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Stability of symmetric film-splitting between counter-rotating cylinders

Published online by Cambridge University Press:  26 April 2006

D. J. Coyle ,
C. W. Macosko  and
L. E. Scriven
D. J. Coyle
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA General Electric Company, Corporate Research and Development, Schenectady, NY 12301, USA.
C. W. Macosko
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
L. E. Scriven
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
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Abstract

The ribbing instability, an extremely common cause of non-uniform liquid films in coating operations, is investigated both theoretically and experimentally. The Navier–Stokes system for the two-dimensional flow in symmetric film-splitting in forward roll coating is solved by finite-element analysis. Stability of the flow with respect to three-dimensional disturbances is examined by applying linear stability theory in a consistent finite-element approach, taking Fourier components in the transverse direction. The resulting generalized asymmetric eigenproblem is solved for the growth rates of disturbances as functions of wavenumber. The theory accurately predicts the critical capillary number and wavenumber at the transition to large-amplitude ribs. A sensitive experimental technique for detecting the ribs was developed that relies on low-angle reflection of a focused strip of white light off the meniscus between the rolls. This allowed detection of much smaller amplitude ribs, and much smaller critical capillary numbers were measured. The results indicate that the transition to ribbing is an imperfect bifurcation due to end effects, and clarify earlier discordances in the literature.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 216 , July 1990 , pp. 437 - 458
Copyright
© 1990 Cambridge University Press

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