TY - JOUR

T1 - Transient granular shock waves and upstream motion on a staircase

AU - van der Weele, Ko

AU - Kanellopoulos, Giorgos

AU - Tsiavos, Christos

AU - van der Meer, Devaraj

PY - 2009

Y1 - 2009

N2 - A granular cluster, placed on a staircase setup, is brought into motion by vertical shaking. Molecular dynamics simulations show that the system goes through three phases. After a rapid initial breakdown of the cluster, the particle stream organizes itself in the form of a shock wave moving down the steps of the staircase. As this wave becomes diluted, it transforms into a more symmetric flow, in which the particles move not only downwards but also toward the top of the staircase. This series of events is accurately reproduced by a dynamical model in which the particle flow from step to step is modeled by a flux function. To explain the observed scaling behavior during the three stages, we study the continuum version of this model (a nonlinear partial differential equation) in three successive limiting cases. (i) The first limit gives the correct t−1/3 decay law during the rapid initial phase, (ii) the second limit reveals that the transient shock wave is of the Burgers type, with the density of the wave front decreasing as t−1/2, and (iii) the third limit shows that the eventual symmetric flow is a slow diffusive process for which the density falls off as t−1/3 again. For any finite number of compartments, the system finally reaches an equilibrium distribution with a bias toward the lower compartments. For an unbounded staircase, however, the t−1/3 decay goes on forever and the distribution becomes increasingly more symmetric as the dilution progresses.

AB - A granular cluster, placed on a staircase setup, is brought into motion by vertical shaking. Molecular dynamics simulations show that the system goes through three phases. After a rapid initial breakdown of the cluster, the particle stream organizes itself in the form of a shock wave moving down the steps of the staircase. As this wave becomes diluted, it transforms into a more symmetric flow, in which the particles move not only downwards but also toward the top of the staircase. This series of events is accurately reproduced by a dynamical model in which the particle flow from step to step is modeled by a flux function. To explain the observed scaling behavior during the three stages, we study the continuum version of this model (a nonlinear partial differential equation) in three successive limiting cases. (i) The first limit gives the correct t−1/3 decay law during the rapid initial phase, (ii) the second limit reveals that the transient shock wave is of the Burgers type, with the density of the wave front decreasing as t−1/2, and (iii) the third limit shows that the eventual symmetric flow is a slow diffusive process for which the density falls off as t−1/3 again. For any finite number of compartments, the system finally reaches an equilibrium distribution with a bias toward the lower compartments. For an unbounded staircase, however, the t−1/3 decay goes on forever and the distribution becomes increasingly more symmetric as the dilution progresses.

KW - METIS-257971

KW - IR-73650

U2 - 10.1103/PhysRevE.80.011305

DO - 10.1103/PhysRevE.80.011305

M3 - Article

VL - 80

JO - Physical review E: covering statistical, nonlinear, biological, and soft matter physics

JF - Physical review E: covering statistical, nonlinear, biological, and soft matter physics

SN - 2470-0045

IS - 1

M1 - 011305

ER -