Elastic Rayleigh–Taylor and Richtmyer–Meshkov instabilities in spherical geometry

Abstract

An exact theoretical analysis is presented, mainly concerning linearly elastic Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities at spherical interfaces by using the decomposition method. A stability analysis is performed to investigate the growth rates of instability with the degree of spherical harmonics n, dimensionless radius Br, Atwood number At, and elastic ratio T. Generally, spherical RT and RM instabilities resemble their planar counterparts when Br or n is large. However, low mode perturbations behave quite differently from planar ones, which is also observed in experiments in inertial confinement fusion and in theoretical analysis in geophysics. An analytical expression is obtained to study the dependence of the cutoff radius. There exist two kinds of it: a small one at which the most-active mode cannot be activated and a larger one that is similar to the planar one. In addition, the elastic parameter strongly affects the evolution of a mode with a lower number. The method is extended to study the RT instability at the interface between an elastic solid and a viscous fluid, which is relevant to some situations encountered experimentally. Finally, RM instability in elastic solids is also investigated, in particular, with regard to the evolution of RM instability with low-mode asymmetries. This method is able to provide guidelines for further studying spherical RT and RM instabilities in complex situations, including the finite thickness effects, elastoplastic properties, and Bell–Plesset effects.