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A local model for the limiting configuration of interfacial solitary waves

Published online by Cambridge University Press:  25 June 2021

X. Guan
Institute of Mechanics, Chinese Academy of Sciences, Beijing100190, PR China Department of Mathematics, University College London, LondonWC1E 6BT, UK
J.-M. Vanden-Broeck
Department of Mathematics, University College London, LondonWC1E 6BT, UK
Z. Wang*
Institute of Mechanics, Chinese Academy of Sciences, Beijing100190, PR China
F. Dias
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin4, Ireland
Email address for correspondence:


The limiting configuration of interfacial solitary waves between two homogeneous fluids consisting of a sharp $120^{\circ }$ angle with an enclosed bubble of stagnant heavier fluid on top is investigated numerically. We use a boundary integral equation method to compute the almost limiting profiles which are nearly self-intersecting and thus extend the work of Pullin & Grimshaw (Phys. Fluids, vol. 31, 1988, pp. 3550–3559) by obtaining the overhanging solutions for very small density ratios. To further study the local configuration of the limiting profile, we propose a reduced model that replaces the $120^{\circ }$ angle with two straight solid walls intersecting at the bottom of the bubble. Using a series truncation method, a one-parameter family of solutions depending on the angle between the two solid walls (denoted by $\gamma$) is found. When $\gamma = {2{\rm \pi} }/{3}$, it is shown that the simplified model agrees well with the near-limiting wave profile if the density ratio is small, and thus provides a good local approximation to the assumed limiting configuration. Interesting solutions for other values of $\gamma$ are also explored.

JFM Papers
© The Author(s), 2021. Published by Cambridge University Press

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