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Stability of constrained cylindrical interfaces and the torus lift of Plateau–Rayleigh

Published online by Cambridge University Press:  18 March 2010

J. B. BOSTWICK  and
P. H. STEEN
J. B. BOSTWICK
Affiliation:
Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
P. H. STEEN*
Affiliation:
School of Chemical and Biomolecular Engineering and Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA
*
†Email address for correspondence: [email protected]
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Abstract

Surface tension acting at a cylindrical interface holds an underlying liquid in motionless equilibrium. This static base state is subject to dynamic capillary instability, including Plateau–Rayleigh breakup. If the interface is partially supported by a cylindrical cup-like solid, the extent of the wetting contact can significantly influence the dynamics and the stability of the configuration. The equation for the motion of small disturbances is formulated as an eigenvalue equation on linear operators. A solution is constructed on a constrained function space using a Rayleigh–Ritz procedure. The influence of the extent-of-constraint on the dispersion relation and on modal structures is reported. In the extreme, the support reduces to a wire, aligned axially, and just touching the interface. From prior work, this constraint is known to stabilize the Plateau–Rayleigh limit by some 13%. We report the wavenumber of maximum growth and estimate the time to breakup. The constraint is then bent in-plane to add a weak secondary curvature to the now nearly cylindrical base state. This is referred to as the torus lift of the cylinder. The static stability of these toroidal equilibria, calculated using a perturbation approach, shows that the position of constraint is crucial – constraint can stabilize (outside) or destabilize (inside). The combined influence of secondary curvature and wire constraint on the Plateau–Rayleigh limit is tracked. Finally, attention is restricted to constraints that yield a lens-like cylindrical meniscus. For these lenses, the torus lift is used as apparatus along with a symmetrization procedure to prove a large-amplitude static stability result. Our study is conveniently framed by a classic paper on rivulets by Davis (J. Fluid Mech., vol. 98, 1980, p. 225).

Type
Papers
Information
Journal of Fluid Mechanics , Volume 647: Dedicated to Professor S. H. Davis on his 70th birthday , 25 March 2010 , pp. 201 - 219
Copyright
Copyright © Cambridge University Press 2010

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