Shape oscillations of a spherical bubble or drop, for which part of its interface is fixed due to contact with a solid support, are studied analytically using variational methods. Linear oscillations and irrotational flow are assumed. The present analysis is parallel to those of Strani and Sabetta [“Free vibrations of a drop in partial contact with a solid support,” J. Fluid Mech. 141, 233–247 (1984)] https://doi.org/10.1017/S0022112084000811; and Bostwick and Steen [“Capillary oscillations of a constrained liquid drop,” Phys. Fluids 21, 032108 (2009)] https://doi.org/10.1063/1.3103344 but is also able to determine the response of bubbles or drops to movements imposed on their supports or to variations of their volumes. The analysis leads to equations of motion with a simple structure, from which the eigenmodes and frequency response to periodic forcing are easily determined.
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See supplementary material at http://dx.doi.org/10.1063/1.4810959 for more details of some principal steps of the analysis. They include some useful relations for Legendre polynomials, detailed derivations of the linearized volume constraint and of the constraint of Strani and Sabetta,21,22 calculations of the change in interfacial area, of the kinetic energy, and of the energy dissipation rate. Detailed expressions for the elements of matrices M, D, K and vectors f and fs are provided. The solution for the linearized static shape is given. Details about the eigenproblem evaluation are given.
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2013
AIP Publishing LLC
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