Coarsening of Faraday Heaps: Experiment, Simulation, and Theory

When a layer of granular material is vertically shaken, the surface spontaneously breaks up in a landscape of small Faraday heaps that merge into larger ones on an ever increasing time scale. This coarsening process is studied in a linear setup, for which the average life span of the transient state with $N$ Faraday heaps is shown to scale as $N^{-3}$. We describe this process by a set of differential equations for the peak positions; the calculated evolution of the landscape is in excellent agreement with both the experiments and simulations. The same model explains the observational fact that the number of heaps towards the end of the process decreases approximately as $N(t)\propto t^{-1/2}$.

Figure 1

Coarsening of a vertically vibrated 1D granular bed, as recorded in our experiments. It takes roughly two minutes to evolve from a flat landscape to a single Faraday heap. Every image is taken at the same point during the vibration cycle, when the container moves upward and the bed is pressed against the floor.

Figure 2

Evolution of the heap pattern from $t=4\mathrm{s}$ to 18 s obtained by (a) experiment, (b) simulation, and (c) the analytical model. A stroboscopic video of the evolving heap pattern as obtained by experiment, simulation, and model is supplied in the supplementary material in 11.

Figure 3

Experiment: Number of heaps $N$ as a function of time. The black dots are the measured times when the $N$-heap state gives way to the ($N-1$)-state; the open circles represent the average over all 19 experimental runs. No data for $N=2$ are shown [8]. Inset: The averaged data on a doubly logarithmic scale reveals no clear scaling behavior for $N(t)$, which is explained in the text. The solid line corresponds to the approximation Eq. (7).

Figure 4

Part of a typical heap pattern, indicating the key parameters used in the coarsening model. The dashed profile indicates the position of the heap after one time step $dt$.

Figure 5

Validation of Eq. (1) from the simulation results in Fig. 2b. The slope of the fitted line gives the factor $C=0.2\times {10}^{-3}{\mathrm{m}}^2/\mathrm{s}$. The data in this figure are taken from the eight heaps in Fig. 2b, with each heap being indicated by a different marker.

Figure 6

Mean life span of the $N$ heap state ${\tau }_N$ as a function of the number of heaps $N$. The grey circles indicate the experimental data. The black circles indicate the data averaged over 10 000 runs of the model starting with 100 initial heaps. Inset: Number of heaps as a function of time (cf. Fig. 3).