Period Tripling Causes Rotating Spirals in Agitated Wet Granular Layers

Pattern formation of a thin layer of vertically agitated wet granular matter is investigated experimentally. Rotating spirals with three arms, which correspond to the kinks between regions with different colliding phases, are the dominating pattern. This preferred number of arms corresponds to period tripling of the agitated granular layer, unlike predominantly subharmonic Faraday crispations in dry granular matter. The chirality of the spatiotemporal pattern corresponds to the rotation direction of the spirals.

Figure 1

The experimental setup with sample images of a spiral pattern. Vertical or horizontal information of the patterns formed is obtained by a high speed CCD camera mounted above the sample illuminated by a laser sheet or a low angle LED ring correspondingly. The camera is triggered by a multipulse generator. Snapshot (a) is captured with the laser sheet illumination. The red half transparent line is a surface profile determined by an image processing procedure. The spiral arm corresponds to a kink between regions with different heights. Snapshot (b) corresponds to the low angle illumination (averaged over three synchronized images to enhance the contrast).

Figure 2

Snapshots of migrating fronts (left panel) and a rotating spiral (right panel) captured at a fixed phase of continuous vibration cycles (indicated as numbers) showing the period tripling behavior. Parameters: $f=80\mathrm{Hz}$, $W=1.6\%$, $m=96\mathrm{g}$ and $\Gamma =36$ (left), 37 (right).

Figure 3

Phase diagram for the patterns (shaded with squares) measured by varying $\Gamma$ while keeping the vibration frequency for $m=129\mathrm{g}$ and $W=1.6\%$. The region with $\Gamma >50$ is unexplored. The snapshot (averaged over three synchronized images) shows two spirals rotating in opposite direction captured at $f=70\mathrm{Hz}$ and $\Gamma =23$. The red dash curves highlight the spiral arms and yellow arrows denote the rotation directions. The inset shows the order parameter $I_{\mathrm{rms}}$ vs $\Gamma$ at $f=80\mathrm{Hz}$ for various time step $\Delta t$ and both directions of $\Gamma$ variations.

Figure 4

Time space plots of the surface profiles obtained with laser profilometry. The long time behavior (a) presents the traveling of the kinks, i.e., spiral arms. The short time behavior (b), corresponding to the region marked with dash line in (a), shows the periodicity of the height fluctuations. Other parameters: $f=80\mathrm{Hz}$, $\Gamma =32$, frame rate $320\mathrm{Hz}$, $H=0$ chosen as the lowest surface height.

Figure 5

(a) The covariance between two subsequent images taken as the collision with the container occurs. The brightness corresponds to the mobility of particles. Sketches (b) and (c) indicate the spatiotemporal chirality of the pattern. The black line in (c) indicates the positions of the vibrating bottom, the dash lines correspond to the center of mass in regions I, II, and III. $H$ denotes the height.

Figure 6

Periodicity as a function of driving acceleration for $f=80\mathrm{Hz}$, obtained experimentally from the surface height fluctuations (a), and estimated by a single particle bouncing model (b). Errors in (a) correspond to the variance of the Gaussian profiles fitted to the spectrum of the height fluctuations. Red solid symbols highlight period 3 regimes above the threshold. Other parameters are the same as in Fig. 3.