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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
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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

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 714 , 10 January 2013 , pp. 312 - 335
Copyright
©2013 Cambridge University Press

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