Q. Jl Mech. Appl. Math. (2005) 58 (2), 229–256 © Oxford University Press 2005; all rights reserved.
On three-dimensional stability of a uniform, rigidly rotating film on a rotating cylinder
( Division of Applied Mathematics, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD )
We study the linear stability of rigidly rotating films on a rotating circular cylinder in the absence of gravity. Three-dimensional disturbances are examined, particularly in the limit in which the liquid volume fraction is small, it being assumed throughout that there is a continuous film coating the surface of the cylinder.
The most unstable mode for thin film flows on the inside surface of a cylinder, that is, rimming flows, is shown to be purely axial (ring instability), and the critical wave number is identified for given values of the reduced Reynolds number and reciprocal Weber number S. For thin film flows on the outside surface of a cylinder, that is, coating flows, the critical wave number (that with maximal growth rate) is also identified for small reduced Reynolds numbers. For S > 1, the disturbance is purely axial (ring instability). For 0 < S < 1, however, the most unstable disturbance is shown to be either a purely axial or a purely azimuthal mode. Numerical results suggest that a stripe instability (azimuthal mode) is likely to occur. While agreeing with some experimental evidence, this result conflicts with results from others, suggesting that effects ignored in this study, gravity for example, may play a significant role in mode selection.
Received 23 July 2004. Revise 23 December 2004.
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