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New gravity–capillary waves at low speeds. Part 1. Linear 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: [email protected]
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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 invariably predict a waveless surface at 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. By applying techniques in exponential asymptotics to this problem, we have discovered the existence of new classes of gravity–capillary waves, from which the usual linear solutions form but a special case. In this paper, we present the initial theory for deriving these waves through a study of gravity–capillary flow over a linearized step. This will be done using two approaches: in the first, we derive the surface waves using the standard method of Fourier transforms; in the second, we derive the same result using exponential asymptotics. Ultimately, these two methods give the same result, but conceptually, they offer different insights into the study of the low-Froude-number, low-Bond-number problem.

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. 367 - 391
Copyright
©2013 Cambridge University Press 

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