Dispersive nonlinear waves in two-layer flows with free surface. I. Model derivation and general properties

Studies in Applied Mathematics

Volumn 119, Issue 3, 2007, Pages 191-211

  Barros, R ^a   Gavrilyuk, S L ^a,b   Teshukov, V M ^a  

^a AIX MARSEILLE UNIVERSITY   (France)

^b AIX MARSEILLE UNIVERSITÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

DISPERSION (WAVES); LAGRANGE MULTIPLIERS;

DISPERSIVE WAVES; FREE SURFACES; LAGRANGIAN; MODEL DERIVATIONS; MULTI-DIMENSIONAL MODEL; NONLINEAR WAVES; PROPERTY; TWO LAYER FLUID; TWO-LAYER; TWO-LAYER FLOWS;

FLUID MECHANICS;

EID: 34548288908     PISSN: 00222526     EISSN: 14679590     Source Type: Journal    
DOI: 10.1111/j.1467-9590.2007.00383.x     Document Type: Article

References (19)

1. C. H. Su and C. S. Gardner, Korteweg-de Vries Equation and Generalizations. III. Derivation of the Korteweg-de Vries Equation and Burgers Equation, J. Math. Phys. 10 : 536 539 (1969).
2. A. E. Green, N. Laws, and P. M. Naghdi, On the Theory of Water Waves, Proceedings of the Royal Society of London A 338 : 43 55 (1974).
3. A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech. 78 : 237 246 (1976).
4. J. Miles and R. Salmon, Weakly dispersive nonlinear gravity waves, J. Fluid Mech. 157 : 519 531 (1985).
5. R. Salmon, Hamiltonian Fluid Mechanics, Ann. Rev. Fluid Mech 20 : 225 256 (1988).
6. R. Salmon, Lectures on Geophysical Fluid Dynamics, Oxford University Press, New York, Oxford, 1998.
7. Y. A. Li, Linear stability of solitary waves of the Green-Naghdi equations, Comm. Pure Appl. Math. 54 : 501 536 (2001).
8. S. L. Gavrilyuk and V. M. Teshukov, Linear stability of parallel inviscid flows of shallow water and bubbly fluid, Stud. Appl. Math. 113 : 1 29 (2004).
9. S. L. Gavrilyuk and V. M. Teshukov, Generalized vorticity for bubbly liquid and dispersive shallow water equations, Continuum Mech. Thermodyn. 13 : 365 382 (2001).
10. V. M. Teshukov and S. L. Gavrilyuk, Three-Dimensional nonlinear dispersive waves on shear flows, Stud. Appl. Math. 116 : 241 255 (2006).
11. G. A. El, R. H. J. Grimshaw, and N. F. Smyth, Unsteady undular bores in fully nonlinear shallow-water theory, Physics of Fluids 18 : 027104 (2006 17 pages).
12. W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech. 396 : 1 36 (1999).
13. R. Liska, L. Margolin, and B. Wendroff, Nonhydrostatic two-layer models of incompressible flow, Computers Math. Applic. 29 : 25 37 (1995).
14. G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.
15. W. Craig, P. Guyenne, and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math. 58 : 1587 1641 (2005).
16. P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964.
17. J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik, Bd. VIII/1, Springer, Berlin, 1959, pp. 125 263.
18. V. L. Berdichevsky, Variational Principles of Continuum Mechanics, Nauka, Moscow, 1983 (in Russian).
19. S. L. Gavrilyuk and H. Gouin, A new form of governing equations of fluids arising from Hamilton's principle, Int. J. Eng. Sci. 37 : 1495 1520 (1999).