Rayleigh–Taylor and Richtmyer–Meshkov instabilities of flat and curved interfaces

In this work we discuss a non-trivial effect of the interfacial curvature on the stability of uniformly and suddenly accelerated interfaces, such as liquid rims. The new stability analysis is based on operator and boundary perturbation theories and allows us to treat the Rayleigh–Taylor and Richtmyer–Meshkov instabilities as a single phenomenon and thus to understand the interrelation between these two fundamental instabilities. This leads, in particular, to clarification of the validity of the original Richtmyer growth rate equation and its crucial dependence on the frame of reference. The main finding of this study is the revealed and quantified influence of the interfacial curvature on the growth rates and the wavenumber selection of both types of instabilities. Finally, the systematic approach taken here also provides a generalization of the widely accepted ad hoc idea, due to Layzer (Astrophys. J., vol. 122, 1955, pp. 1–12), of approximating the potential velocity field near the interface.