Theoretical analysis of the effect of strength on oscillations and Rayleigh-Taylor instabilities on a collapsing spherical surface, supported by a study on Bell-Plesset oscillations

Abstract

Small perturbations on a spherical interface, between two materials of different densities, oscillate in amplitude as the radius of the interface increases or decreases. The historical approach to this problem has been to solve Laplace's equation for the velocity potential in the domains on either side of the interface and equate the pressure at the interface. This paper considers the effects of yield strength and shear modulus on oscillations on a collapsing spherical surface between a higher density, strong material, and a lower-density, weak material.

The strong material is assumed to be incompressible, flow in an elasto-plastic manner and lie on the yield surface. The yield surface is taken to be defined by the von Mises yield criterion and the flow to follow the Prandtl-Reuss rules. The Navier-Stokes equation provides the starting point for the analysis and yields the necessary stress terms for inclusion in Laplace's equation. The analysis is limited to a first-order approximation. The material strength model is assumed to be constant. This paper will outline the theoretical analysis and show a comparison of the analytical results with simulations carried out using a hydrocode. The theoretical analysis will be shown to give good agreement with the calculations over a range of different initial wavelengths and strength parameters. When the yield strength is high, the amplitudes of the oscillations decay monotonically to zero; at even higher yield strengths oscillations are completely inhibited and the amplitudes increase, due to geometric convergence effects. Criteria for these phenomena are derived and shown to agree approximately with calculations made using the theoretical analysis.

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