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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: [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 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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References

Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
Berk, H. L., Nevins, W. M. & Roberts, K. V. 1982 New Stokes’ line in WKB theory. J. Math. Phys. 23 (6), 988–1002.CrossRefGoogle Scholar
Berry, M. V. 1989 Uniform asymptotic smoothing of Stokes discontinuities. Proc. R Soc. Lond. A 422, 7–21.Google Scholar
Boyd, J. P. 1998 Weakly Non-local Solitary Waves and Beyond-all-orders Asymptotics. Kluwer.CrossRefGoogle Scholar
Chapman, S. J., Howls, C. J., King, J. R. & Olde Daalhuis, A. B. 2007 Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation. Nonlinearity 20 (10), 2425.CrossRefGoogle Scholar
Chapman, S. J., King, J. R. & Adams, K. L. 1998 Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Proc. R. Soc. Lond. A 454, 2733–2755.CrossRefGoogle Scholar
Chapman, S. J., Lawry, J. M. H., Ockendon, J. R. & Tew, R. H. 1999 On the theory of complex rays. SIAM Rev. 41 (3), 417–509.CrossRefGoogle Scholar
Chapman, S. J. & Mortimer, D. B. 2005 Exponential asymptotics and Stokes lines in a partial differential equation. Proc. R. Soc. Lond. A 461, 2385–2421.Google Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. 2002 Exponential asymptotics and capillary waves. SIAM J. Appl. Maths 62 (6), 1872–1898.Google Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.CrossRefGoogle Scholar
Costin, O. 2008 Asymptotics and Borel Summability, vol. 141. Chapman & Hall/CRC.CrossRefGoogle Scholar
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary-gravity waves. Annu. Rev. Fluid Mech. 31, 301–346.CrossRefGoogle Scholar
Dingle, R. B. 1973 Asymptotic Expansions: Their Derivation and Interpretation. Academic.Google Scholar
Forbes, L. K. 1983 Free-surface flow over a semi-circular obstruction, including the influence of gravity and surface tension. J. Fluid Mech. 127, 283–297.CrossRefGoogle Scholar
Grandison, S. & Vanden-Broeck, J.-M. 2006 Truncation approximations for gravity-capillary free-surface flows. J. Engng Maths 54, 89–97.CrossRefGoogle Scholar
Howls, C. J., Langman, P. J. & Daalhuis, A. B. O. 2004 On the higher-order Stokes phenomenon. Proc. R. Soc. Lond. A 460, 2285–2303.CrossRefGoogle Scholar
King, A. C. & Bloor, M. I. G. 1987 Free-surface flow over a step. J. Fluid Mech. 182, 193–208.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Dover.Google Scholar
Lustri, C. J., McCue, S. W. & Binder, B. J. 2012 Free surface flow past topography: a beyond-all-orders approach. Eur. J. Appl. Maths 1 (1), 1–27.Google Scholar
Lustri, C. J., McCue, S. W. & Chapman, S. J. 2013 Exponential asymptotics of free surface flow due to a line source. IMA J. Appl. Maths (in press).CrossRefGoogle Scholar
Olde Daalhuis, A. B. 2004 On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. J. Comput. Appl. Maths 169, 235–246.CrossRefGoogle Scholar
Rayleigh, Lord 1883 The form of standing waves on the surface of running water. Proc. Lond. Math. Soc. 15, 69–78.CrossRefGoogle Scholar
Scullen, D. C. 1998 Accurate computation of steady nonlinear free-surface flows. PhD thesis, University of Adelaide.Google Scholar
Stoker, J. J. 1957 Water Waves. Interscience.Google Scholar
Trinh, P. H. 2010 Exponential asymptotics and free-surface flows. PhD thesis, University of Oxford.Google Scholar
Trinh, P. H. & Chapman, S. J. 2013a Do waveless ships exist? Part 2. The multi-cornered hull. J. Fluid Mech. (submitted).Google Scholar
Trinh, P. H. & Chapman, S. J. 2013b New theoretical gravity–capillary waves. Part 1. Linear theory. J. Fluid Mech. 724, 367–391.CrossRefGoogle Scholar
Trinh, P. H., Chapman, S. J. & Vanden-Broeck, J.-M. 2011 Do waveless ships exist? Results for single-cornered hulls. J. Fluid Mech. 685, 413–439.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 2010 Gravity-Capillary Free-Surface Flows. Cambridge University Press.CrossRefGoogle Scholar
White, R. B. 2005 Asymptotic Analysis of Differential Equations. Imperial College.CrossRefGoogle Scholar