Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-07-03T08:35:12.016Z Has data issue: false hasContentIssue false
  • English
  • Français

Coupled oscillations of deformable spherical-cap droplets. Part 1. Inviscid motions

Published online by Cambridge University Press:  02 January 2013

J. B. Bostwick  and
P. H. Steen
J. B. Bostwick
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA Department of Theoretical & Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
P. H. Steen*
Affiliation:
Department of Theoretical & Applied Mechanics, Cornell University, Ithaca, NY 14853, USA School of Chemical and Biomolecular Engineering and Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA
*
†Email address for correspondence: phs7@cornell.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A spherical drop is constrained by a solid support arranged as a latitudinal belt. This belt support splits the drop into two deformable spherical caps. The edges of the support are given by lower and upper latitudes yielding a ‘spherical belt’ of prescribed extent and position: a two-parameter family of constraints. This is a belt-constrained Rayleigh drop. In this paper we study the linear oscillations of the two coupled spherical-cap surfaces in the inviscid case, and the viscous case is studied in Part 2 (Bostwick & Steen, J. Fluid Mech., vol. 714, 2013, pp. 336–360), restricting to deformations symmetric about the axis of constraint symmetry. The integro-differential boundary-value problem governing the interface deformation is formulated as a functional eigenvalue problem on linear operators and reduced to a truncated set of algebraic equations using a Rayleigh–Ritz procedure on a constrained function space. This formalism allows mode shapes with different contact angles at the edges of the solid support, as observed in experiment, and readily generalizes to accommodate viscous motions (Part 2). Eigenvalues are mapped in the plane of constraints to reveal where near-multiplicities occur. The full problem is then approximated as two coupled harmonic oscillators by introducing a volume-exchange constraint. The approximation yields eigenvalue crossings and allows post-identification of mass and spring constants for the oscillators.

