Relaxation Dynamics of a Granular Pile on a Vertically Vibrating Plate

Nonlinear relaxation dynamics of a vertically vibrated granular pile is experimentally studied. In the experiment, the flux and slope on the relaxing pile are measured by using a high-speed laser profiler. The relation of these quantities can be modeled by the nonlinear transport law assuming the uniform vibrofluidization of an entire pile. The fitting parameter in this model is only the relaxation efficiency, which characterizes the energy conversion rate from vertical vibration into horizontal transport. We demonstrate that this value is a constant independent of experimental conditions. The actual relaxation is successfully reproduced by the continuity equation with the proposed model. Finally, its specific applicability toward an astrophysical phenomenon is shown.

Figure 1

A schematic illustration of crater relaxation caused by meteorite-impact-induced seismic shaking on small astronomical objects.

Figure 2

(a) A schematic illustration of the experimental system. (b) Snapshots of a relaxing pile during vibration. (c) Raw data of radial profiles taken by a laser profiler. A gray curve shows a profile of an initial pile with the angle of repose, which is taken before turning on a vibration generator. Black profiles are taken at $t=0$, 0.26, 0.5, 1.0, 2.0, 3.0, 5.0 s from top to bottom. Note that the data between gray and top black profiles are not used in this study as the vibration amplitude $A$ is not constant. (d) An image of the NDT model. In the main text, the forces applied to the filled thin slice are discussed.

Figure 3

(a) Flux $q$ versus slope $|\nabla h|$. The vertical solid line corresponds to $|\nabla h|=\tan {\theta }_{\mathrm{c}}$. Colors represent analysis points. (b) Depth-averaged velocity $q/h(=\overline{v_{\mathrm{t}}})$ versus slope $|\nabla h|$. Colors are identical to those in (a). A dashed curve shows the best fitting by Eq. (4). Note that $\mu$ is fixed at $\tan {\theta }_{\mathrm{c}}$. Inset: Analysis results for various vibration conditions. The axises are the same as the main plot. All the data are also fitted by Eq. (4). (c) Relaxation efficiency $c$ as a function of the maximum vibration velocity $v_{\text{vib}}$. Colors and symbols represent vibration frequency $f$ and materials used (Table. 1). Solid and dashed lines depict $c=0.068$ with $1\sigma =0.014$.

Figure 4

Comparison between profiles taken by a laser profiler and those computed by the NDT model. The initial shape for the model computation is approximated by $h-h_0\sim ({r_0}^{1.6}-r^{1.6})$ with $r_0=40\mathrm{mm}$. The time step and spatial resolution of the numerical calculation are ${10}^{-5}\mathrm{s}$ and 0.5 mm, respectively. Experimental and computed curves agree well in relatively steep regimes. When the shape of the pile get flattened to some extent, granular media reach a jammed state, where the relaxation almost halts.