Impact dynamics of granular jets with noncircular cross sections

Using high-speed photography, we investigate two distinct regimes of the impact dynamics of granular jets with noncircular cross sections. In the steady-state regime, we observe the formation of thin granular sheets with anisotropic shapes and show that the degree of anisotropy increases with the aspect ratio of the jet's cross section. Our results illustrate the liquidlike behavior of granular materials during impact and demonstrate that a collective hydrodynamic flow emerges from strongly interacting discrete particles. We discuss the analogy between our experiments and those from the Relativistic Heavy Ion Collider, where similar anisotropic ejecta from a quark-gluon plasma have been observed in heavy-ion impact.

Figure 1

A schematic of the experiment setup. The orientation of the coordinate system is also shown.

Figure 2

Shape of granular jet's cross section. (a) Collision zone between two spherical ions (gray area). The left ion (solid line) moves in the $z$ direction out of the plane of the paper. The right ion (dashed line) moves in the negative $z$ direction into the plane of the paper. The spherical ions are subjected to Lorentz contraction in the $z$ direction and become disks in the laboratory frame (see Sec. 3c). $x_1$ and $y_1$ define the semiaxis of the collision zone. $b$ is the impact parameter. (b) Schematic of the protocol for machining slots of different aspect ratios. The relation between $\alpha$ and $c=y_1/x_1$ is shown in the lower right inset. (c) Slots of different aspect ratios. The dimension of each slot is given in Table 1.

Figure 3

Impact dynamics of granular jets with $\sigma =16.6$ mm${}^2$ and $c=4$ [(a) and (b)] and $\sigma =25.8$ mm${}^2$ and $c=2$ [(c) and (d)]. [(a) and (c)] First transient regime. [(b) and (d)] Second steady-state regime. The focused stream in (a) is indicated by an arrow. In (c) and (d), the target holder partially blocks the view of granular ejecta. A downward-moving stream can be seen from the side view of the impact process. The cross section and orientation of the slots are indicated at the upper left corner of (a) and (c). The linear size of real slots is 5 times smaller than those shown. The long axis of the slot is along the $y$ axis as indicated in (a) and (c).

Figure 4

Particle motion behind a transparent target with $\sigma =16.6$ mm${}^2$ and $c=4$. (a) Mean velocity field in the steady-state regime from the PIV measurement. (b) Mean radial speed in the azimuthal plane (in color). For comparison, the shape and orientation of a slot with $c=4$ (dashed line) is overlaid on the image. The direction of gravity is indicated in (a). Streamlines from the velocity field in (a) are displayed. (see also Supplementary Movie 4 [20]). (c) Radial velocity of particles at different radial locations, $r$, as a function of time. (d) Azimuthal distribution of the mean radial velocity of particles for circular rings at different $r$. $r$ is indicated by the color of the lines. The magnitude of velocity is indicated by the polar axis of the plot. Displayed data was time-averaged over 27.5 ms.

Figure 5

Quantitative measurement of ejecta dynamics. (a) Three concentric material rings (red curves) in the ejecta sheet. The image is from a granular jet with $\sigma =16.6$ mm${}^2$ and $c=4$. (b) A schematic showing the quantitative measurement of $r\left(\varphi \right)$ and $P\left(\varphi \right)$. The thick red line indicates the shape of the material ring at the end of the second regime. The blue arrow indicates the location of the ring, $r\left(\varphi \right)$. The yellow wedge indicates the differential area around $\varphi$, $dA$, which is proportional to $P\left(\varphi \right)$ (see the main text). (c) The material rings extracted from (a). (d) Scaling of the three rings by their characteristic lengths.

Figure 6

Dynamics of anisotropic ejecta. (a) Azimuthal momentum distribution $P\left(\varphi \right)$ at different $t$ for the granular jet of $\sigma =16.6$ mm${}^2$ and $c=4$. (b) Second Fourier coefficient of $P\left(\varphi \right)$, ${\upsilon }_2$, as a function of $t$ for granular jets of different $c$. Since the area enclosed inside a material ring, $A$, increases monotonically with $t$ [see (c)], we use $A$ as the time axis. The aspect ratio, $c$, of experimental curves increases from 1 to 6 from bottom to top. The solid lines are fits using Eq. (7). (c) Enclosed area inside a material ring, $A$, as a function of $t$. Different curves are for the jets of different $c$. The symbols are the same as those used in (b). Solid lines are second-order polynomial fits [Eq. (4)].

Figure 7

Impact of granular jets with cross-section area $\sigma =16.6$ mm${}^2$ and the following aspect ratio: (a) $c=1$, (b) $c=3$, (c) $c=4$, and (d) $c=6$. The cross section and orientation of the slots are indicated at the upper left corner of each plot. The linear size of real slots is 2.5 times smaller than those shown in the plot.

Figure 8

Azimuthal momentum distribution and the anisotropy of ejecta for granular jets of cross-section area $\sigma =16.6$ mm${}^2$ and different aspect ratios. (a) Azimuthal momentum distribution, $P\left(\varphi \right)$, for granular jets of different $c$. The distribution is taken from the large $t$ limit where the ejecta has achieved a steady shape. (b) Anisotropy ${\upsilon }_2$ (black squares) and higher harmonic anisotropy ${\upsilon }_4$ (triangle) as a function of $c$. ${\upsilon }_2/\varepsilon$ is also shown (red disks). The data are averaged over three to four experimental trials. Solid lines are polynomial fits to guide the eye. The horizontal dashed lines indicate the upper and lower limits of ${\upsilon }_2/\varepsilon$ observed in the RHIC experiments [27].