Jean-Marc Vanden-Broeck

Fingers in a Hele-Shaw Cell with surface tension

McLean and Saffman’s model for the fingering in a Hele–Shaw cell is solved numerically. The results suggests that a countably infinite number of solutions exist for nonzero surface tension. This set of solutions contains the solution previously obtained by McLean and Saffman.

Gravity-capillary solitary waves in water of infinite depth and related free-surface flows

Two-dimensional free-surface flows due to a pressure distribution moving at a constant velocity U at the surface of a fluid of infinite depth are considered. Both gravity g and surface tension T are included in the dynamic boundary condition. The velocity U is assumed to be smaller than (4 gT /ρ) ¼ , so that there are no waves in the far field. Here ρ is the density of the fluid. The problem is solved numerically by a boundary integral equation technique. It is shown that for some values of U , four different flows are possible. Three of these flows are interpreted as perturbations of solitary waves in water of infinite depth. It is found that both elevation and depression solitary waves are possible in water of infinite depth. The numerical results for depression waves confirm and extend the solutions previously computed by Longuet-Higgins (1989).

Gravity-Capillary Free Surface Flows

Free-surface problems occur in many aspects of science and of everyday life, for example in the waves on a beach, bubbles rising in a glass of champagne, melting ice, pouring flows from a container and sails billowing in the wind. Consequently, the theory of gravity–capillary free-surface flows continues to be a fertile field of research in applied mathematics and engineering. Concentrating on applications arising from fluid dynamics, Vanden-Broeck draws upon his years of experience in the field to address the many challenges involved in attempting to describe such flows mathematically. Whilst careful numerical techniques are implemented to solve the basic equations, an emphasis is placed upon the reader developing a deep understanding of the structure of the resulting solutions. The author also reviews relevant concepts in fluid mechanics to enable readers from other scientific fields to develop a working knowledge of free-boundary problems.

Broadening of interfacial solitary waves

Progressing interfacial gravity waves are considered for two fluids of differing densities confined in a channel of finite vertical extent and infinite horizontal extent. An integrodifferential equation for the unknown shape of the interface is derived. This equation is discretized and the resulting algebraic equations are solved using Newton’s method. It is found that, for a range of heights and densities of the two fluids, the system supports a branch of solitary waves. Progression along the branch produces a broadening of the wave. With increased broadening both the amplitude and the wave speed approach limiting values. The results are in good agreement with analytical studies and indicate the existence of internal surges.

Free surface flow due to a sink

Abstract : When fluid is withdrawn from a reservoir, the free surface may be drawn down. In order to investigate this phenomenon we consider two-dimensional free surface flows without waves, produced by a submerged sink in a reservoir. Numerical solutions are obtained for various configurations. For a sink the horizontal bottom of a layer of fluid, there are solutions for all values of the Froude number F greater than some particular value. However, when the fluid is sufficiently deep, there is an additional solution for one special value of F 1. We were led to look for these solutions by our experiences with other free surface flows with gravity, such as flows over weirs in channels and flows around lips of teapot spouts. In those cases we found that in fluids of infinite depth there was a flow only for a special value of the appropriate Froude number. This kind of flow also occurred in fluids of finite depth, but in addition there were solutions for all Froude numbers greater than some particular value. The present results show that this is also the case for free surface flows produced by sinks.

Open channel flows with submerged obstructions

Free-surface flows past a submerged triangular obstacle at the bottom of a channel are considered. The flow is assumed to be steady, two-dimensional and irrotational; the fluid is treated as inviseid and incompressible and gravity is taken into account. The problem is solved numerically by series truncation. It is shown that there are solutions for which the flow is suberitical upstream and supercritical downstream and other flows for which the flow is supercritical both upstream and downstream. The latter flows have limiting configurations with a stagnation point on the free surface with a 120° angle at it. It is found that solutions exist for triangular obstacles of arbitrary size. Local solutions are constructed to describe the flow near the apex when the height of the triangular obstacle is infinite.

Nonlinear three-dimensional gravity–capillary solitary waves

Steady three-dimensional fully nonlinear gravity–capillary solitary waves are calculated numerically in infinite depth. These waves have decaying oscillations in the direction of propagation and monotone decay perpendicular to the direction of propagation. They travel at a velocity

Free-surface flow over an obstruction in a channel

U

Axisymmetric bubble or drop in a uniform flow

smaller than the minimum velocity

Accurate Computations for Steep Solitary Waves.

c_{min}