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Dissipative descent: rocking and rolling down an incline

Published online by Cambridge University Press:  15 October 2007

N. J. BALMFORTH ,
J. W. M. BUSH ,
D. VENER  and
W. R. YOUNG
N. J. BALMFORTH
Affiliation:
Departments of Mathematics and Earth & Ocean Science, UBC, Vancouver BC, V6K 1Z2, Canada
J. W. M. BUSH
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA
D. VENER
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA
W. R. YOUNG
Affiliation:
Scripps Institution of Oceanography, UCSD, La Jolla, CA 92093-0213, USA
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Abstract

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. For 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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 590 , 10 November 2007 , pp. 295 - 318
Copyright
Copyright © Cambridge University Press 2007

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References

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