Rolling stones: The motion of a sphere down an inclined plane coated with a thin liquid film

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Fall 2010 entry - Anna Wang

References

J. Bico, J. Ashmore-Chakrabarty, G. H. McKinley, and H. A. Stone, "Rolling stones: The motion of a sphere down an inclined plane coated with a thin liquid film," Physics of Fluids 21 (8) (2009)

Introduction

The motion of a solid sphere on an inclined plane was famously studied by Galileo Galilei in 1602. More recently, the same experiment - but completely submerged in fluid - has been used to help understand many phenomena ranging from lubrication to the flow of vesicles. Bico et al examine the intermediate configuration, where the solid sphere rolls down an inclined plane which has been lubricated with a thin layer of viscous liquid.

Experimental setup

Figure 1. (a) Schematic of setup and the relevant parameters. (b) Photograph of a typical experiment.

The experimental apparatus consists of an inclined glass plate (40cm long, incline angle α) which has been carefully coated in 50-500μm of oil (density ρf, surface tension σ, viscosity μ, thickness h) by blade, and high precision ball bearings of varying densities ρs and radii a (Figure 1). The rotation and translation of the ball bearings is monitored for various incline angles α.

Characterisation of the system

Figure 2. The linear and rotational velocities appear to stay steady throughout this typical experiment.

By extracting the translational (V) and rotational velocities (Ω) from the positional data (Figure 2), sliding coefficient aΩ/V is calculated to be <1; ie the spheres simultaneous rotate and slide down the incline, and there is no solid-solid friction between the ball bearing and the incline.

Figure 3. Spheres of different density and radii occupy different positions. The white arrow shows a sphere in an overhang position.

A complimentary experiment was performed where the ball bearings were placed on the inner face of a rotating drum, also coated in viscous liquid (Figure 3). As the ball bearing rolls in its own track in the drum rather than on a ‘fresh’ layer of oil down the plane the situations are not entirely comparable, but the steadiness of the rolling/sliding phenomenon was demonstrated as even the balls in overhang position retained their position for several hours.


Results and conclusions

Three different regimes

Figure 4. Three different regimes were observed as the slope α was increased. (a) the non-dimensional velocity or capillary number capillary length <math>Ca</math> = µV/σ is plotted against inclination for two different ball bearings (small and large symbols). (b) the view from underneath the glass shows a circular meniscus for <math>Ca</math> < 1, cusp shape for <math>Ca</math> > 1 and an overhang case (α > 90̊) showing a single ridge.

The expected increase in sphere velocity with inclination angle exhibited three different regimes as shown in Figure 3. The transition from having a circular meniscus around the ball bearing to having a cusp shape always occurred for <math>Ca</math> ~ 1, indicating a strong interplay between viscous and capillary stresses in the meniscus at the transition. The overhang regime transition was exhibited when α > 90̊.

Dependence on physical parameters

Figure 5. Analysis collapses the experimental data (α < 90̊). Results are for experiments conducted with silicone oil inless otherwise specified. Inset: <math>Ca</math> vs radius for castor oil experiments.
Figure 6. The empirical scaling relationship for the overhang regime.

Different empirical power-law dependencies on various non-dimensional groups were combined to show the scaling law in Figure 5 for experiments in the first two regimes. For the overhang regime, the ratio <math>Ca</math>/<math>Bo</math>1.6 was found to scale linearly with inclination angle. <math>Bo</math> = ρsga2/σ, comparing the capillary adhesion force to the weight of the sphere. The rotation of the spheres appears to be critical to their adhesion - a viscous restoring force from the steady detachment in the rear part of the meniscus may be the mechanism for the dynamic adhesion of spheres.

Sliding vs rotation

Figure 7. Sliding coefficient aΩ/V as a function of <math>Ca</math>.

The sliding ratio is close to 1 for lower velocities, and erratic motion revealed that some solid-solid friction occurs between the sphere and the wall in this regime. When <math>Ca</math> > 1, the sliding coefficient remained steady ~0.6, and increased slightly with ratio a/h, in accordance with intuition that a thicker lubricating layer increases the sliding.

Comments

The complexity of the three-dimensional free-surface flow prevented the authors from providing theoretical insights into these scaling laws and result. Future numerical simulations may be of use.