Rayleigh-Taylor instability at spherical interfaces between viscous fluids: Fluid/vacuum interface

Abstract

For a spherical interface of radius R separating two different homogeneous regions of incompressible viscous fluids under the action of a radially directed acceleration, we perform a linear stability analysis in terms of spherical surface harmonics Yn to derive the dispersion relation. The instability behavior is investigated by computing the growth rates and the most-unstable modes as a function of the spherical harmonic degree n. This general methodology is applicable to the entire parameter space spanned by the Atwood number, the viscosity ratio, and the dimensionless number B=(aRρ22/μ22)1/3 R (where aR, ρ2, and μ2 are the local radial acceleration at the interface, and the density and viscosity of the denser overlying fluid, respectively). While the mathematical formulation herein is general, this paper focuses on instability that arises at a spherical viscous fluid/vacuum interface as there is a great deal to be learned from the effects of one-fluid viscosity and sphericity alone. To quantify and understand the effect that curvature and radial acceleration have on the Rayleigh-Taylor instability, a comparison of the growth rates, under homologous driving conditions, between the planar and spherical interfaces is performed. The derived dispersion relation for the planar interface accounts for an underlying finite fluid region of thickness L and normal acceleration aR. Under certain conditions, the development of the most-unstable modes at a spherical interface can take place via the superposition of two adjacent spherical harmonics Yn and Yn+1. This bimodality in the evolution of disturbances in the linear regime does not have a counterpart in the planar configuration where the most-unstable modes are associated with a unique wave number.