Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-22T14:31:39.438Z Has data issue: false hasContentIssue false
  • English
  • Français

A reformulation and applications of interfacial fluids with a free surface

Published online by Cambridge University Press:  17 July 2009

T. S. HAUT  and
M. J. ABLOWITZ
T. S. HAUT
Affiliation:
Department of Applied Mathematics, University of Colorado at Boulder, 526 UCB, Boulder, CO 80209-0526, USA
M. J. ABLOWITZ*
Affiliation:
Department of Applied Mathematics, University of Colorado at Boulder, 526 UCB, Boulder, CO 80209-0526, USA
*
†Email address for correspondence: ablowitz@colorado.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A non-local formulation, depending on a free spectral parameter, is presented governing two ideal fluids separated by a free interface and bounded above either by a free surface or by a rigid lid. This formulation is shown to be related to the Dirichlet–Neumann operators associated with the two-fluid equations. As an application, long wave equations are obtained; these include generalizations of the Benney–Luke and intermediate long wave equations, as well as their higher order perturbations. Computational studies reveal that both equations possess lump-type solutions, which indicate the possible existence of fully localized solitary waves in interfacial fluids with sufficient surface tension.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 631 , 25 July 2009 , pp. 375 - 396
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ablowitz, M. J. & Clarkson, P. A. 1991 Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press.CrossRefGoogle Scholar
Ablowitz, M. J., Fokas, A. S. & Musslimani, Z. H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–342.CrossRefGoogle Scholar
Ablowitz, M. J. & Musslimani, Z. H. 2005 A spectral renormalization method to compute self-localized solutions to nonlinear systems. Opt. Lett. 30 (16), 2140–2142.CrossRefGoogle ScholarPubMed
Ablowitz, M. J. & Segur, H. 1981 Solitons and the Inverse Scattering Transform. SIAM.CrossRefGoogle Scholar
Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 241–270.CrossRefGoogle Scholar
Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.CrossRefGoogle Scholar
Benjamin, T. B. & Bridges, T. J. 1997 Reappraisal of the Kelvin–Helmholtz problem. I. Hamiltonian structure. J. Fluid Mech. 333, 301–325.CrossRefGoogle Scholar
Benjamin, T. B. & Olver, P. J. 1982 Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185.CrossRefGoogle Scholar
Benney, D. J. & Luke, J. 1964 Interactions of permanent waves of finite amplitude. J. Math. Phys. 43, 309–313.CrossRefGoogle Scholar
Berger, K. M. & Milewski, P. A. 2000 The generation and evolution of lump solitary waves in surface-tension-dominated flows. SIAM J. Appl. Math. 61, 731–750.CrossRefGoogle Scholar
Bona, J. L., Lannes, D. & Saut, J.-C. 2008 Asymptotic models for internal waves. J. Math. Pures Appliques 89, 538–566.CrossRefGoogle Scholar
Bridges, T. J. & Donaldson, N. M. 2007 Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves. Phys. Fluids 19, 072111.CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1996 Weakly nonlinear internal waves in a two-fluid system. J. Fluid Mech. 386, 1–36.Google Scholar
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 313, 83–103.CrossRefGoogle Scholar
Craig, W. & Groves, M. D. 1994 Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19, 367–389.CrossRefGoogle Scholar
Craig, W., Groves, M. D., Schneider, G. & Toland, J. F. 2002 Recent developments in the mathematical theory of water waves introduction. Roy. Soc. of London Phil. Tr. A 360 (1799), 2107–2109.CrossRefGoogle Scholar
Craig, W., Guyenne, D. & Kalisch, H. 2005 a Hamiltonian long wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 58, 1587–1641.CrossRefGoogle Scholar
Craig, W., Guyenne, P., Nicholls, D. & Sulem, C. 2005 b Hamiltonian long wave expansions for water waves over a rough bottom. Proc. R. Soc. A 461, 1–35.Google Scholar
Craig, W. & Nicholls, D. P. 2000 Travelling two and three-dimensional capillary gravity water waves. SIAM J. Math. Anal. 32 (2), 323–359.CrossRefGoogle Scholar
Craig, W., Schanz, U. & Sulem, C. 1997 The modulational regime of three-dimensional water waves and the Davey–Stewartson system. Annales de l'I. H. P. 14 (section C), 615–667.Google Scholar
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83.CrossRefGoogle Scholar
Fokas, A. S. 2000 On the integrability of linear and nonlinear partial differential equations. J. Math. Phys. 41 (6), 4188–4237.CrossRefGoogle Scholar
Fokas, A. S. 2007 A Unified Approach to Boundary Value Problems. CBMS-SIAM.Google Scholar
Groves, M. D. & Sun, S. M. 2008 Fully localized solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem. Arch. Rat. Math. Anal. 188, 1–91.CrossRefGoogle Scholar
Haut, T. S. 2008 Nonlocal formulations of ideal fluids and applications. PhD thesis, University of Colorado, Boulder.Google Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.CrossRefGoogle Scholar
Joseph, R. I. 1977 Solitary waves in finite depth fluid. J. Phys. A: Math. Gen. 10, L225L227.CrossRefGoogle Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersive media. Sovi. Phys. Dokl. 15, 539–541.Google Scholar
Korteweg, D. J. & deVries, G. 1895 On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary wave. Phil. Mag. 39, 422–443.CrossRefGoogle Scholar
Kubota, T., Ko, D. R. S. & Dobbs, L. D. 1978 Weakly nonlinear internal gravity waves in stratified fluids of infinite depth. J. Hydrodynamics 12, 157–165.Google Scholar
Matsuno, Y. 1992 Nonlinear evolutions of surface gravity waves of fluid of finite depth. Phys. Rev. Lett. 69, 609–611.CrossRefGoogle ScholarPubMed
Matsuno, Y. 1994 Higher-order nonlinear evolution equation for interfacial waves in a two-layer fluid system. Phys. Rev. E 49, 2091–2095.CrossRefGoogle Scholar
Ono, H. 1975 Algebraic solitary waves in stratified flow. J. Phys. Soc. Jpn 39, 1082–1091.CrossRefGoogle Scholar
Pego, R. L. & Quintero, J. R. 1999 Two-dimensional solitary waves for a Benney–Luke equation. Physica D 132, 476–496.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–1994.CrossRefGoogle Scholar