Singularities in water waves and the Rayleigh–Taylor problem

• Published in: Journal of fluid mechanics Vol. 651; pp. 211 - 239
• Main Authors: FONTELOS, M. A., DE LA HOZ, F.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 25.05.2010
• Subjects: Asymptotic methods; Capillary waves; Density; Exact sciences and technology; Fluid dynamics; Fluid mechanics; Fundamental areas of phenomenology (including applications); Gravity; Hydrodynamic stability; Interfacial instability; Physics; Surface tension; Surface-tension-driven instability; Water waves; Interfaces; Singularity; Rayleigh-Taylor instability; Capillary waves; Non viscous fluid; Digital simulation; Modelling; Surface tension; Incompressible fluid
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S0022112009992710

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.