Rayleigh-Taylor instability at spherical interfaces between viscous fluids: The fluid/fluid interface

Through the computation of the most-unstable modes, we perform a systematic analysis of the linear Rayleigh-Taylor instability at a spherical interface separating two different homogeneous regions of incompressible viscous fluids under the action of a radially directed acceleration over the entire parameter space. Using the growth rate as the dependent variable, the parameter space is spanned by the spherical harmonic degree n and three dimensionless variables: the Atwood number A, the viscosity ratio s, and the dimensionless variable B=(a(R rho 2)(2)/mu(2)(2))R-1/3, where a(R), rho(2), and mu(2) are the local radial acceleration at the interface and the density and viscosity of the denser overlying fluid, respectively. To understand the effect of the various parameters on the instability behavior and to identify similarities and differences between the planar and spherical configurations, we compare the most-unstable growth rates alpha P* (planar) and alpha S* (spherical) under homologous driving conditions. For all A, when s << 1, the planar configuration is more unstable than the spherical (alpha P*>alpha S*) within the interval 0 < B < infinity. However, as s increases to O(1), there is a region for small values of B where alpha S*>alpha P*, whereas for larger values of B, alpha P*>alpha S* once again. When s similar to 2, the maximum of alpha S* for the n = 1 mode is greater than alpha S* for any other mode (n >= 2). For s similar to O(10), alpha S*>alpha P* for all A within 0 < B < infinity. We find that the instability behavior between the planar and spherical systems departs from each other for s greater than or similar to 2 and diverges considerably for s >> 1. In the limit when s -> infinity, the planar configuration reduces to the trivial solution alpha P*equivalent to 0 for all B and A, whereas alpha S* has a non-zero limiting value for the n = 1 mode but vanishes for all the other modes (n >= 2). We derive an equation for alpha S* in this limit and obtain closed form solutions for the maximum of alpha S* and the value of B at which this occurs. Finally, we compare the most-unstable growth rates between the exact dispersion relation and three different approximations to highlight their strengths and weaknesses.