Dissipation of interfacial Marangoni waves and their resonance with capillary-gravity waves

Abstract

We investigate theoretically the effects of capillarity and gravity on interfacial Marangoni waves (M-waves). Previous studies which neglected these effects have shown the decay rate of M-waves to vary monotonically. In this work, we disprove it by deriving a new dispersion relation (accounting for these effects) for the M-waves, and show that their decay rate passes through a minimum, rather than varying monotonically. This result, obtained numerically elsewhere, was explained using a hypothetical directional packaging process. Our explanation here is consistent with the bigger picture, which includes both capillary-gravity waves (CG-waves) and M-waves, which coexist at the contaminated interface of two fluids. It is known that the CG-waves undergo maximum dissipation at a critical value of interfacial elasticity, which provides the restoring force for the M-waves. Here, we complete the story; we show that the M-waves undergo minimum dissipation at a critical value of capillarity/gravity, which provide the restoring forces for the CG-waves. Previously, it has been hypothesized that the maximum decay rate for CG-waves is caused due to their resonance with M-waves. Here, we show this hypothesis to be true for a temporally-damped wave system, and that the resonance (in addition to causing the maximum decay rate for CG-waves) also causes the minimum decay rate for M-waves.