A Stability Criterion for Two-Fluid Interfaces and Applications

We derive a new stability criterion for two-fluid interfaces that ensures the existence of “stable” local solutions that do not break down too fast due to Kelvin–Helmholtz instabilities. It can be seen both as a two-fluid generalization of the Rayleigh–Taylor criterion and as a nonlinear version of the Kelvin stability condition. We show that gravity can control the inertial effects of the shear up to frequencies that are high enough for the surface tension to play a relevant role. This explains why surface tension is a necessary condition for well-posedness while the (low frequency) main dynamics of interfacial waves are unaffected by it. In order to derive a practical version of this criterion, we work with a nondimensionalized version of the equations and allow for the possibility of various asymptotic regimes, such as the shallow water limit. This limit being singular, we have to derive a new symbolic analysis of the Dirichlet–Neumann operator that includes an infinitely smoothing “tail” accounting for the contribution of the bottom. We then validate our criterion by comparison with experimental data in two important settings: air–water interfaces and internal waves. The good agreement we observe allows us to discuss the scenario of wave breaking and the behavior of water-brine interfaces, and to propose a formula for the maximal amplitude of interfacial waves. We also show how to rigorously justify two-fluid asymptotic models used for applications and how to relate some of their properties to Kelvin–Helmholtz instabilities.