Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-01T04:02:16.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Coupled oscillations of deformable spherical-cap droplets. Part 2. Viscous 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: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A spherical drop is constrained by a solid support arranged as a latitudinal belt. The spherical belt splits the drop into two deformable spherical caps. The edges of the belt support are given by lower and upper latitudes, yielding a support of prescribed extent and position: a two-parameter family of geometrical constraints. In this paper we study the linear oscillations of the two coupled surfaces in the viscous case, the inviscid case having been dealt with in Part 1 (Bostwick & Steen, J. Fluid Mech., vol. 714, 2013, pp. 312–335), restricting to axisymmetric disturbances. For the viscous case, limiting geometries are the spherical-bowl constraint of Strani & Sabetta (J. Fluid Mech., vol. 189, 1988, pp. 397–421) and free viscous drop of Prosperetti (J. Méc., vol. 19, 1980b, pp. 149–182). In this paper, a boundary-integral approach leads to an integro-differential boundary-value problem governing the interface disturbances, where the constraint is incorporated into the function space. Viscous effects arise due to relative internal motions and to the no-slip boundary condition on the support surface. No-slip is incorporated using a modified set of shear boundary conditions. The eigenvalue problem is then reduced to a truncated set of algebraic equations using a spectral method in the standard way. Limiting cases recover literature results to validate the proposed modification. Complex frequencies, as they depend upon the viscosity parameter and the support geometry, are reported for both the drop and bubble cases. Finally, for the drop, an approximate boundary between over- and under-damped motions is mapped over the constraint parameter plane.

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. 336 - 360
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. 1992 Nonlinear oscillations of viscous liquid drops. J. Fluid Mech. 241, 169–198.Google Scholar
Basaran, O. & DePaoli, D. 1994 Nonlinear oscillations of pendant drops. Phys. Fluids 6, 2923–2943.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bauer, H. F. & Chiba, M. 2004 Oscillations of captured spherical drop of frictionless liquid. J. Sound Vib. 274, 725–746.CrossRefGoogle Scholar
Bauer, H. F. & Chiba, M. 2005 Oscillations of captured spherical drop of viscous liquid. J. Sound Vib. 285, 51–71.Google 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 1. Inviscid motions. J. Fluid Mech 714, 312–335.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98, 225–242.Google Scholar
Hirsa, A. H., López, 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.CrossRefGoogle 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
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
López, C. A. & Hirsa, A. H. 2008 Fast focusing using a pinned-contact liquid lens. Nat. Photonics 2 9, 610–613.Google Scholar
López, C. A., Lee, C. C. & Hirsa, A. H. 2005 Electrochemically activated adaptive liquid lens. Appl. Phys. Lett. 87, 134102.Google Scholar
MacRobert, T. M. 1967 Spherical Harmonics. Pergamon.Google Scholar
Malouin, B. A., Vogel, M. J. & Hirsa, A. H. 2010 Electromagnetic control of coupled droplets. Appl. Phys. Lett. 96, 214104.Google Scholar
Maxwell, J. 1898 Encylopedia Britannica, 9th edn. Adam and Charles Black.Google Scholar
Miller, C. A. & Scriven, L. E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 417–435.Google Scholar
Olles, J. D., Vogel, M. J., Malouin, B. A. & Hirsa, A. H. 2011 Axisymmetric oscillation modes of a double droplet system. Opt. Express 19, 19399–19406.Google Scholar
Padrino, J. C., Funada, T. & Joseph, D. D. 2007 Purely irrotational theories for the viscous effects on the oscillations of drops and bubbles. Intl J. Multiphase Flow 34, 61–75.Google Scholar
Prosperetti, A. 1980a Free oscillations of drops and bubbles: the initial-value problem. J. Fluid Mech. 100, 333–347.Google Scholar
Prosperetti, A. 1980b Normal-mode analysis for the oscillations of a viscous liquid drop in an immiscible liquid. J. Méc. 19, 149–182.Google Scholar
Prosperetti, A. 1980c Viscous effects on perturbed spherical flows. Q. J. Appl. Maths 34, 339–352.Google Scholar
Ramalingam, S. K. & Basaran, O. A. 2010 Axisymmetric oscillation modes of a double droplet system. Phys. Fluids 22, 112111.Google Scholar
Rayleigh, Lord 1879 On the capillary phenomenon of jets. Proc. R. Soc. Lond. 29, 71–97.Google Scholar
Reid, W. H. 1960 The oscillations of a viscous liquid drop. Q. J. Appl. Maths 18, 86–89.Google Scholar
Rodot, H. & 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
Rodot, H., Bisch, C. & Lasek, A. 1979 Zero-gravity simulation of liquids in contact with a solid surface. Acta. Astronaut 6, 1083–1092.Google Scholar
Segner, J. A. 1751 Comment. Soc. Reg. Goetting.Google Scholar
Slater, D. M., López, C. A., Hirsa, A. H. & Steen, P. H. 2008 Chaotic motions of a forced droplet-droplet oscillator. Phys. Fluids 20, 092107.Google Scholar
Smith, W. R. 2010 Modulation equations for strongly nonlinear oscillations of an incompressible viscous drop. J. Fluid Mech. 654, 141–159.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.CrossRefGoogle Scholar
Theisen, E. A., Vogel, M. J., Hirsa, C. A., López, 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 systems. J. Fluid Mech. 115, 453–474.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.Google 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