Viscous Surface Waves Research Articles

Global in time behavior of viscous surface waves: horizontally periodic motion

Global in time behavior of viscous surface waves: horizontally periodic motion

A finite element model of interaction between viscous free surface waves and submerged cylinders

This paper describes the simulation of the flow of a viscous incompressible Newtonian liquid with a free surface. The Navier–Stokes equations are formulated using a streamline upwind Petrov–Galerkin scheme, and solved on a Q-tree-based finite element mesh that adapts to the moving free surface of the liquid. Special attention is given to fitting the mesh correctly to the free surface and solid wall boundaries. Fully non-linear free surface boundary conditions are implemented. Test cases include sloshing free surface motions in a rectangular tank and progressive waves over submerged cylinders.

Rotating spirals in a Faraday experiment.

We report experiments and model simulations of patterns of parametrically excited capillary ripples in a large aspect-ratio cell with thin horizontal layer of viscous fluid subjected to sinusoidal vertical oscillations. We found stable rotating spirals with different topological charges in the parameter range where steady straight rolls were observed previously. Formation of multiarmed spirals via dislocations approaching a target core was observed. The direction of the spiral rotation depends on its chirality and is always consistent with wave propagation towards the core. Wave drift towards the spiral core is associated with the shear flow which is generated near the walls by rapidly decaying viscous surface waves. {copyright} {ital 1996 The American Physical Society.}

Large time existence of small viscous surface waves without surface tension

The motion of a three-dimensional viscous, imcompressible fluid is governed by the Navier-Stokes equations. We study the case where the fluid is in an ocean of infinite extent and finite depth with a free surface on top. This gives rise to a nonlinear free boundary problem. The given data are the initial velocity field and the initial free surface. In general, given smooth data, the solution will develop singularities in finite time; however, the effect of viscosity and surface tension tends to prevent the ingulitrities. It was previously known that when both are present, small, appropriately smooth solutions do not develop singularities; that is, smooth solutions exist globally in time. In this paper, we show that viscosity alone will prevent the formation of singularitics, even without surface tension; i.e., small smooth data which satisfy certain natural compatibility conditions, smooth solutions exist for all time. Uniqueness of the solution for any finite time interval is also proved.

Large-time regularity of viscous surface waves

Etude du mouvement d'un fluide visqueux incompressible contenu dans un ocean tridimensionnel de dimensions finies, limite en bas par un sol solide et en haut par une atmosphere a pression constante. La surface superieure se modifie en fonction du mouvement du fluide