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Flow and instability of thin films on a cylinder and sphere

Published online by Cambridge University Press:  18 March 2010

DAISUKE TAKAGI  and
HERBERT E. HUPPERT
DAISUKE TAKAGI
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
HERBERT E. HUPPERT*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the dynamics of thin films driven by gravity on the outer surface of a cylinder and sphere. The surface is rigid, stationary and the axis of the cylinder is horizontal. An instantaneous release of a constant volume of fluid at the top of the cylinder or sphere results initially in a two-dimensional or axisymmetric current respectively. The resultant flow of a thin film of fluid is described using lubrication theory when gravity and viscous forces govern the dynamics. We show that the thickness of the flow remains uniform in space and decreases in time like t−1/2 near the top of both the cylinder and the sphere. Analytic solutions for the extent of the flow agree well with our experiments until the advancing front splits into a series of rivulets. We discuss scalings of the flow at the onset of the instability as a function of the Bond number, which characterizes the relative importance of gravity and surface tension. The experiments, conducted within an intermediate range of Bond numbers, suggest that the advancing front becomes unstable after it has propagated a critical distance, which depends primarily and monotonically on the volume of fluid and not on the viscosity of fluid. Releasing a sufficiently large volume of fluid ensures that rivulets do not develop on either a cylinder or sphere.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 647: Dedicated to Professor S. H. Davis on his 70th birthday , 25 March 2010 , pp. 221 - 238
Copyright
Copyright © Cambridge University Press 2010

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