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Singularities in water waves and the Rayleigh–Taylor problem

Published online by Cambridge University Press:  30 April 2010

M. A. FONTELOS  and
F. DE LA HOZ
M. A. FONTELOS*
Affiliation:
Instituto de Ciencias Matemáticas (ICMAT, CSIC-UAM-UCM-UC3M), C/ Serrano 123, 28006 Madrid, Spain
F. DE LA HOZ
Affiliation:
Departamento de Matemática Aplicada, Escuela Universitaria de Ingeniería Técnica Industrial, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Plaza de la Casilla 3, 48012 Bilbao, Spain
*
†Email address for correspondence: marco.fontelos@uam.es
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Abstract

We describe, by means of asymptotic methods and direct numerical simulation, the structure of singularities developing at the interface between two perfect, inviscid and irrotational fluids of different densities ρ1 and ρ2 and under the action of gravity. When the lighter fluid is on top of the heavier fluid, one encounters the water-wave problem for fluids of different densities. In the limit when the density of the lighter fluid is zero, one encounters the classical water-wave problem. Analogously, when the heavier fluid is on top of the lighter fluid, one encounters the Rayleigh–Taylor problem for fluids of different densities, with this being the case when one of the densities is zero for the classical Rayleigh–Taylor problem. We will show that both water-wave and Rayleigh–Taylor problems develop singularities of the Moore-type (singularities in the curvature) when both fluid densities are non-zero. For the classical water-wave problem, we propose and provide evidence of the development of a singularity in the form of a logarithmic spiral, and for the classical Rayleigh–Taylor problem no singularities were found. The regularizing effects of surface tension are also discussed, and estimates of the size and wavelength of the capillary waves, bubbles or blobs that are produced are provided.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 651 , 25 May 2010 , pp. 211 - 239
Copyright
Copyright © Cambridge University Press 2010

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