Wave damping by a thin layer of viscous fluid

Abstract

The rate of damping of surface gravity-capillary waves is investigated, in a system which consists of a thin layer of a Newtonian viscous fluid of thickness d floating on a Newtonian fluid of infinite depth. The surface and interfacial tensions, elasticities and viscosities are taken into account. In particular, an approximate dispersion relation is derived for the case where kd and (/ ϩ) 1/2 d are both small, where k is the wavenumber, is the angular frequency and ϩ is the kinematic viscosity of the upper fluid. If d→0 while ϩ d remains finite, published dispersion relations for viscoelastic surface films of extremely small ͑e.g., monomolecular͒ thickness are reproduced, if we add the surface and interfacial tensions, elasticities and viscosities together, and then add an additional 4 ϩ ϩ d to the surface viscosity, where ϩ is the density of the upper fluid. A simple approximation is derived for the damping rate and associated frequency shift when their magnitudes are both small. An example is given of what may happen with a slick of heavy fuel oil on water: a slick 10 m thick produces a damping rate only slightly different from that of a film of essentially zero thickness, but the effect of the finite thickness becomes very noticeable if it is increased to 0.1-1 mm.

Drift of an Inextensible Sheet Caused by Surface Waves

“…The role of waves is perhaps the least investigated, and if the object in question is a slick of surfactant, the problem becomes even more complicated because the wavefield is modified by the slick in the whole area it covers. The slick acts to restrict motion tangential to the surface and strong shear is produced in the viscous surface boundary layer, which ultimately leads to more rapid damping of the waves [1][2][3]. The momentum lost as oscillatory motion is regained as an Eulerian mean current by the diffusion of vorticity from the boundary layer [4].…”

Interaction of waves, surface currents, and turbulence: the application of surface-following coordinate systems

“…The same effect is reproduced (see Fig. 10) in the case of a surface layer of fluid of finite thickness (Jenkins and Jacobs, 1997;Jenkins and Dysthe, 1997), and the same mathematical technique may be used to calculate the effect of a layer of sea ice.…”

“…The effect of the viscous boundary layer, with characteristic depth (ν/σ) 1/2 , may be parameterized in models for the evolution of the mean current by adjusting the interfacial boundary conditions (Jenkins, 1986(Jenkins, , 1987(Jenkins, , 1989b, except in the case where the interface has a non-zero elasticity or viscoelasticity, where the mass transport in the boundary layer may be significant (Weber and Saetra, 1995) and the viscous damping of the waves is substantially enhanced by energy dissipation within this boundary layer. In the latter case one may also consider the effect of a thin layer of viscous (viscoelastic) material, for example, oil or ice, floating at the surface, under a similar mathematical formulation (Jenkins and Jacobs, 1997).…”

Unpolarized infrared emissivity of oil films on sea surfaces

“…The first one, namely the Lombardini et al damping model [10], is rather simple, as it depends on only 2 hydrodynamic parameters, but in return is independent of the film thickness H. The second one, namely the Jenkins and Jacobs damping model [12], is more sophisticated, as it depends on 9 hydrodynamic parameters and depends on the film thickness H, but is valid only for thin films. Here, we concentrate on the first model, and represent in Fig.…”