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Third-order resonance effects and the nonlinear stability of drop oscillations

Published online by Cambridge University Press:  21 April 2006

Ramesh Natarajan  and
Robert A. Brown
Ramesh Natarajan
Affiliation:
Department of Chemical Engineering and Materials Processing Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Robert A. Brown
Affiliation:
Department of Chemical Engineering and Materials Processing Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

The three-dimensional nonlinear oscillations of an isolated, inviscid drop with surface tension are studied by a multiple timescale analysis and pre-averaging applied to the variational principle for the appropriate Lagrangian. Amplitude equations are derived which describe the generic cubic resonance caused by the spatial degeneracy of the eigenfrequencies of the linear normal modes. This resonant coupling leads to the instability of the finite amplitude axisymmetric oscillations to small non-axisymmetric perturbations, as is demonstrated here for the three-and four-lobed normal modes. Solutions to the interaction equations that describe finite amplitude, non-axisymmetric travelling-wave solutions are also obtained and their stability is investigated. A non-generic cubic resonance between the two-lobed and four-lobed oscillatory modes leads to quasi-periodic motions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 183 , October 1987 , pp. 95 - 121
Copyright
© 1987 Cambridge University Press

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