Dissipative descent: rocking and rolling down an incline

N. J. BALMFORTH, J. W. M. BUSH, D. VENER, W. R. YOUNG
2007 Journal of Fluid Mechanics  
We consider the dynamics of a hollow cylindrical shell that is filled with viscous fluid and another, nested solid cylinder, and allowed to roll down an inclined plane. A mathematical model is compared to simple experiments. Two types of behaviour are observed experimentally: on steeper slopes, the device accelerates; on shallower inclines, the cylinders rock and roll unsteadily downhill, with a speed that is constant on average. The theory also predicts runaway and unsteady rolling motions.
more » ... the rolling solutions, however, the inner cylinder cannot be suspended in the fluid by the motion of the outer cylinder, and instead falls inexorably toward the outer cylinder. Whilst 'contact' only occurs after an infinite time, the system slows progressively as the gap between the cylinders narrows, owing to heightened viscous dissipation. Such a deceleration is not observed in the experiments, suggesting that some mechanism limits the approach to contact. Coating the surface of the inner cylinder with sandpaper of different grades changes the rolling speed, consistent with the notion that surface roughness is responsible for limiting the acceleration. † The magic trick is marketed with the explanation: "A small, metallic gold ball just over 2 cm in diameter . . . this ball does roll, but it does so incredibly slowly. To an audience, it seems baffling why it should roll down a slope apparently in slow motion. You can pick the ball up, and it seems heavy, possibly solid. No clues if you shake it. However inside the ball, which is actually hollow, there is a viscous liquid and a smaller ball which is very heavy. When the Snail Ball rolls slowly down an incline, it is the smaller, heavier ball inside that determines the pace, and this is slow because of the viscous liquid."
doi:10.1017/s0022112007008051 fatcat:opvwfumdirdslfb3bbqox6pwru