Complex analysis of resonance conditions for coupled capillary and dilational waves

Abstract

Capillary and dilational waves occur at a surfactant–covered interface between two fluids. These wave modes, two physically realizable roots of the same dispersion relation, can resonate when the surfactant rheological parameters possess a critical combination of values. Recent numerical studies have shown that this resonant behaviour is marked by two separate phenomena: (i) a peak in the capillary mode damping rate together with a concurrent trough in the dilational mode damping rate, or (ii) a mixed–mode behaviour. The latter phenomenon suggests that, as the values of the rheological parameters approach a critical combination, neither wave mode exhibits primarily capillary nor dilational behaviour. In this paper we show that both of these phenomena are manifestations of the occurrence of a root coalescence point, which marks resonant interaction, in the fully complex wave mode solution space resulting from complexification of the dispersion relation. In our analysis, we consider six rheological parameters which are important for this problem and which we term the static surface tension, surface normal shear viscosity, surface dilational elasticity, surface relaxational elasticity, surface tangential shear viscosity, and surface tangential dilational viscosity. Many of these parameters are difficult to measure experimentally and impossible to measure in the field. We demonstrate in this paper that it is possible to obtain analytically an approximate value over a specific frequency range of each of the first three parameters and an apparent viscosity combining the last three parameters from simple measurements of capillary wavenumber and frequency.