the 2D system to be studied in the absence of jamming. High-speed digital imaging and particle tracking software allow the system to be studied for a variety of flow speeds and shear rates at the boundary wall. The apparatus can also be used to study flow past an obstacle in analogy to a wind tunnel for granular flows.   

Non-equilibrium Steady-State of Granular Fluids

We experimentally investigate the non-equilibrium steady-state (NESS) structure of a uniformly heated quasi-2D granular fluid as a function of the filling fraction. For the first time we experimentally show that the structure (as measured by radial distribution function, bond order parameter, and shape factor) of a NESS behaves identically to that of it's equilibrium counterpart. In our driven dissipative system there is a constant energy flow through the system making it far from equilibrium, yet it behaves as though it were in equilibrium with a maximization principle like entropy. It appears that homogeneity and stationarity are more important than energy conservation for producing a thermodynamic state. The existence of a thermodynamics for NESS has profound consequences for granular systems and more broadly for any systems, which are out of equilibrium due to energy flux, and may lead to a fundamental justification for a theory of gradients in a near-NESS analogous to Navier-Stokes equations for near-equilibrium systems.   

Caging Dynamics in a quasi-2D granular fluid

We report a novel experimental investigation of the dynamics of a uniformly heated, horizontal and quasi-2D granular fluid. Our study is done as a function of filling fraction, $\phi$, in the region prior to crystallization which we observe at $\phi_s=0.719\pm0.007$. We perform a statistical analysis based on two quantities that are typically employed in colloidal/molecular systems: the Mean Square Displacement (MSD) and the Self Intermediate Scattering Function (SISF). These are calculated from the trajectories obtained by tracking all particles inside a representative imaging window of the full system. At low $\phi$ the classic diffusive behavior of a disordered fluid is observed. As the filling fraction is increased towards $\phi_s$, the MSD (or SISF) develops a two-step increase (or decrease) analogous to what is commonly observed in glassy systems. This plateau at intermediate timescales is a signature of the slowing down of the motion of particles due to temporary trapping inside the cages formed by their neighbors. This caging is increasingly more pronounced as $\phi_s$ is approached from below. For $\phi>\phi_s$, each particle becomes fully arrested by its six neighbors, for the whole time accessible experimentally. Moreover, the relaxation time extracted from the SISF, as a function of $\phi$, is well described by the classic Vogel-Fulcher’s law common of many glass formers.   

Multiple Solutions of Rapid Granular Chute Flow

The free-surface flow of highly agitated particles on an inclined chute is analysed using a continuum model that adopts the kinetic theory of rapid granular flows. Steady, fully developed profiles for the volume fraction, velocity and granular temperature of the flowing grains may be found as solutions to the governing equations. These are calculated using a pseudospectral method that exploits the asymptotic form of the solutions at large heights. The character of the steady solutions is determined by a relatively small number of dimensionless parameters, which includes the slope inclination and material properties of the grains. The pseudospectral approximation lends itself to parametric continuation, which allows us to efficiently track the form of the solutions as we vary the controlling parameters. In particular, we investigate the depth of steady flows, here defined as the centre of mass, as the volume flux of material is varied and find that, in some parameter regimes, three flow depths occur for a given volume flux of material. For such flows, we consider the linear stability of the multiple solutions to small perturbations.   

Direct quadrature method of moments for Boltzmann equation with dissipative dynamics

Computational modeling of transport phenomena, such as those in rarefied gases, granular flows and plasmas, is particularly challenging due to the presence of a collision operator in the governing Boltzmann equation that is non-linear and integro-differential in nature. Earlier numerical approaches in the literature have been found to be computationally expensive and/or lacking in sufficient accuracy. In order to address these issues, we propose a new approach for the Boltzmann equation using direct quadrature method of moments. In this approach, the velocity distribution function is represented as a set of Dirac-delta functions, with associated weights and locations. The evolution of these weights and their locations is derived from the Boltzmann equation using constraints on generalized moments of velocity. The collision integral can be simplified into a more manageable form using appropriate coordinate transformations and multinomial expansions. The approach is designed to preserve mass, momenta and correct rates of dissipation of energy and selected generalized moments of velocity. The criteria for selection of moment constraints and the performance of this approach will be evaluated.