A Stability Criterion for TwoFluid Interfaces and

Authors: David Lannes

Publish Date: 2013/03/08

Volume: 208, Issue: 2, Pages: 481-567

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Abstract

We derive a new stability criterion for twofluid 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 twofluid 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 wellposedness 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 waterbrine interfaces and to propose a formula for the maximal amplitude of interfacial waves We also show how to rigorously justify twofluid asymptotic models used for applications and how to relate some of their properties to Kelvin–Helmholtz instabilities

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