Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity

This paper presents existence theories for several families of small-amplitude solitarywave solutions to the classical water-wave problem in the presence of surface tension and with an arbitrary distribution of vorticity. Moreover, the established local bifurcation diagram for irrotational solitary waves is shown to remain qualitatively unchanged for any choice of vorticity distribution. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite numer of degrees of freedom. Homoclinic solutions to the reduced system, which correspond to solitary water waves, are detected by a variety of dynamical systems methods

Solitary gravity water waves with an arbitrary distribution of vorticity

This paper presents an existence theory for small-amplitude solitary-wave solutions to the classical water-wave problem in the absence of surface tension and with an arbitrary distribution of vorticity. The hydrodynamic problem is formulated as an in nite-dimensional Hamiltonian system in which the horizontal spatial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with one degree of freedom. The phase portrait of the reduced system contains a homoclinic orbit, and the corresponding solution of the water-wave problem is a solitary wave of elevation

Traveling water waves — the ebb and flow of two centuries

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions

Steady water waves

Steady water wave

Mathematical Theory of Water Waves

The water-wave problem is the study of the two- and three-dimensional flow of a perfect fluid bounded above by a free surface subject to the forces of gravity and surface tension. From a mathematical viewpoint, the water-wave equations pose surprisingly deep and subtle challenges for mathematical analysis. The governing equations are widely accepted and there has been substantial research into their validity and limitations. However, a rigorous theory of their solutions is extremely complex due not only to the fact that the water-wave problem is a classical free-boundary problem (where the problem domain, specifically the water surface, is one of the unknowns), but also because the boundary conditions (and, in some cases, the equations) are strongly nonlinear. In contrast to other meetings on water waves, which usually focus upon modelling and numerical issues, this workshop was devoted to the rigorous mathematical theory for the exact hydrodynamic equations

Mathematical Theory of Water Waves

Water waves, that is waves on the surface of a fluid (or the interface between different fluids) are omnipresent phenomena. However, as Feynman wrote in his lecture, water waves that are easily seen by everyone, and which are usually used as an example of waves in elementary courses, are the worst possible example; they have all the complications that waves can have. These complications make mathematical investigations particularly challenging and the physics particularly rich. Indeed, expertise gained in modelling, mathematical analysis and numerical simulation of water waves can be expected to lead to progress in issues of high societal impact (renewable energies in marine environments, vorticity generation and wave breaking, macro-vortices and coastal erosion, ocean shipping and near-shore navigation, tsunamis and hurricane-generated waves, floating airports, ice-sea interactions, ferrofluids in high-technology applications, ...). The workshop was mostly devoted to rigorous mathematical theory for the exact hydrodynamic equations; numerical simulations, modelling and experimental issues were included insofar as they had an evident synergy effect

Solitary waves on falling liquid films in the inertia-dominated regime

We offer new insights and results on the hydrodynamics of solitary waves on inertia-dominated falling liquid films using a combination of experimental measurements, direct numerical simulations (DNS) and low-dimensional (LD) modelling. The DNS are shown to be in very good agreement with experimental measurements in terms of the main wave characteristics and velocity profiles over the entire range of investigated Reynolds numbers. And, surprisingly, the LD model is found to predict accurately the film height even for inertia-dominated films with high Reynolds numbers. Based on a detailed analysis of the flow field within the liquid film, the hydrodynamic mechanism responsible for a constant, or even reducing, maximum film height when the Reynolds number increases above a critical value is identified, and reasons why no flow reversal is observed underneath the wave trough above a critical Reynolds number are proposed. The saturation of the maximum film height is shown to be linked to a reduced effective inertia acting on the solitary waves as a result of flow recirculation in the main wave hump and in the moving frame of reference. Nevertheless, the velocity profile at the crest of the solitary waves remains parabolic and self-similar even after the onset of flow recirculation. The upper limit of the Reynolds number with respect to flow reversal is primarily the result of steeper solitary waves at high Reynolds numbers, which leads to larger streamwise pressure gradients that counter flow reversal. Our results should be of interest in the optimisation of the heat and mass transport characteristics of falling liquid films and can also serve as a benchmark for future model development