Periodic Waves and Ligaments on the Surface of a Viscous Exponentially Stratified Fluid in a Uniform Gravity Field (original) (raw)

On capillary gravity-wave motion in two-layer fluids

Journal of Engineering Mathematics, 2011

Generalized expansion formulae for the velocity potentials associated with plane gravity-wave problems in the presence of surface tension and interfacial tension are derived in both the cases of finite and infinite water depths in two-layer fluids. As a part of the expansion formulae, orthogonal mode-coupling relations, associated with the eigenfunctions of the velocity potential, are derived. The dispersion relations are analyzed to determine the characteristics of the two propagating modes in the presence of surface and interfacial tension in both the cases of deep-water and shallow-water waves. The expansion formulae are then generalized to deal with boundary-value problems satisfying higher-order boundary conditions at the free surface and interface. As applications of the expansion formulae, the solutions associated with the source potential, forced oscillation and reflection of capillary-gravity waves in the presence of interfacial tension are derived.

Gravity-Capillary Waves in Layered Fluid

Nonlinear Oscillations, 2003

UDC 532.595 In this work, we analyze the stability of a gravity wave generated on the separation surface of two immiscible liquids inside a moving container and perturbed by a capillary wave. Such a phenomenon is experimentally observed when the amplitude and the frequency of the motion imposed to the container attain certain values. The evolution of the system is described by the variational principle. We assume that the motion of the system is decomposed into two modes: the gravity mode and the capillary mode. With suitable scaling assumptions, it is possible to show that the evolution of the gravity mode is determined by the forcing motion, while the capillary mode is excited by the nonlinear interactions between the capillary and gravity modes. Finally, an analytic dispersion relation is obtained for the pulsation of the capillary mode. This relation is a function of several quantities, all depending on the capillary wave number and the characteristics of the exciting motion.

Wave resistance for capillary gravity waves: Finite-size effects

EPL (Europhysics Letters), 2011

We study theoretically the capillary-gravity waves created at the water-air interface by an external surface pressure distribution symmetrical about a point and moving at constant velocity along a linear trajectory. Within the framework of linear wave theory and assuming the fluid to be inviscid, we calculate the wave resistance experienced by the perturbation as a function of its size (compared to the capillary length). In particular, we analyze how the amplitude of the jump occurring at the minimum phase speed cmin = (4gγ/ρ) 1/4 depends on the size of the pressure distribution (ρ is the liquid density, γ is the water-air surface tension, and g is the acceleration due to gravity). We also show how for pressure distributions broader than a few capillary lengths, the result obtained by Havelock for the wave resistance in the particular case of pure gravity waves (i.e., γ = 0) is progressively recovered.

NONLINEAR GRAVITY AND CAPILLARY-GRAVITY WAVES

Annual Review of Fluid Mechanics, 1999

This review deals primarily with the bifurcation, stability, and evolution of gravity and capillary-gravity waves. Recent results on the bifurcation of various types of capillary-gravity waves, including two-dimensional solitary waves at the minimum of the dispersion curve, are reviewed. A survey of various mechanisms (including the most recent ones) to explain the frequency downshift phenomenon is provided. Recent significant results are given on "horseshoe" patterns, which are three-dimensional structures observable on the sea surface under the action of wind or in wave tank experiments. The so-called short-crested waves are then discussed. Finally, the importance of surface tension effects on steep waves is studied.

An intermediate wavelength, weakly nonlinear theory for the evolution of capillary gravity waves

2011

A new nonlinear evolution equation is derived for surface solitary waves propagating on a liquid-air interface where the wave motion is induced by a harmonic forcing. Instead of the traditional approach involving a base state of the long wave limit, a base state of harmonic waves is assumed for the perturbation analysis. This approach is considered to be more appropriate for channels of length just a few multiples of the depth. The dispersion relation found approaches the classical long wave limit. The weakly nonlinear dispersive waves satisfy a KdV-like nonlinear evolution equation with steeper nonlinearity.

Breaking and Cascade of Internal Gravity Waves in a Continuously Stratified Fluid

Fluid mechanics and its applications, 1993

The evolution of a few large scale high frequency standing internal waves confined to a vertical plane is studied numerically. The growth of nonlinear interactions leads to a transfer of energy toward small vertical scales and lower frequencies: the result is a steep energy decrease due to wave breaking. Induced mixing is evaluated. A parametric forcing is also introduced in order to compare with laboratory experiments. Wave breaking also occurs but as opposed to the unforced case different phases are next observed: internal wave growth due to constructive forcing alternate with energy decrease.

Stationary Gravity Waves with the Zero Mean Vorticity in Stratified Fluid

Studies in Applied Mathematics, 2011

A new approach to the description of stationary plane waves in ideal density stratified incompressible fluid is considered without the application of Boussinesq approximation. The approach is based on the equation derived by Dubreil-Jacotin [4, 5, 6] and Long [7] with the additional assumption that the mean vorticity of the flow is zero. It is shown that in the linear approximation the spectrum of eigenmodes and dispersion equations corresponding to these eigenmodes can be found in the closed analytical forms for many particular relationships between the fluid density and stream function. Examples are presented for waves of infinitesimal amplitude. Exact expression for the velocity of solitary wave of any amplitude is derived.

Gravity waves with effect of surface tension and fluid viscosity

Journal of Hydrodynamics, Ser. B, 2006

The potential flow in a viscous fluid due to a point impulsive source is considered within the framework of linear Stokes equations. The combined effect of fluid viscosity and surface tension on the potential function below the water surface is studied. Dependent on the wavenumbers associated with the level of the effect due to surface tension, the oscillations can be grouped as gravity-dominant waves and capillary-dominant waves. It is shown that the wave form of gravity-dominant oscillations is largely modified by the surface tension while the wave amplitude of capillary-dominant oscillations is mostly reduced by the fluid viscosity.