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New gravity–capillary waves at low speeds. Part 2. Nonlinear geometries

Published online by Cambridge University Press:  29 April 2013

Philippe H. Trinh  and
S. Jonathan Chapman
Philippe H. Trinh*
Affiliation:
Program in Applied and Computational Mathematics, Princeton University, Washington Road, Princeton, NJ 08544, USA Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, 24-29 St. Giles’, Oxford OX1 3LB, UK
S. Jonathan Chapman
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, 24-29 St. Giles’, Oxford OX1 3LB, UK
*
†Email address for correspondence: trinh@maths.ox.ac.uk
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Abstract

When traditional linearized theory is used to study gravity–capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions are waveless to every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In Part 1 (Trinh & Chapman, J. Fluid Mech., vol. 724, 2013b, pp. 367–391), we showed how exponential asymptotics could be used to study the problem when the size of the obstruction is first linearized. In this paper, we extend the analysis to the nonlinear problem, thus allowing the full geometry to be considered at leading order. When applied to the classic problem of flow over a step, our analysis reveals the existence of six classes of gravity–capillary waves, two of which share a connection with the usual linearized solutions first discovered by Rayleigh. The new solutions arise due to the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 724 , 10 June 2013 , pp. 392 - 424
Copyright
©2013 Cambridge University Press 

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