Linear Stability of Self-Similar Flow. 4. Convective Instability of a Spherical Cloud Expanding into Vacuum.

The well-known class of self-similar solutions for an ideal polytropic gas sphere of radius Rt expanding into a vacuum with velocity ur,t r R-dotR is shown to be convectively unstable. The physical mechanism results from the buoyancy force experienced by anisentropic distributions in the inertial gravitational field. An equation for the perturbed displacement, derived from the linearized fluid equations in Lagrangian coordinates, is solved by separation of variables. Because the basic state is nonsteady, the perturbations do not grow exponentially, but can be expressed in terms of hypergeometric functions.