Kink-Induced Transport and Segregation in Oscillated Granular Layers

We use experiments and molecular dynamics simulations of vertically oscillated granular layers to study horizontal particle segregation induced by a kink (a boundary between domains oscillating out of phase). Counterrotating convection rolls carry the larger particles in a bidisperse layer along the granular surface to a kink, where they become trapped. The convection originates from avalanches that occur inside the layer, along the interface between solidified and fluidized grains. The position of a kink can be controlled by modulation of the container frequency, making possible systematic harvesting of the larger particles.

Figure 1

Segregation and controlled transport of $650\mu \mathrm{m}$ diameter black glass spheres by a kink in an experiment with a ten-particle deep oscillated layer of $165\mu \mathrm{m}$ bronze spheres at $a_{\mathrm{m}\mathrm{a}\mathrm{x}}=5g$ and $f=92\mathrm{H}\mathrm{z}$. The motion of the kink is controlled by modulation of the phase difference $\Delta \varphi$ between the primary oscillation signal and an added small subharmonic sinusoidal perturbation (see text). (a) A kink (dashed line) sweeps across the layer after a rapid change in $\Delta \varphi$ of $2\pi /3$; glass spheres close to the kink move toward it and remain trapped in the kink at the surface of the layer, while far from the kink they diffuse; these two types of trajectories are illustrated by the solid white lines. The numbers in each frame denote plate oscillations after $\Delta \varphi$ is changed. (b) Sinusoidal oscillation of a kink, with $\Delta \varphi (t)=f_{ms}/f_{mr}\sin (2\pi f_{mr}t)$, $f_{ms}=0.17\mathrm{H}\mathrm{z}$, and $f_{mr}=0.1\mathrm{H}\mathrm{z}$. The large trapped particles follow the motion of the kink.

Figure 2

Grains form a pair of convection rolls, flowing downward at the kink, as illustrated by these projections of a layer with a kink at different times $ft$ during a cycle ($ft=0$ when the container is at its equilibrium position, horizontal dashed line, moving upward), obtained from a simulation; the container bottom oscillates between $z=0$ and $2A$ with $a_{\mathrm{m}\mathrm{a}\mathrm{x}}=5.2g$ and $f=69\mathrm{H}\mathrm{z}$. The vertical position of the center of mass as a function of time of the left (right) side of the kink is shown as a dashed (solid) line in the inset of the top panel. The two bold vertical arrows on either side of the kink indicate the directions of motion of these two domains. The small arrows show the discrete grain velocity $\mathbf{u}\mathbf{(}\mathbf{r}(t),t\mathbf{)}$ averaged over the shorter horizontal direction. Each circle represents a grain, and the container bottom is indicated by horizontal gray lines. The region shown is far from the rigid sidewalls; the $x$ axis is parallel to the longer horizontal direction.

Figure 3

A schematic diagram showing trapping of larger particles at a kink in a layer of smaller particles. The larger particles move toward the kink due to the surface flow of the convective motion of the smaller particles. The larger particles do not flow down in between the convection rolls, but remain there, due to their size. Dashed arrows represent the convection rolls formed by the smaller particles, and solid arrows indicate the direction of the motion of large particles.

Figure 4

The trapping zone width (defined in the inset) for various $a_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $f$, obtained in simulations of eight-particle deep monodisperse layers of $165\mu \mathrm{m}$ spheres. The width is linearly proportional to $V_{\mathrm{m}\mathrm{a}\mathrm{x}}$, as shown by the dashed line (a least square fit). The symbols correspond to ●, $a_{\mathrm{m}\mathrm{a}\mathrm{x}}=4.9g$; ○, $5g$; $*$, $5.1g$; □, $5.2g$; △, $5.3g$. For each $a_{\mathrm{m}\mathrm{a}\mathrm{x}}$, the data points correspond to $f=86$, 78, 69, 60, and 52 Hz, respectively. Inset: The trapping zone is defined as the region where the vertically integrated $|\mathbf{u}{|}^2$ (gray solid line, in an arbitrary unit) is larger than some small value (dash-dotted line).

Figure 5

Simulations reveal an avalanche occurring inside the layer, along the interface between the solidified and fluidized parts: (a) Instantaneous grain velocities $\mathbf{v}\mathbf{(}\mathbf{r}(t),t\mathbf{)}$ in the laboratory frame (indicated by arrows), averaged over the shorter horizontal direction, when the left side of a kink is being pushed up by the bottom plate (a horizontal solid line) for the same case as in Fig. 2. A gray background represents the layer. (b) Instantaneous velocities in the container frame, during the same time. As the left side is pushed up, it solidifies on the container bottom, while the right side is fluidized and cascades down onto the left side, as shown by gray arrows ($a_{\mathrm{m}\mathrm{a}\mathrm{x}}=5.2g$, $f=69\mathrm{H}\mathrm{z}$).