A theory for the impact of a wave breaking onto a permeable barrier with jet generation

We model a water wave impact onto a porous breakwater. The breakwater surface is modelled as a thin barrier composed of solid matter pierced by channels through which water can flow freely. The water in the wave is modelled as a finite-length volume of inviscid, incompressible fluid in quasi-one-dimensional flow during its impact and flow through a typical hole in the barrier. The fluid volume moves at normal incidence to the barrier. After the initial impact the wave water starts to slow down as it passes through holes in the barrier. Each hole is the source of a free jet along whose length the fluid velocity and width vary in such a way as to conserve volume and momentum at zero pressure. We find there are two types of flow, depending on the porosity, ß , of the barrier. If ß : 0 = ß < 0.5774 then the barrier is a strong impediment to the flow, in that the fluid velocity tends to zero as time tends to infinity. But if ß : 0.5774 = ß = 1 then the barrier only temporarily holds up the flow, and the decelerating wave water passes through in a finite time. We report results for the velocity and impact pressure due to the incident wave water, and for the evolving shape of the jet, with examples from both types of impact. We account for the impulse on the barrier and the conserved kinetic energy of the flow. Consideration of small ß gives insight into the sudden changes in flow and the high pressures that occur when a wave impacts a nearly impermeable seawall

Cavity formation on the surface of a body entering water with deceleration

The two-dimensional water entry of a rigid symmetric body with account for cavity formation on the body surface is studied. Initially the liquid is at rest and occupies the lower half plane. The rigid symmetric body touches the liquid free surface at a single point and then starts suddenly to penetrate the liquid vertically with a time-varying speed. We study the effect of the body deceleration on the pressure distribution in the flow region. It is shown that, in addition to the high pressures expected from the theory of impact, the pressure on the body surface can later decrease to sub-atmospheric levels. The creation of a cavity due to such low pressures is considered. The cavity starts at the lowest point of the body and spreads along the body surface forming a thin space between a new free surface and the body. Within the linearised hydrodynamic problem, the positions of the two turnover points at the periphery of the wetted area are determined by Wagner’s condition. The ends of the cavity’s free surface are modelled by the Brillouin–Villat condition. The pressure in the cavity is assumed to be a prescribed constant, which is a parameter of the model. The hydrodynamic problem is reduced to a system of integral and differential equations with respect to several functions of time. Results are presented for constant deceleration of two body shapes: a parabola and a wedge. The general formulation made also embraces conditions where the body is free to decelerate under the total fluid force. Contrasts are drawn between results from the present model and a simpler model in which the cavity formation is suppressed. It is shown that the expansion of the cavity can be significantly slower than the expansion of the corresponding zone of sub-atmospheric pressure in the simpler model. For forced motion and cavity pressure close to atmospheric, the cavity grows until almost complete detachment of the fluid from the body. In the problem of free motion of the body, cavitation with vapour pressure in the cavity is achievable only for extremely large impact velocities

Simple waves and shocks in a thin film of a perfectly soluble anti-surfactant solution

We consider the dynamics of a thin film of a perfectly soluble anti-surfactant solution in the limit of large capillary and Peclet numbers in which the governing system of nonlinear equations is purely hyperbolic. We construct exact solutions to a family of Riemann problems for this system, and discuss the properties of these solutions, including the formation of both simple-wave and uniform regions within the flow, and the propagation of shocks in both the thickness of the film and the gradient of the concentration of solute

On dynamic interactions between body motion and fluid motion

This contribution on dynamic fluid-body interactions concentrates on applying mathematical/analytical ideas to complement direct numerical studies. The typical body may be of given shape or flexible depending on the context. In the background there are numerous real-world motivations in industry, biomedical and environmental applications, many of which involve high flow rates. A review of ideas developed over the last decade for cases of high flow rates first addresses inviscid approaches to one or more bodies free to move within a channel flow, a skimming sharp-edged body on a free surface, the sinking of a body in water and the rocking or rolling of a body on a solid surface, before moving on to more recent viscous-inviscid approaches for channel flows and boundary layers. The beginnings of certain current research projects are also outlined. These concern models of liftoff of a body from a solid surface, the impact of a smooth body during skimming and viscous-inviscid effects in the presence of more than one freely moving body. Linear and nonlinear mathematical properties as appropriate are described

Free and moving boundary problems in hydrodynamics.

Discusses well posed and ill posed problems in free surface flows in porous media (with or without gravity) and the two dimensional Hele Shaw cell analogue. Examines models for these flows, related heat and mass transfer problems, and studies local stability analyses. Establishes well posedness criteria. Discusses the rectangular dam problem and considers exact and approximate methods of solution. Develops an asymptotic approximation for cases where change in fluid level is small compared to overall dam dimensions. Derives explicit solutions from a linear formulation, and also treats the problem as a variational inequality. Studies Hele Shaw flow without gravity and presents a class of explicit solutions for unsteady flow. These solutions break down in finite time. Examines surface tension effects and proposes a numerical method for ill posed Hele Shaw flow. Presents numerical results. (C.J.U.

A note on the two-phase Hele-Shaw problem

We discuss some techniques for finding explicit solutions to immiscible two-phase flow in a Hele-Shaw cell, exploiting properties of the Schwartz function of the interface between the fluids. We also discuss the question of the well-posedness of this problem

Kochina and Hele-Shaw in modern mathematics, natural science and industry

A brief review of P. Ya. Kochina's studies, namely, her investigations of free-boundary problems for harmonic functions, is presented. Her ideas have had implications for many areas of quantitative science, including materials science, the environment, medicine and finance. Within mathematics, they have stimulated many new developments in the areas of complex analysis, asymptotic analysis, and partial differential equations with free boundaries. © 2002 Elsevier Science Ltd. All rights reserved

Kochina and Hele-Shaw in modern mathematics, science and industry

The review of the activity of Kochina P.Ya. on free boundary problems for harmonic functions is presented. The Kochina models for fluid flows with free boundaries in the filtration theory are analyzed as well as the application of the Hele-Shaw cell for flow visualization is considered. The Kochina models and the Hele-Shaw cell are widely used in materials science, ecology, medicine and finance