Solid-on-solid contact in a sphere-wall collision in a viscous fluid

We study experimentally the collision between a sphere falling through a viscous fluid and a solid plate below. It is known that there is a well-defined threshold Stokes number above which the sphere rebounds from such a collision. Our experiment tests for direct contact between the colliding bodies and, contrary to prior theoretical predictions, shows that solid-on-solid contact occurs even for Stokes numbers just above the threshold for rebounding. The dissipation is fluid dominated, though details of the contact mechanics depend on the surface and bulk properties of the solids. Our experiments and a model calculation indicate that mechanical contact between the two colliding objects is generic and will occur for any realistic surface roughness.

Figure 1

(a) Schematic diagram of the experimental setup. The voltage is measured across a resistor $R_s$ of $220\mathrm{k}\mathrm{\Omega }$. (b) Sphere approaching a plane, where $r$ and $z$ are the radial and vertical directions, $D=2R$ is the diameter of the sphere, $g$ is the acceleration due to gravity, and $h(r,t)$ is the time-dependent distance between the sphere and the bottom wall.

Figure 2

Probability density $P\left(\xi \right)$ of roughness of spheres as measured with a contact profilometer. To obtain the roughness $\xi$, the path traced by profilometer as a function of distance has been subtracted from the gross curvature of the sphere. The inset shows typical profiles $\xi \left(s\right)$ of roughness as a function of the position $s$ along the surface, for all three spheres used.

Figure 3

Voltage versus time graphs for increasing $\text{St}$ with ac and dc voltages. (a) At $\text{St}=5.8$ with an ac applied voltage, no bounce occurs, but electrical contact is made in a finite time. This is the generic behavior for $\text{St}<{\text{St}}_c$. (b) At $\text{St}=7.1$ and (c) at $\text{St}=6.7$ with dc and ac voltages, respectively, we see a collision, and then a period of no contact, followed by permanent contact. The contact has much lower resistance than the series resistance, i.e., $R_c\ll R_s$. This is the generic behavior for $\text{St}>{\text{St}}_c$. For $\text{St}>{\text{St}}_c$, we also see the case shown in (d) where there is a high-resistance contact $(R_c$ is of the same order as $R_s)$ (more details are in Appendix pp2).

Figure 4

Contact fraction $\varphi$, contact time $\delta T_c$, and flight time $T_F$ vs Stokes number for the 15.4-mm etched ball and 16-mm and 12-mm unetched balls. Here we report data only for low-resistance contacts $\left(R_c\ll R_s\right)$ as in Figs. 3 and 3, however the trends are unaffected if we include more resistive contacts (see Appendix pp2) as in Fig. 3. Error bars are the standard deviation of measurements in (a) and standard error of measurements in (b) and (c) taken at fixed $\text{St}$. Wherever the error bars are not visible, errors are smaller than the symbol size.

Figure 5

Flight time vs contact time for $D=16\mathrm{mm}$ unetched and 12-mm and 15.4-mm etched balls. Notice the increase in the scatter at lower $\text{St}$.

Figure 6

Velocity $\dot{h}$ versus height $h_0$ of a rigid smooth sphere approaching a rigid plane calculated from the equation of motion (3). The results shown are for a steel sphere of diameter 15.4 mm, dropped from different heights, yielding the labeled nominal values of $\text{St}$. The solid curves are obtained by using Eqs. (4) and (5) to get the net pressure force and then numerically solving Eq. (3). The curves for various Stokes numbers cross a given height with different velocities; a typical example is shown by the vertical dash-dotted line at $h_0=100$ nm. At this height, the velocity for the curve labeled $\text{St}=12.25$ is three orders of magnitude higher than those for the curves shown for lower $\text{St}$. This implies that even for extremely smooth surfaces, a physical contact will occur prior to bouncing for a high enough Stokes number. The dashed curves at $\text{St}=25.8$ and $\text{St}=5.4$ are obtained by approximating the sphere as a paraboloid near $r=0$ to compare with previous calculations in DSH86 = Ref. [2] that use this approximation.

Figure 7

A fit for the 15.4-mm etched sphere on the path traced by the profilometer as a function of position along a line on the sphere's surface is shown in red. The fit gives us the global curvature of the path and the local roughness of the sphere is obtained by subtracting the surface profile from this fit. The roughness $\xi$ is shown in blue in the inset.

Figure 8

(a) Low-resistance contact and (b) high-resistance contact at 50 mV, 100 kHz, and $\text{St}=19.7$. (c) and (d) are the zoomed-in versions of contacts shown in (a) and (b), respectively. The voltage shown in black is measured by a digital oscilloscope across a 220-$\mathrm{k}\mathrm{\Omega }$ resistor. In (a) and (c) the output voltage is equal to the source voltage (shown in red) of 50 mV, suggesting that there is no voltage drop across the contact and the contact has very low resistance. However, in (b) and (d) the output voltage is lower than the source voltage, suggesting that some contacts obtained during the experiments have high resistance. Notice the zero phase shift in both cases.

Figure 9

There appears to be no systematic effect of the electrical voltage on (b) the duration of contact or (c) the intervals between bounces. This experiment was carried out at 300 kHz for $\text{St}=19.7$. (a) The total measured contact fraction remains close to one. The fraction of low-resistance contacts (black squares) is lower. Error bars are the standard deviation of measurements in (a) and standard error of measurements in (b) and (c) taken at fixed Stokes numbers. Wherever the error bars are not visible, errors are smaller than the symbol size.

Figure 10

Variation in flight time with frequency. No systematic trend was observed and the contact fraction was found to be equal to one for all values of frequencies.

Figure 11

Craters visible inside the marked circle. The size of the crater on the plate is around $700\mu \mathrm{m}$ for $\text{St}=28$.