Applications of degenerate bifurcation equations to the taylor problem and the water wave problem

1990 ◽  
pp. 117-127
Author(s):  
Hisashi Okamoto
Keyword(s):  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Taylor Problem ◽  
Degenerate Bifurcation

Singularities in water waves and the Rayleigh–Taylor problem

2010 ◽  
Vol 651 ◽  
pp. 211-239 ◽  
Author(s):  
M. A. FONTELOS ◽  
F. DE LA HOZ
Keyword(s):  
Numerical Simulation ◽  
Water Waves ◽  
Water Wave ◽  
Asymptotic Methods ◽  
Logarithmic Spiral ◽  
Capillary Waves ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Taylor Problem ◽  
Regularizing Effects

We describe, by means of asymptotic methods and direct numerical simulation, the structure of singularities developing at the interface between two perfect, inviscid and irrotational fluids of different densities ρ1 and ρ2 and under the action of gravity. When the lighter fluid is on top of the heavier fluid, one encounters the water-wave problem for fluids of different densities. In the limit when the density of the lighter fluid is zero, one encounters the classical water-wave problem. Analogously, when the heavier fluid is on top of the lighter fluid, one encounters the Rayleigh–Taylor problem for fluids of different densities, with this being the case when one of the densities is zero for the classical Rayleigh–Taylor problem. We will show that both water-wave and Rayleigh–Taylor problems develop singularities of the Moore-type (singularities in the curvature) when both fluid densities are non-zero. For the classical water-wave problem, we propose and provide evidence of the development of a singularity in the form of a logarithmic spiral, and for the classical Rayleigh–Taylor problem no singularities were found. The regularizing effects of surface tension are also discussed, and estimates of the size and wavelength of the capillary waves, bubbles or blobs that are produced are provided.

scholarly journals An alternative approach to study irrotational periodic gravity water waves

2021 ◽  
Vol 72 (4) ◽  
Author(s):  
Biswajit Basu ◽  
Calin I. Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Stagnation Points ◽  
Alternative Approach ◽  
New Formulation ◽  
Bounded Below ◽  
Surface Dynamic

AbstractWe are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

scholarly journals A remark on the two dimensional water wave problem with surface tension

2019 ◽  
Vol 266 (9) ◽  
pp. 5748-5771
Author(s):  
Shuanglin Shao ◽  
Hsi-Wei Shih
Keyword(s):  
Surface Tension ◽  
Water Wave ◽  
Two Dimensional ◽  
Wave Problem ◽  
Water Wave Problem

Unsteady non-linear waves in sloping channels

1958 ◽  
Vol 247 (1249) ◽  
pp. 186-198 ◽  
Keyword(s):  
Exact Solution ◽  
Water Wave ◽  
Legendre Function ◽  
Elliptic Integral ◽  
Vertical Wall ◽  
Auxiliary Function ◽  
Level Line ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Non Linear

It is shown in general that the exact solution to every non-degenerate unsteady water-wave problem in a straight channel inclined at arbitrary slope, governed by the non-linear hydraulic equations, can be obtained in terms of the complete elliptic integral of the second kind, E . By means of a non-Newtonian reference frame, every such wave problem for a sloping channel can be replaced by an associated problem for a horizontal channel. For the latter, the partial differential equations become reducible and thus permit hodograph inversion. The Riemann integration method for the resulting Euler-Poisson equation yields an auxiliary function for these hydraulic problems which is transformable into a Legendre function and then into the elliptic integral. In particular, the procedure is applied to obtain the exact solution for the water wave in a sloping channel produced by sudden release of the triangular wedge of water (the reservoir) initially at rest behind a vertical wall. The behaviour of the solution is exhibited for convenience in two level-line charts, and representative wave profiles and velocity distributions are presented.

A mathematical analysis of tsunami generation in shallow water due to seabed deformation

2011 ◽  
Vol 141 (3) ◽  
pp. 551-608 ◽  
Author(s):  
Tatsuo Iguchi
Keyword(s):  
Velocity Field ◽  
Shallow Water ◽  
Water Wave ◽  
Initial Velocity ◽  
Tsunami Generation ◽  
Wave Problem ◽  
Initial Displacement ◽  
Water Wave Problem ◽  
Submarine Earthquakes ◽  
Tsunami Model

In numerical computations of tsunamis due to submarine earthquakes, it is frequently assumed that the initial displacement of the water surface is equal to the permanent shift of the seabed and that the initial velocity field is equal to zero and the shallow-water equations are often used to simulate the propagation of tsunamis. We give a mathematically rigorous justification of this tsunami model starting from the full water-wave problem by comparing the solution of the full problem with that of the tsunami model. We also show that, in some cases, we have to impose a non-zero initial velocity field, which arises as a nonlinear effect.

Hamiltonian long-wave approximations to the water-wave problem

Wave Motion ◽  
1994 ◽  
Vol 19 (4) ◽  
pp. 367-389 ◽  
Author(s):  
Walter Craig ◽  
Mark D. Groves
Keyword(s):  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Long Wave
Sign in / Sign up

Export Citation Format

Share Document