Bounds for Arbitrary Steady Gravity Waves on Water of Finite Depth

Necessary conditions for the existence of arbitrary bounded steady waves are proved (earlier, these conditions, that have the form of bounds on the Bernoulli constant and other wave characteristics, were established only for Stokes waves). It is also shown that there exists an exact upper bound such that if the free-surface profile is less than this bound at infinity (positive, negative, or both), then the profile asymptotes the constant level corresponding to a unform stream (supercritical or subcritical). Finally, an integral property of arbitrary steady waves is obtained. A new technique is proposed for proving these results; it is based on modified Bernoulli’s equation that along with the free surface profile involves the difference between the potential and its vertical average.     

Local Bifurcation for Steady Periodic Capillary Water Waves with Constant Vorticity

We study periodic capillary waves at the free surface of water in a flow with constant vorticity over a flat bed. Using bifurcation theory the local existence of waves of small amplitude is proved even in the presence of stagnation points in the flow. We also derive the dispersion relation.     

Least Supersolution Approach to Regularizing Free Boundary Problems

In this paper, we study a free boundary problem obtained as a limit as ε → 0 to the following regularizing family of semilinear equations , where β _ε approximates the Dirac delta in the origin and F is a Lipschitz function bounded away from 0 and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces that are uniform in ε. This allows to prove that the free boundary of a limit has the “right” weak geometry, in the measure theoretical sense. By the construction of some barriers with curvature, the classification of global profiles of the blow-up analysis is carried out and the limit functions are proven to be viscosity and pointwise solution ( almost everywhere) to a free boundary problem. Finally, the free boundary is proven to be a C ^1,α surface around almost everywhere point. An erratum to this article can be found at     

Instabilities in a Two-Dimensional Combustion Model with Free Boundary

We prove instability of the planar travelling wave solution in a two-dimensional free boundary problem modelling the propagation of near- equidiffusional premixed flames in the whole plane. We reduce the problem to a fixed boundary fully nonlinear parabolic system. The spectrum of the linearized operator contains an interval [0,ω ^c ], ω ^c > 0, so we cannot construct backward solutions. We use an argument about stability of dynamical systems in Banach spaces in order to prove pointwise instability of the moving front. (Accepted: January 31, 2000)?Published online August 21, 2000     

Periodic Traveling Gravity Water Waves with Discontinuous Vorticity

We use elliptic theory to prove the existence of steady two-dimensional periodic waterwaves of large amplitude in a flowwith an arbitrary bounded but discontinuous vorticity. This is achieved by developing a local and global bifurcation construction of weak solutions of the elliptic partial differential equations that are relevant to this hydrodynamical context.     

Steady Water Waves with Multiple Critical Layers: Interior Dynamics

We study small-amplitude steady water waves with multiple critical layers. Those are rotational two-dimensional gravity-waves propagating over a perfect fluid of finite depth. It is found that arbitrarily many critical layers with cat’s-eye vortices are possible, with different structure at different levels within the fluid. The corresponding vorticity depends linearly on the stream function.     

Bound on the Slope of Steady Water Waves with Favorable Vorticity

We consider the angle \({\theta}\) of inclination (with respect to the horizontal) of the profile of a steady two dimensional inviscid symmetric periodic or solitary water wave subject to gravity. Although \({\theta}\) may surpass 30° for some irrotational waves close to the extreme wave, Amick (Arch Ration Mech Anal 99(2):91–114, 1987) proved that for any irrotational wave the angle must be less than 31.15°. Is the situation similar for periodic or solitary waves that are not irrotational? The extreme Gerstner wave has infinite depth, adverse vorticity and vertical cusps (θ = 90°). Moreover, numerical calculations show that even waves of finite depth can overturn if the vorticity is adverse. In this paper, on the other hand, we prove an upper bound of 45° on \({\theta}\) for a large class of waves with favorable vorticity and finite depth. In particular, the vorticity can be any constant with the favorable sign. We also prove a series of general inequalities on the pressure within the fluid, including the fact that any overturning wave must have a pressure sink.     

A Film-Flow Model to Describe Free Water Transport during Drying of a Hygroscopic Capillary Porous Medium

In this work, a two-phase film-flow model in a hygroscopic capillary tube is developed and extended to describe the two-phase capillary viscous transport in a network of parallel capillary tubes in terms of relative permeabilities. This film-flow approach is further considered to predict the longitudinal moisture transport in oak wood during drying. Numerical results obtained from this prediction are compared with data of convective drying experiments performed on samples of this wood. The comparison seems to confirm the physical relevance of a film-flow model to correctly represent the moisture transfer until the hygroscopic regime is reached.     

Steady Periodic Water Waves with Constant Vorticity: Regularity and Local Bifurcation

This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow.     

Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology

We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation

Flat-Plate Boundary Layer Receptivity to a Steady Free-Stream Vortex Disturbance

A single trailing vortex developed behind a micro-wing immersed in a free stream was used to study the vortex receptivity of the boundary layer on a flat plate. As a result of the interaction, in the boundary layer there develop longitudinal-velocity disturbances which grow almost linearly in the longitudinal coordinate. The parameters of the excited steady disturbances agree with the data of previous experiments performed under natural conditions and dealing with an indirect scenario of laminar-turbulent transition at high free-stream turbulence. It is shown that the leading edge of the plate does not play a decisive role in the mechanism of growth of disturbances of this kind and the receptivity is non-local in nature.     

Swept-Wing Boundary Layer Receptivity to a Steady Free-Stream Vortex Disturbance

A single trailing vortex developed behind a micro-wing immersed in a free stream was used to study the vortex receptivity of a swept-wing boundary layer. As a result of the interaction, longitudinal-velocity disturbances develop in the boundary layer. On the swept wing, disturbance transformation occurs near the leading edge and is accompanied by the formation of a wave packet consisting of waves typical of cross-flow instability. Disturbances with other characteristics are also detected. These disturbances may be attributable to distributed boundary-layer receptivity to the free-stream vortex disturbance considered.     

重力作用下液体自由面的数值解

给出一种重力作用下静止液体自由面的数值解方法。根据流体力学Young-Laplace公式和微分几何学主曲率方程,并利用液体自由面的对称性质,导出重力作用下静止液体自由面母线的常微分控制方程。为了使该方程便于利用现有的Runge-Kutta法,根据问题的物理背景,将体积边界条件转化为固-液接触面半径条件;采用一种坐标变换,避免了该方程数值计算中刚性问题的出现。最后给出了算例。