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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: phs7@cornell.edu
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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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References

REFERENCES

Basaran, O. & DePaoli, D. 1994 Nonlinear oscillations of pendant drops. Phys. Fluids 6, 2923–2943.CrossRefGoogle Scholar
Bostwick, J. B. & Steen, P. H. 2009 Capillary oscillations of a constrained liquid drop. Phys. Fluids 21, 032108.CrossRefGoogle Scholar
Brown, R. A. & Scriven, L. E. 1980 On the multiple equilibrium shapes and stability of an interface pinned on a slot. J. Colloid Interface Sci. 78, 528–542.Google Scholar
Byrne, C. J., Theisen, E. A., Reed, B. L. & Steen, P. H. 2006 Capillary puddle vibrations linked to casting-defect formation in planar-flow melt spinning. Metall. Mater. Trans. B 37B, 445–456.CrossRefGoogle Scholar
Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98, 225–242.CrossRefGoogle Scholar
Deegan, R. D., Brunet, P. & Eggers, J. 2008 Complexities of splashing. Nonlinearity 21, C1–C11.Google Scholar
Gillette, R. D. & Dyson, D. C. 1972 Stability of axisymmetric liquid-fluid interfaces towards general disturbances. Chem. Engng J. 3, 196–199.Google Scholar
Kreyszig, E. 1991 Differential Geometry. Dover.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Langbein, D. 1990 The shape and stability of liquid menisci at solid edges. J. Fluid Mech. 213, 251–265.Google Scholar
Lowry, B. J. & Thiessen, D. B. 2007 Fixed contact line helical interfaces in zero gravity. Phys. Fluids 19, 022102.Google Scholar
Lyubimov, D. V., Lyubimova, T. P. & Shklyaev, S. V. 2006 Behaviour of a drop on an oscillating solid plate. Phys. Fluids 18, 012101.CrossRefGoogle Scholar
Majumbar, S. R. & Michael, D. H. 1976 The equilibrium and stability of two-dimensional pendent drop. Proc. R. Soc. Lond. A 351, 89–115.Google Scholar
May, H. J. & Lowry, B. J. 2008 Microgravity/microscale double-helical fluid containment. Proc. R. Soc. A 464, 855–875.Google Scholar
Michael, D. H. 1981 Meniscus stability. Annu. Rev. Fluid Mech. 13, 189–215.Google Scholar
Myshkis, A. D., Babskii, V. G., Slobozhanin, N. D. & Tyuptsov, A. D. 1987 Low-Gravity Fluid Mechanics. Springer.Google Scholar
Plateau, J. A. F. 1863 Experimental and theoretical researches on the figures on equilibrium of a liquid mass withdrawn from the action of gravity. A. Rep. Board Reg. Smithson. Inst. pp. 207–285.Google Scholar
Rayleigh, Lord 1879 On the capillary phenomenon of jets. Proc. R. Soc. Lond. 29, 71–97.Google Scholar
Renardy, Y., Popinet, S., Duchemin, L., Renardy, M., Zaleski, S., Josserand, C., Drumright-Clarke, M. A., Richard, D., Clanet, C. & Quèrè, D. 2003 Pyramidal and toroidal water drops after impact on a solid surface. J. Fluid Mech. 484, 69–83.CrossRefGoogle Scholar
Roy, R. V. & Schwartz, L. W. 1999 On the stability of liquid ridges. J. Fluid Mech. 391, 293–318.CrossRefGoogle Scholar
Russo, M. J. & Steen, P. H. 1986 Instability of rotund capillary bridges to general disturbances: experiment and theory. J. Colloid Interface Sci. 113, 154–163.CrossRefGoogle Scholar
Segel, L. A. 1987 Mathematics Applied to Continuum Mechanics. Dover.Google Scholar
Slobozhanin, L., Alexander, J. I. D. & Resnick, A. 1997 Bifurcation of the equilibrium states of a weightless liquid bridge. Phys. Fluids 9, 1893–1905.CrossRefGoogle Scholar
Steen, P. H. & Karcher, C. 1997 Fluid mechanics of spin casting. Annu. Rev. Fluid Mech. 29, 373–397.CrossRefGoogle Scholar
Steiner, J. 1882 Gesammelte Werke. Reimer.Google Scholar
Strani, M. & Sabetta, F. 1984 Free vibrations of a drop in partial contact with a solid support. J. Fluid Mech. 141, 233.CrossRefGoogle Scholar
Thiessen, D. B., Bernate, J. & Marston, P. L. 2005 Capillary-driven flow in supported capillary pipes. AIChE Annual Meeting. Oct. 30–Nov. 4, 2005, Cincinnati, OH (paper 124f).Google Scholar
Weiland, R. H. & Davis, S. H. 1981 Moving contact lines and rivulet instabilities. Part 2. Long waves on flat rivulets. J. Fluid Mech. 107, 261–280.CrossRefGoogle Scholar