Soluble model for the analysis of stability in an imploding compressible liner

A soluble model of the development of the linear pertubations about a time-varying state of a compressible medium is presented. A Lagrangian description is employed to rederive the equations for the self-similar motion of an ideal fluid and to obtain the linearized equations of motion for pertubations about a general time-varying basic state. The resulting formalism is applied in cylindrical geometry to calculate the growth of flute-like modes associated with a similarity solution modeling the implosion and expansion of a fluid liner. A complete solution is obtained for the perturbed motion. The only modes for which the perturbation amplitudes grow faster than the unperturbed liner radius during both implosion and expansion are divergence- and curl-free. Numerical and analytical results are obtained for these and shown to reduce, in the short-wavelength limit, to the Rayleigh--Taylor instability found previously for incompressible time-independent basic states. In addition, a new kind of instability is found: a class of overstable internal modes (sound waves), which are ''pumped up'' in amplitude during implosion, but decay during the expansion phase