Resonance damping of gravity–capillary waves on water covered with a visco-elastic film of finite thickness: A reappraisal

Abstract

A new approach to the problem of damping of gravity-capillary waves (GCW) on water covered with a layer of viscous liquid (a film) of finite thickness with two elastic boundaries is developed. It is shown that the rotational component of GCW can be described formally as a "forced" longitudinal, or Marangoni wave (MW) and the potential component of GCW plays a role of the "external force". The resonance -like excitation of the "forced" MW is demonstrated when the GCW and MW frequencies and wave numbers are approximately close to each other. For a film which is thinner than the viscous boundary layers in film a single "forced" MW exists being located within the boundary layer beneath the water surface. For a thick film the "forced" MW is characterized by the existence of two spatially separated MW modes: one is localized in the boundary layer below the upper, air-film interface and another within the boundary layers in the vicinity of the water-film interface. Then at different elasticities of the interfaces a double peak dependence of the GCW damping coefficient on wave number can occur due to the resonance with the two "forced" MW modes. The dependence of the damping coefficient on film thickness is characterized by a strong maximum appearing when the film and boundary layer thickness values are comparable to each other. The developed theory is consistent with existing numerical studies and experiment.

Application of a Robin boundary condition to surface waves

“…This phenomenon is sometimes referred to as "negative resonance". Similar behaviors appear in more complex setups, for example, the case of finite layer thickness with two elastic interfaces, for which two damping maxima are possible (Ermakov and Khazanov, 2022).…”

Dilational Rheology of Fluid/Fluid Interfaces: Foundations and Tools

“…These can be generated upon the application of mechanically, thermally, or electrically driven perturbations of the interface, when their propagation happens along the fluid/fluid interface. During their propagation, the waves are dampened by the action of surface forces that try to restore the flatness of the interface [22,[67][68][69][70][71].…”

“…This makes understanding the deformation and flow of fluid/fluid interfaces under the application of mechanical stresses, i.e., the rheological properties of the interface, a matter of key importance for science and technology [9,10]. For instance, the understanding and control of the rheological response of fluid/fluid interfaces plays a very important role in the control of emulsion stability [11,12], foamability and foam stability (resistance against drainage) [13], lung surfactant performance [4,14], aerosol formation [15], tear film stability [16,17], encapsulation process [18], coffee ring formation [19], tertiary oil recovery [20], or remote sensing [21,22].…”