Capillary wave scattering from a surfactant domain

Abstract

The study of capillary wave scattering by a circular region with different interfacial properties from the rest of an otherwise homogeneous interface is motivated by experiments on wave attenuation at a monolayer-covered air–water interface where domains of one surface phase are dispersed in a second surface phase. Here the scattering function is calculated for an incident wave of frequency ω (wavevector k0) scattering from an isolated circular domain of radius a with surface tension σ1 which is imbedded in an otherwise infinite interface of surface tension σ0. The underlying fluid is treated as irrotational and the three-dimensional flow problem coupling the heterogeneous surface to the underlying liquid is reduced to a set of dual integral equations, which are solved numerically. With this solution the scattering amplitudes and the total scattering cross sections are calculated as a function of the surface tension ratio σ0/σ1 and incident wavenumber k0a. The analogous problem of a discontinuous change in bending rigidity is also considered and the solution to the complete viscous problem is outlined in the Appendix. Experimental implications of these results are discussed.

Surfactant Scavenging and Surface Deposition by Rising Bubbles

“…Note that [11] excludes the deposition of surfactantladen bubbles arriving at the surface; this effect is included later in [14]. The boundary condition at the bulk surface is a balance between the flux of surfactant diffusing away from the surface into the bulk and the flux of surfactant desorbing from the surface.…”

“…For a summary of much of the experimental work in this area we refer the reader to (7)(8)(9)(10). A theory of capillary wave scattering from a surfactant domain was developed by (11), and (12, 13) carefully quantified marine surface-active materials.…”

“…The average flux of surfactant to the bulk surface by bubbling is then 4a 2 nu b ␥ b (0). To determine the average steady-state surfactant concentration on the bulk surface we add the flux of surfactant from the bulk fluid, [11], to the flux of surfactant by bubbling. The result is…”

Capillary–gravity wave transport over spatially random drift

“…The physical origin of γ arises from causality, but can also be explicitly derived from considerations of an infinitesimally small viscous dissipation (Chou, Lucas & Stone (1995)). Although we have assumed γ → 0, for our model to be valid, the viscosity need only be small enough such that surface waves are not attenuated before they have a chance to multiply scatter and enter the transport or diffusion regimes under consideration.…”

“…Water waves can also scatter from regions of underlying vorticity regions smaller than the wavelength Fabrikant & Raevsky (1994) and Cerda & Lund (1993). Boundary conditions that vary on capillary length scales, as well as wave interactions with structures comparable to or smaller than the wavelength can also lead to wave scattering (Chou, Lucas & Stone (1995), Gou, Messiter & Schultz (1993)), attenuation (Chou & Nelson (1994), Lee et al (1993)), and Bragg reflections (Chou (1998), Naciri & Mei (1988)). Nonetheless, water wave propagation over random static underlying currents that vary on both large and small length scales, and their interactions, have received relatively less attention.…”