Thomas C. Halsey

Granular flow down an inclined plane: Bagnold scaling and rheology

We have performed a systematic, large-scale simulation study of granular media in two and three dimensions, investigating the rheology of cohesionless granular particles in inclined plane geometries, i.e., chute flows. We find that over a wide range of parameter space of interaction coefficients and inclination angles, a steady-state flow regime exists in which the energy input from gravity balances that dissipated from friction and inelastic collisions. In this regime, the bulk packing fraction (away from the top free surface and the bottom plate boundary) remains constant as a function of depth z, of the pile. The velocity profile in the direction of flow vx(z) scales with height of the pile H, according to vx(z) proportional to H(alpha), with alpha=1.52+/-0.05. However, the behavior of the normal stresses indicates that existing simple theories of granular flow do not capture all of the features evidenced in the simulations.

Geometry of frictionless and frictional sphere packings

We study static packings of frictionless and frictional spheres in three dimensions, obtained via molecular dynamics simulations, in which we vary particle hardness, friction coefficient, and coefficient of restitution. Although frictionless packings of hard spheres are always isostatic (with six contacts) regardless of construction history and restitution coefficient, frictional packings achieve a multitude of hyperstatic packings that depend on system parameters and construction history. Instead of immediately dropping to four, the coordination number reduces smoothly from z=6 as the friction coefficient mu between two particles is increased.

How Sandcastles Fall

Exxon Research and Engineering, Route 22 East, Annandale, N.J. 08801(February 1, 2008)Capillary forces significantly affect the stability of sandpiles. We analyze the stability of sandpileswith such forces, and find that the critical angle is unchanged in the limit of an infinitely largesystem; however, this angle is increased for finite-sized systems. The failure occurs in the bulk ofthe sandpile rather than at the surface. This is related to a standard result in soil mechanics. Theincrease in the critical angle is determined by the surface roughness of the particles, and exhibitsthree regimes as a function of the added-fluid volume. Our theory is in qualitative agreement withthe recent experimental results of Hornbaker et al., although not with the interpretation they makeof these results.81.05.Rm, 68.45.Gd, 91.50.Jc

Granular gravitational collapse and chute flow

We argue that inelastic grains in a flow under gravitation tend to collapse into states in which the relative normal velocities of two neighboring grains is zero. If the time scale for this gravitational collapse is shorter than inverse strain rates in the flow, we propose that this collapse will lead to the formation of granular eddies, large-scale condensed structures of particles moving coherently with one another. The scale of these eddies is determined by the gradient of the strain rate. Applying these concepts to chute flow of granular media (gravitationally driven flow down inclined planes), we predict the existence of a bulk flow region whose rheology is determined only by flow density. This theory yields the Pouliquen flow rule, correlating different chute flows; it also accounts for the different flow regimes observed.

The double layer impedance at a rough surface: Theoretical results

Abstract The roughness of an electrode surface can dramatically influence the double layer impedance at that electrode. We review a Greens function approach to this problem. For nearly flat surfaces, this approach can be developed perturbatively. In the more general case, the impedance problem can be mapped onto a problem involving the behavior of random walks near the surface of the electrode. For self-similar surfaces, we develop a scaling theory for this random walk problem. We explore the consequences of this theory for the double layer impedance. We find a stretched-exponential impedance, with the amplitude and exponent of the stretched-exponential decay being related to the multifractal properties of the self-similar surface. At high frequencies, this behavior is similar to the experimentally observed “constant phase angle” (CPA) response. We discuss experimental consequences of these results.

Singular Shape of a Fluid Drop in an Electric or Magnetic Field

Beyond a threshold, electric or magnetic fields cause a dielectric or ferromagnetic fluid drop, respectively, to develop conical tips. We analyze the appearance of the conical tips and the associated shape transition of the drop using a loca-force balance as well as a global-energy argument. We find that a conical interface is possible only when the dielectric constant (or permeability) of the fluid exceeds a critical value ec = 17.59. For a fluid with e > ec, a conical interface is possible at two angles, one stable and one unstable. We calculate the critical field required to sustain a drop with stable conical tips. Such a drop is energetically favored at sufficiently high field. Our results also apply to the formation of conical dimples when a pool of fluid is placed in a normal field.

Velocity correlations in dense gravity-driven granular chute flow

We report numerical results for velocity correlations in dense, gravity-driven granular flow down an inclined plane. For the grains on the surface layer, our results are consistent with experimental measurements reported by Pouliquen. We show that the correlation structure within planes parallel to the surface persists in the bulk. The two-point velocity correlation function exhibits exponential decay for small to intermediate values of the separation between spheres. The correlation lengths identified by exponential fits to the data show nontrivial dependence on the averaging time Deltat used to determine grain velocities. We discuss the correlation length dependence on averaging time, incline angle, pile height, depth of the layer, system size, and grain stiffness and relate the results to other length scales associated with the rheology of the system. We find that correlation lengths are typically quite small, of the order of a particle diameter, and increase approximately logarithmically with a minimum pile height for which flow is possible, hstop, contrary to the theoretical expectation of a proportional relationship between the two length scales.

Gravity-driven dense granular flows

The authors report and analyze the results of numerical studies of dense granular flows in two and three dimensions, using both linear damped springs and Hertzian force laws between particles. Chute flow generically produces a constant density profile that satisfies scaling relations suggestive of a Bagnold grain inertia regime. The type for force law has little impact on the behavior of the system. Failure is not initiated at the surface, consistent with the absence of surface flows and different principal stress directions at vs. below the surface.

The double layer impedance as a probe of surface roughness

Abstract We examine the effects of surface roughness on the double layer impedance between an electrolyte and a solid electrode. This impedance can be related to the statistics of random walks that are reflected from the electrode surface. We discuss three distinct behaviors for these random walk statistics, and we relate each behavior to the impedance as well as to the geometrical properties of the electrode surface. The case of a self-similar surface is presented in detail, and numerical results for an electrode in the shape of a diffusion-limited aggregate (DLA) are shown.

Fractal measures and their singularities: The characterization of strange sets☆

Abstract We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α, which determines the strength of their singularities; and f , which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of α values and the function f ( α ). We apply this formation to the 2 ∞ cycle of period doubling, to the devils staircase of mode locking, and to trajectories on 2-tori with golden-mean winding numbers. In all cases the new formalism allows an introduction of smooth functions to characterize the measures. We believe that this formalism is readily applicable to experiments and should result in new tests of global universality.