JFM classification

Drops and Bubbles: Bubble dynamics Interfacial Flows (free surface): Capillary flows Drops and Bubbles: Drops
Type
Papers
Information
Journal of Fluid Mechanics , Volume 714 , 10 January 2013 , pp. 312 - 335
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arfken, G. B. & Weber, H. J. 2001 Mathematical Methods for Physicists. Harcourt Academic Press.Google Scholar
Basaran, O. & DePaoli, D. 1994 Nonlinear oscillations of pendant drops. Phys. Fluids 6, 2923–2943.Google Scholar
Bauer, H. F. & Chiba, M. 2004 Oscillations of captured spherical drop of frictionless liquid. J. Sound Vib. 274, 725–746.Google Scholar
Bauer, H. F. & Chiba, M. 2005 Oscillations of captured spherical drop of viscous liquid. J. Sound Vib. 285, 51–71.Google Scholar
Beck, J. L. 1972 Convection in a box of porous material saturated with fluid. Phys. Fluids 15, 1377–1383.CrossRefGoogle Scholar
Bhandar, A. S. & Steen, P. H. 2005 Liquid-bridge mediated droplet switch: a tristable capillary system. Phys. Fluids 17, 127107.Google Scholar
Bisch, C., Lasek, A. & Rodot, H. 1982 Compartement hydrodynamique de volumes liquides spheriques semi-libres en apesanteur simulee. J. Mec. Theor. Appl. 1, 165–184.Google Scholar
Bostwick, J. B. & Steen, P. H. 2009 Capillary oscillations of a constrained liquid drop. Phys. Fluids 21, 032108.Google Scholar
Bostwick, J. B. & Steen, P. H. 2010 Stability of constrained cylindrical interfaces and the torus lift of Plateau-Rayleigh. J. Fluid Mech. 647, 201–219.Google Scholar
Bostwick, J. B. & Steen, P. H. 2013 Coupled oscillations of deformable spherical-cap droplets. Part 2. Viscous motions. J. Fluid Mech 714, 336–360.Google Scholar
Brunet, P., Eggers, J. & Deegan, R. D. 2009 Motion of a drop driven by substrate vibrations. Eur. Phys. J. Special Topics 166, 11–14.Google Scholar
Charlson, G. S & Sani, R. L 1970 Thermoconvective instability in a bounded cylindrical fluid layer. Intl. J. Heat Mass Transfe 13, 1479–1496.Google Scholar
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics, Vol. I. Wiley-Interscience.Google Scholar
Daniel, S., Sircar, S., Gliem, J. & Chaudhury, M. K. 2004 Racheting motion of liquid drops on gradient surfaces. Langmuir 20, 4085–4092.Google Scholar
DePaoli, W. D., Feng, J. Q., Basaran, O. A. & Scott, T. C. 1995 Hysteresis in forced oscillations of pendant drops. Phys. Fluids 7, 1181–1183.Google Scholar
Fayzrakhmanova, I. S. & Straube, A. V. 2009 Stick-slip dynamics of an oscillated sessile drop. Phys. Fluids 21, 072104.Google Scholar
Gañan, A. & Barerro, A. 1990 Free oscillations of liquid captive drops. Microgravity Sci. Technol. 3, 70–86.Google Scholar
Hammack, J. L. & Henderson, D. M. 1993 Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25, 55–97.Google Scholar
Hirsa, A. H., Lopez, C. A., Laytin, M. A., Vogel, M. J. & Steen, P. H. 2005 Low-dissipation capillary switches at small scales. Appl. Phys. Lett. 86, 014106.Google Scholar
James, A., Smith, M. K. & Glezer, A. 2003a Vibration-induced drop atomization and the numerical simulation of low-frequency single-droplet ejection. J. Fluid Mech. 476, 29–62.Google Scholar
James, A., Vukasinovic, B., Smith, M. K. & Glezer, A. 2003b Vibration-induced drop atomization and bursting. J. Fluid Mech. 476, 1–28.Google Scholar
Lopez, C. A. & Hirsa, A. H. 2008 Fast focusing using a pinned-contact liquid lens. Nat. Photonics 2 9, 610–613.Google Scholar
Lopez, C. A., Lee, C. C. & Hirsa, A. H. 2005 Electrochemically activated adaptive liquid lens. Appl. Phys. Lett. 87, 134102.CrossRefGoogle Scholar
Lyubimov, D. V., Lyubimova, T. P. & Shklyaev, S. V. 2004 Non-axisymmetric oscillations of a hemispheric drop. Fluid Dyn. 39, 851–862.CrossRefGoogle Scholar
Lyubimov, D. V., Lyubimova, T. P. & Shklyaev, S. V. 2006 Behavior of a drop on an oscillating solid plate. Phys. Fluids 18, 012101.Google Scholar
Malouin, B. A., Vogel, M. J. & Hirsa, A. H. 2010 Electromagnetic control of coupled droplets. Appl. Phys. Lett. 96, 214104.Google Scholar
Noblin, X., Buguin, A. & Brochard-Wyart, F. 2004 Vibrated sessile drops: transition between pinned and mobile contact lines. Eur. Phys. J. E 14, 395–404.Google Scholar
Noblin, X., Buguin, A. & Brochard-Wyart, F. 2005 Triplon modes of puddles. Phys. Rev. Lett. 94, 166102.Google Scholar
Noblin, X., Kofman, R. & Celestini, F. 2009 Ratchet-like motion of a shaken drop. Phys. Rev. Lett. 19, 194504.Google Scholar
Olles, J. D., Vogel, M. J., Malouin, B. A. & Hirsa, A. H. 2011 Axisymmetric oscillation modes of a double droplet system. Optics Express 19, 19399–19406.Google Scholar
Prosperetti, A 2012 Linear oscillations of constrained drops, bubbles and plane liquid surfaces. Phys. Fluids 24, 032109.Google Scholar
Ramalingam, S. K. & Basaran, O. A. 2010 Axisymmetric oscillation modes of a double droplet system. Phys. Fluids 22, 112111.CrossRefGoogle Scholar
Ramalingam, S., Ramkrishna, D. & Basaran, O. A. 2012 Free vibrations of a spherical drop constrained at an azimuth. Phys. Fluids 24 (8), 082102.CrossRefGoogle Scholar
Rayleigh, Lord 1879 On the capillary phenomenon of jets. Proc. R. Soc. Lond. 29, 71–97.Google Scholar
Rodot, H. A. & Bisch, C. 1984 Oscillations de volumes liquides semi-libres en microgravite-experience es326 dans spacelab 1. 5th European Symp. on Material Sciences under Microgravity, Paper ESA SP-222, pp. 23–29.Google Scholar
Segel, L. A. 1987 Mathematics Applied to Continuum Mechanics. Dover.Google Scholar
Strani, M. & Sabetta, F. 1984 Free vibrations of a drop in partial contact with a solid support. J. Fluid Mech. 141, 233–247.Google Scholar
Strani, M. & Sabetta, F. 1988 Viscous oscillations of a supported drop in an immiscible fluid. J. Fluid Mech. 189, 397–421.Google Scholar
Theisen, E. A., Vogel, M. J., Hirsa, C. A., Lopez, A. H. & Steen, P. H. 2007 Capillary dynamics of coupled spherical-cap droplets. J. Fluid Mech. 580, 495–505.Google Scholar
Trinh, E. & Wang, T. G. 1982 Large-amplitude free and driven drop-shape oscillation: experimental results. J. Fluid Mech. 122, 315–338.Google Scholar
Trinh, E., Zwern, A. & Wang, T. G. 1982 An experimental study of small-amplitude drop oscillations in immiscible liquid system. J. Fluid Mech. 115, 453–474.CrossRefGoogle Scholar
Tsamopoulos, J. A. & Brown, R. A. 1983 Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519–537.Google Scholar
Vogel, M. J., Ehrhard, P. & Steen, P. H. 2005 The electroosmotic droplet switch: Countering capillarity with electrokinetics. Proc. Natl Acad. Sci. 102, 11974–11979.Google Scholar
Vogel, M. J. & Steen, P. H. 2010 Capillarity-based switchable adhesion. Proc. Natl Acad. Sci. 107, 3377–3381.Google Scholar
Vukasinovic, B., Smith, M. K. & Glezer, A. 2007 Dynamics of a sessile drop in forced vibration. J. Fluid Mech. 587, 395–423.CrossRefGoogle Scholar
Wang, T. G., Anilkumar, A. V. & Lee, C. P. 1996 Oscillations of liquid drops: results from usml-1 experiments in space. J. Fluid Mech. 308, 1–14.Google Scholar
Wilkes, E. & Basaran, O. 1999 Hysteretic response of supported drops during forced oscillations. J. Fluid Mech. 393, 333–356.Google Scholar
Wilkes, E. & Basaran, O. 2001 Drop ejection from an oscillating rod. J. Colloid Interface Sci. 242, 180–201.Google Scholar