A three-fluid model for the dissipation of interfacial capillary-gravity waves

We investigate the dissipation of linear, two-dimensional, interfacial waves in a setting comprising three fluids (an upper fluid of semi-infinite depth, a middle fluid-layer of finite thickness, and a lower fluid of semi-infinite depth) separated by two distinct interfaces, which we consider to be elastic. We derive analytic expressions for the dissipation rate of capillary-gravity waves in such a system, in both the barotropic and baroclinic modes of propagation. Using the dissipation rate model formulated herein, we conduct parametric studies of barotropic gravity waves in an air-oil-water system. We consider six different wavenumbers within the range of 0.0165 m(-1) (corresponding to ocean swell) to 44.5 m(-1) (corresponding to a typical laboratory gravity wave) and investigate the effects of three major mechanisms of loss of energy, which are the dissipation due to the (i) dynamics in the upper fluid (air), (ii) elastic interfaces, and (iii) viscous middle fluid (oil) layer of finite thickness. For waves with wavenumbers of 0.0165 m(-1) and 0.04 m(-1), the dominant mechanism for the energy loss is that due to the dynamics in air. For waves with wavenumbers of 1 m(-1) and 4 m(-1), the oil layer acts to increase the dissipation rates significantly but only when its thickness is beyond a threshold value. For waves with wavenumbers of 36.2 m(-1) and 44.5 m(-1), the elastic interfaces cause significant increases in the dissipation rates, when their elasticities change from a value of 0.01 N/m to 0.0225 N/m. The three-fluid model developed herein is applicable to capillary-gravity waves propagating in a generic fluid system with arbitrary values for the densities, viscosities, interfacial elasticities, and with an arbitrary value for the middle fluid-layer thickness within an upper limit. This model is useful in predicting the dissipation rates of waves on the ocean surface, which is (in general) covered with biofilms and oil layers of thicknesses ranging from a few mu m to a few mm, and in predicting the dissipation rates of waves such as swell, for which the dynamics in the upper fluid (air) are important.