A hydraulic jump is a physical phenomenon commonly observed in nature such as in open channel flows or spillways and is dependent upon the relation between the initial upstream fluid speed and a critical speed characterized by a dimensionless number F known as the Froude number. In this paper we prove the existence of hydraulic jumps for stationary water-waves as a consequence of the existence of bifurcation branches of non-flat liquid interfaces originated from each of a sequence of upstream velocities F1 > F2 > ⋯ > Fr > ⋯ (Fr → 0 as r → ∞). We further establish explicitly, for F > 0, F≠Fr, r ∈ ℕ, the existence and uniqueness of the solution of a perfect, incompressible, irrotational free surface flow over a flat bottom, under the influence of gravity; as well as the corresponding hydraulic jump.
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August 2016
Research Article|
August 30 2016
Stationary shapes for 2-d water-waves and hydraulic jumps
M. A. Fontelos;
M. A. Fontelos
a)
1
ICMAT-CSIC
, C/Nicolás Cabrera, No. 13-15 Campus de Cantoblanco, UAM, 28049 Madrid, Spain
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R. Lecaros;
R. Lecaros
b)
2Centro de Modelamiento Matemático (CMM),
Universidad de Chile (UMI CNRS 2807)
, Beauchef 851, Torre Norte, Piso 5, Casilla 170-3, Correo 3, Santiago, Chile
3Departamento de Matemática,
Universidad Técnica Federico Santa María
, Casilla 110-V, Valparaíso, Chile
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J. C. López-Ríos
;
J. C. López-Ríos
c)
4Departamento de Ingeniería Matemática,
Universidad de Chile
, Beauchef 851, Torre Norte, Piso 5, Santiago, Chile
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J. H. Ortega
J. H. Ortega
d)
2Centro de Modelamiento Matemático (CMM),
Universidad de Chile (UMI CNRS 2807)
, Beauchef 851, Torre Norte, Piso 5, Casilla 170-3, Correo 3, Santiago, Chile
4Departamento de Ingeniería Matemática,
Universidad de Chile
, Beauchef 851, Torre Norte, Piso 5, Santiago, Chile
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a)
Electronic mail: marco.fontelos@icmat.es
b)
Electronic mail: rlecaros@dim.uchile.cl
c)
Electronic mail: j.lopezr@dim.uchile.cl
d)
Electronic mail: jortega@dim.uchile.cl
J. Math. Phys. 57, 081520 (2016)
Article history
Received:
October 13 2015
Accepted:
August 10 2016
Citation
M. A. Fontelos, R. Lecaros, J. C. López-Ríos, J. H. Ortega; Stationary shapes for 2-d water-waves and hydraulic jumps. J. Math. Phys. 1 August 2016; 57 (8): 081520. https://doi.org/10.1063/1.4961514
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