Three-dimensional analysis of the collision process of a bead on a granular packing

We present results of the collision process of a bead onto a static granular packing. We provide, in particular, a three-dimensional (3D) extensive characterization of this process from a model experiment that allows us to propel a spherical bead onto a granular packing with a well-controlled velocity and impact angle. A collision typically produces a high-energy particle (rebound particle) and several low-energy grains (ejected particles). The collision process is recorded by means of two fast video cameras. The sequence of images from both cameras are then analyzed via image processing and the trajectories of all particles are reconstructed in 3D space. We show that the incident particle does not remain in the vertical incident plane after the rebound and that the deviation angle increases with increasing impact angle. Concerning the ejected particles, we demonstrated that the ejection angle (measured with respect to the horizontal plane) is surprisingly independent of both the impact angle and velocity of the incident particle, and is very close to 60°. The horizontal component of the ejection speed of the splashed particles is found to be weakly dependent on the incident speed and impact angle, and is relatively isotropic (no particular horizontal direction is favored). This last feature suggests that the bead packing acts as a perfect diffusive medium with respect to energy propagation.

Figure 1

Sequences of images taken from both cameras: (a) incident plane and (b) transverse plane. The two cameras are synchronized and take 1 frame every $2\mathrm{ms}$. The time step between two successive images shown in these sequences is $16\mathrm{ms}$ and the size of the images is $120\times 240{\mathrm{mm}}^2$.

Figure 2

Top view of the experimental setup. The projectile is propelled from left to right along the $\left(OX\right)$ direction and impacts the granular packing at its center. One camera is displayed perpendicularly to the incident plane and the other is at the opposite of the shooting direction. Both cameras are placed at the same distance from the center of the granular packing.

Figure 3

(Color online) Example of reconstruction of trajectories: (a) incident plane, (b) transverse plane. Note that some trajectories are truncated due to the limitation of the recording duration of the cameras.

Figure 4

Definition of the rebound angles (${\theta }_r$ and ${\phi }_r$) and ejection angles ($\theta$ and $\phi$). ${\theta }_r$ is the angle between the horizontal plane and the rebound direction, whereas ${\phi }_r$ is the angle between the $OX$ axis and the projection of the rebound direction onto the horizontal plane $XY$. The angles $\theta$ and $\phi$ characterize similarly the ejection direction of the splashed particles.

Figure 5

(a) Variation of the mean actual rebound angle ${\theta }_r$ and the mean 2D angle ${\theta }_{r,2\mathrm{D}}$ as a function of the impact angle for an incident speed $V_i\approx 26\mathrm{m}∕\mathrm{s}$ (except for the experiment performed at ${\theta }_i=40°$ for which the impact speed was $V_i\approx 18\mathrm{m}∕\mathrm{s}$). (b) Distribution of the azimuthal angle for different impact angles. (c) Variation of the root mean square ${\sigma }_{{\phi }_r}$ of the azimuthal angle ${\phi }_r$ as a function of the impact angle ${\theta }_i$.

Figure 6

Mean restitution coefficient $\overline{e}$ and ${\overline{e}}_{2\mathrm{D}}$ versus ${\theta }_i$. The lines correspond to fits of the form $A-B\sin {\theta }_i$ ($A=0.87$ and $B=0.62$, whereas $A_{2\mathrm{D}}=0.87$ and $B_{2\mathrm{D}}=0.67$).

Figure 7

Mean number of ejected particles renormalized by $(1-{\overline{e}}^2)$ versus the impact velocity for various impact angles.

Figure 8

Distribution of the horizontal ejection speed $V_x$ (a) and $V_y$ (b) for various impact angles and incident velocities. The dashed lines correspond to the best fits using normal laws. Data were obtained with PVC particles.

Figure 9

Dimensionless mean quadratic velocities ${\overline{V_x}}^2$, ${\overline{V_y}}^2$, and ${\overline{V_y}}^2$ versus dimensionless impact velocity at an impact angle ${\theta }_i=10°$. The upper dashed curve corresponds to a power law fit giving by Eq. (9).

Figure 10

Sum of the kinetic energy of the splashed particles versus the fraction of energy communicated to the granular bed $(1-{\overline{e}}^2)E_i$. Note that both quantities were made dimensionless by $mgd$.

Figure 11

(a) Mean ejection angle $\overline{\theta }$ versus the impact angle ${\theta }_i$: (◻) 2D ejection angle ${\overline{\theta }}_{2\mathrm{D}}$ and (◼) actual ejection angle $\overline{\theta }$. (b) Distributions of the azimuthal angle for different impact angles. (c) Root mean square ${\sigma }_{\phi }$ versus impact angle ${\theta }_i$. These data correspond to collision experiments performed at an impact velocity $V_i=26\mathrm{m}∕\mathrm{s}$, except for ${\theta }_i=40°$ where $V_i=18\mathrm{m}∕\mathrm{s}$. Data were obtained with PVC particles.

Figure 12

(a) Typical bivariate distribution $P(x,y)$ of the ejection position of the splashed beads, and corresponding distribution $P\left(x\right)$ (b) and $P\left(y\right)$ (c). [Note that $P\left(x\right)={\int }_{-\infty }^{\infty }dyP(x,y)$ and $P\left(y\right)={\int }_{-\infty }^{\infty }dxP(x,y)$]. ${\theta }_i=40°$ and $V_i=18\mathrm{m}∕\mathrm{s}$. The amplitude of $P(x,y)$ is encoded by a gradation in gray levels separated by isoprobability lines with a step $\delta P=0.1P_{\max }$. Data were obtained with PVC particles.

Figure 13

Distribution $P\left(\Vert \vec{r}-{\vec{r}}_0\Vert \right)$ of the ejection location for different impact velocities at an impact angle ${\theta }_i=10°$. The vector $\vec{r}=x{\vec{e}}_x+y{\vec{e}}_y$ denotes the ejection location and ${\vec{r}}_0$ the position of the peak of the distribution. Continuous lines correspond to normal laws of the form $P\left(\Vert \vec{r}-{\vec{r}}_0\Vert \right)=(1∕2\pi {\sigma }_r^2)\exp [-{(\vec{r}-{\vec{r}}_0)}^2∕2{\sigma }_r^2]$. Data were obtained with PVC particles.

Figure 14

(a) Evolution of the peak position $r_0$ as function of the impact velocity. Inset: Same data with $r_0$ rescaled by $\cos {\theta }_i$. (b) Evolution of the distribution width as a function the impact velocity. Inset: Same data plotted as a function $\sqrt{1-{\overline{e}}^2}{V}_i∕\sqrt{gd}$. Data were obtained with PVC particles.