Coefficient of restitution for wet particles

The influence of a liquid film on the coefficient of restitution (COR) is investigated experimentally by tracing freely falling particles bouncing on a wet surface. The dependence of the COR on the impact velocity and various properties of the particle and liquid is presented and discussed in terms of dimensionless numbers that characterize the interplay between inertial, viscous, and surface forces. In the Reynolds number regime where lubrication theory does not apply, the ratio of the film thickness to the particle size is found to be a crucial parameter determining the COR.

Figure 1

Sketch of experimental setup. The bouncing of the glass bead, initially held by the vacuum nozzle, on the glass container is recorded with a high speed camera (Camera 2). The thickness of the liquid layer is monitored by detecting the laser beam reflected from the liquid surface with Camera 1.

Figure 2

A series of snapshots captured with a frame rate of 450 Hz showing a 4 mm glass bead bouncing on a glass plate covered with a 1 mm water film.

Figure 3

Trajectories of particle bouncing on wet (a) and dry (b) surface after image processing. The image sequence in the upper panel represents a fraction of the wet trajectory (a) with superimposed centers and boundaries of the sphere. The blue (dark gray) line in (a) corresponds to a parabolic fit to the trajectory after the first rebound. The peak positions of the trajectories $h_{\mathrm{peak}}$ obtained from the fits are marked with triangles in (a) and upside down triangles in (b). The inset in (a) shows $h_{\mathrm{peak}}$ as a function of the number of impacts.

Figure 4

Normal restitution coefficient $e_{\mathrm{n}}$ and rebound velocity $v_{\mathrm{rebound}}$ as a function of impact velocity $v_{\mathrm{impact}}$ for the impacts of particles with various diameters $D$ on silicone oil (left column) and water (right column) films with fixed thickness $\delta =1$ mm. The solid lines in the lower panels are linear fits to the data and their representatives are shown in the upper panels as a guide to the eyes. The dashed gray lines in the upper panels represent the normal restitution coefficient $e_{\mathrm{dry}}=0.976\pm 0.001$ for dry impacts, which is obtained by a linear fit of the data for all particle sizes [gray diamonds shown in (b)]. Error bars smaller than the symbol size are not shown.

Figure 5

Critical energy $E_{\mathrm{c}}$ and saturated value of COR $e_{\inf }$, obtained from linear fits in Fig. 4b, as a function of particle diameter $D$. The solid line at $E_{\mathrm{c}}=0$ is a guide to the eyes.

Figure 6

Normal restitution coefficient $e_n$ (a) and rebound velocity $v_{\mathrm{rebound}}$ (b) as a function of impact velocity $v_{\mathrm{impact}}$ for impacts of a glass bead with $D=5.5$ mm on dry and wet surfaces covered with silicone oil M5 and M50. $\delta$ denotes the film thickness. The error bars correspond to the statistical error over 10 runs of experiments for the wet impacts. Solid lines in (b) are linear fits to the corresponding data. Their representatives are shown in (a) as a guide to the eye. For dry impacts, the restitution coefficient $e_{\mathrm{dry}}$ is 0.985 with an error of 0.001.

Figure 7

Critical energy $E_{\mathrm{c}}$ and the saturated value of the COR $e_{\inf }$ as a function of film thickness $\delta$. $E_{\mathrm{c}}$ is obtained from the intersection $v_{\mathrm{c}}$ of the linear fits shown in the lower panel of Fig. 6 with the $x$ axis. $e_{\inf }$ corresponds to the slope of these fits.

Figure 8

Normal restitution coefficient $e_{\mathrm{n}}$ as a function of the Stokes number $\mathrm{St}$ for both silicone oil (a) and water (b) films. Parameters are the same as in Fig. 4.

Figure 9

Normal restitution coefficient as a function of Stokes number $\mathrm{St}$. Panel (a) corresponds to the data shown in Fig. 6. Panel (b) corresponds to the data with the dimensionless film thickness $\delta /D$ roughly constant. The other parameters are the same as in Fig. 6. The error bars correspond to the statistical error of 10 runs of experiments. The solid curve in panel (b) corresponds to the master curve $e_{\mathrm{n}}=0.908(1-14.00/\mathrm{St})$, which is obtained from a linear fit to all the data shown in the inset.

Figure 10

Energy dissipation $\Delta E_{\mathrm{wet}}$ due to various liquid films as a function of impact velocity with the liquid properties the same as in Fig. 6 for both silicone oil M50 (a) and M5 (b) films. The straight lines represent the estimated values of the energy dissipation $\Delta E_{\mathrm{visc}}$ from viscosity (see text for detailed descriptions).

Figure 11

Data points are the same as shown in Fig. 10. Various curves represent the estimated values with the consideration of both viscous damping $\Delta E_{\mathrm{visc}}$ and the energy transfer to the fluid $\Delta E_{\mathrm{acc}}$ (see text for detailed descriptions).