Density Waves in Granular Flow: A Kinetic Wave Approach

It was recently observed that sand flowing down a vertical tube sometimes forms a traveling density pattern in which a number of regions with high density are separated from each other by regions of low density. In this work, we consider this behavior from the point of view of kinetic wave theory. Similar density patterns are found in molecular dynamic simulations of the system, and a well defined relationship is observed between local flux and local density -- a strong indicator of the presence of kinetic waves. The equations of motion for this system are also presented, and they allow kinetic wave solutions. Finally, the pattern formation process is investigated using a simple model of interacting kinetic waves.Comment: RevTeX, HLRZ preprint 46/93, 4 figures available upon reques

Power law tail in the radial growth probability distribution for DLA

Using both analytic and numerical methods, we study the radial growth probability distribution $P(r,M)$ for large scale off lattice diffusion limited aggregation (DLA) clusters. If the form of $P(r,M)$ is a Gaussian, we show analytically that the width $\xi(M)$ of the distribution {\it can not} scale as the radius of gyration $R_G$ of the cluster. We generate about $1750$ clusters of masses $M$ up to $500,000$ particles, and calculate the distribution by sending $10^6$ further random walkers for each cluster. We give strong support that the calculated distribution has a power law tail in the interior ($r\sim 0$) of the cluster, and can be described by a scaling Ansatz $P(r,M) \propto {r^\alpha\over\xi}\cdot g\left( {r-r_0}\over \xi \right)$, where $g(x)$ denotes some scaling function which is centered around zero and has a width of order unity. The exponent $\alpha$ is determined to be $\approx 2$, which is now substantially smaller than values measured earlier. We show, by including the power-law tail, that the width {\it can} scale as $R_G$, if $\alpha > D_f-1$.Comment: 11 pages, LaTeX, 5 figures not included, HLRZ preprint-29/9

Microscopic Motion of Particles Flowing through a Porous Medium

We use Stokesian Dynamics simulations to study the microscopic motion of particles suspended in fluids passing through porous media. We construct model porous media with fixed spherical particles, and allow mobile ones to move through this fixed bed under the action of an ambient velocity field. We first consider the pore scale motion of individual suspended particles at pore junctions. The relative particle flux into different possible directions exiting from a single pore, for two and three dimensional model porous media is found to approximately equal the corresponding fractional channel width or area. Next we consider the waiting time distribution for particles which are delayed in a junction, due to a stagnation point caused by a flow bifurcation. The waiting times are found to be controlled by two-particle interactions, and the distributions take the same form in model porous media as in two-particle systems. A simple theoretical estimate of the waiting time is consistent with the simulations. We also find that perturbing such a slow-moving particle by another nearby one leads to rather complicated behavior. We study the stability of geometrically trapped particles. For simple model traps, we find that particles passing nearby can ``relaunch'' the trapped particle through its hydrodynamic interaction, although the conditions for relaunching depend sensitively on the details of the trap and its surroundings.Comment: 16 pages, 19 figure

Heap Formation in Granular Media

Using molecular dynamics (MD) simulations, we find the formation of heaps in a system of granular particles contained in a box with oscillating bottom and fixed sidewalls. The simulation includes the effect of static friction, which is found to be crucial in maintaining a stable heap. We also find another mechanism for heap formation in systems under constant vertical shear. In both systems, heaps are formed due to a net downward shear by the sidewalls. We discuss the origin of net downward shear for the vibration induced heap.Comment: 11 pages, 4 figures available upon request, Plain TeX, HLRZ-101/9

Scaling Behavior of Granular Particles in a Vibrating Box

Using numerical and analytic methods, we study the behavior of granular particles contained in a vibrating box. We measure, by molecular dynamics (MD) simulation, several quantities which characterize the system. These quantities--the density and the granular temperature fields, and the vertical expansion--obey scaling in the variable $x = Af$. Here, $A$ and $f$ are the amplitude and the frequency of the vibration. The behavior of these quantities is qualitatively different for small and large values of $x$. We also study the system using Navier-Stokes type equations developed by Haff. We develop a boundary condition for moving boundaries, and solve for the density and the temperature fields of the steady state in the quasi-incompressible limit, where the average separation between the particles is much smaller than the average diameter of the particles. The fields obtained from Haff's equations show the same scaling as those from the simulations. The origin of the scaling can be easily understood. The behavior of the fields from the theory is consistent with the simulation data for small $x$, but they deviate significantly for large $x$. We argue that the deviation is due to the breakdown of the quasi-incompressibility condition for large $x$.Comment: LaTeX, 26 pages, 9 figures available upon reques

First Passage Time in a Two-Layer System

As a first step in the first passage problem for passive tracer in stratified porous media, we consider the case of a two-dimensional system consisting of two layers with different convection velocities. Using a lattice generating function formalism and a variety of analytic and numerical techniques, we calculate the asymptotic behavior of the first passage time probability distribution. We show analytically that the asymptotic distribution is a simple exponential in time for any choice of the velocities. The decay constant is given in terms of the largest eigenvalue of an operator related to a half-space Green's function. For the anti-symmetric case of opposite velocities in the layers, we show that the decay constant for system length $L$ crosses over from $L^{-2}$ behavior in diffusive limit to $L^{-1}$ behavior in the convective regime, where the crossover length $L^*$ is given in terms of the velocities. We also have formulated a general self-consistency relation, from which we have developed a recursive approach which is useful for studying the short time behavior.Comment: LaTeX, 28 pages, 7 figures not include

Super-lattice, rhombus, square, and hexagonal standing waves in magnetically driven ferrofluid surface

Standing wave patterns that arise on the surface of ferrofluids by (single frequency) parametric forcing with an ac magnetic field are investigated experimentally. Depending on the frequency and amplitude of the forcing, the system exhibits various patterns including a superlattice and subharmonic rhombuses as well as conventional harmonic hexagons and subharmonic squares. The superlattice arises in a bicritical situation where harmonic and subharmonic modes collide. The rhombic pattern arises due to the non-monotonic dispersion relation of a ferrofluid

Circular Kinks on the Surface of Granular Material Rotated in a Tilted Spinning Bucket

We find that circular kinks form on the surface of granular material when the axis of rotation is tilted more than the angle of internal friction of the material. Radius of the kinks is measured as a function of the spinning speed and the tilting angle. Stability consideration of the surface results in an explanation that the kink is a boundary between the inner unstable and outer stable regions. A simple cellular automata model also displays kinks at the stability boundary

Harmonic forcing of an extended oscillatory system: Homogeneous and periodic solutions

In this paper we study the effect of external harmonic forcing on a one-dimensional oscillatory system described by the complex Ginzburg-Landau equation (CGLE). For a sufficiently large forcing amplitude, a homogeneous state with no spatial structure is observed. The state becomes unstable to a spatially periodic ``stripe'' state via a supercritical bifurcation as the forcing amplitude decreases. An approximate phase equation is derived, and an analytic solution for the stripe state is obtained, through which the asymmetric behavior of the stability border of the state is explained. The phase equation, in particular the analytic solution, is found to be very useful in understanding the stability borders of the homogeneous and stripe states of the forced CGLE.Comment: 6 pages, 4 figures, 2 column revtex format, to be published in Phys. Rev.