Horizontal flow and surface patterns in a vertically vibrated annular granular layer

A set of experiments was carried out on the motion of granular materials in a vertically vibrated annular system with a sawtooth-shaped base. We observed the coexistence of granular flow and surface patterns such as periodic subharmonic waves and kink pairs. Different patterns can transit each other as control parameters vary. The flow varies with space and time, and oppositely directed flows can occur at different levels and different moments. The magnitude and direction of the flow depend on the parameters defining the system in a complex manner. The motion of the patterns relates to the granular flow in a different way from that in which the wave in an ordinary fluid relates to the moving fluid. A preliminary explanation is given to our experimental findings.

Figure 1

Phase diagram showing surface patterns. The shaded area represents a transitional region between anticlockwise kink pairs and clockwise kink pairs.

Figure 2

Typical surface patterns recorded using a high-speed digital camera (Redlake MASD Motionscope PCI 2000 SC) and processed using an image technique [19] for $f=18$ Hz: (a) periodic subharmonic wave, (b) anticlockwise kink pair, and (c) superposition of clockwise kink pair and subharmonic wave. The rectangle on each surface pattern represents the measure window in which the granular flow is measured. $h$ is the height of the measure window. ${\lambda }_1$ and ${\lambda }_2$ are the wavelengths of subharmonic waves. The arrows in each measure window represent the schematic spatial flow distribution. The circles and the arrows in the circles represent the nodes and the directions in which they move, respectively.

Figure 3

$v_k$ as a function of $\Gamma$ for different $f$'s. The markers represent the experimental results. Each curve represents a linear fit for the experimental results.

Figure 4

Velocity fields for $f=20$ Hz. (a)–(d) A node region of a clockwise kink pair for $\Gamma =5.4$. (a')–(d') One wavelength from a subharmonic wave for $\Gamma =4.0$. (a”)–(d”) An arm region of a clockwise kink pair for $\Gamma =5.0$. The lowest arrow in each subgraph represents the motion of the base in a vertical direction.

Figure 5

Total flow in the granular layer and trajectory of the granular layer as a function of time for $f=20$ Hz and two different $\Gamma$'s: (a) $f/2$ subharmonic wave and (b) $f/2$ clockwise kink pair. In each subgraph, the upper figure represents the total flow versus time and the lower figure represents the trajectories of the layer (solid line, measured for vertical position of the lower boundary of the layer) and the base (dashed line) versus time. Here, $t=0$ corresponds to the moment when the base reaches its highest point and $t=T/2$ the lowest point. Four driving periods are shown. The markers $(\circ$ and $•)$ on the lines indicate 20 sample points in every driving period.

Figure 6

Stratified horizontal flow in the granular layer as a function of height for several specific moments in one vibrating period for $f=20$ Hz: (a) $f/2$ subharmonic wave and (b) $f/2$ clockwise kink pair. The inset of each subgraph shows the motions of the layer and the base in a vertical direction. The moments $t_1,t_2,t_3,t_4$, and $t_5$ correspond to those in Fig. 5, respectively.

Figure 7

Time-averaged total flow in the granular layer as a function of $\Gamma$ for different $f$'s. The markers represent the results of the experiments. Typical error bars are shown. Each curve represents a smoothing spline fit for the experimental results.

Figure 8

Relation between $v_k$ and $v_f$ as $\Gamma$ increases for different $f$'s. The arrows on each curve indicate the direction in which $\Gamma$ increases. The values of $\Gamma$ as parameters are marked next to some of the points.