Instability in a spherical fluid shell

Summary

The model considered is a collapsing or expanding spherical shell of incompressible fluid with constant total energy. The stability of its surfaces is studied by the usual perturbation method. There is a non-uniform acceleration through the shell which satisfies Taylor's criterion for stability at both surfaces. The inner surface however fails to satisfy Birkhoff's condition during collapse and is in general algebraically unstable. The stability of the outer surfaces is found to depend on the ratio of shell thickness to radius. For a thin shell the ratio of the initial perturbation amplitudes on the two surfaces is found to govern the motion at each surface, while for a thick shell, and for harmonics of order higher than the second, the two surfaces are independent, the inner surface being unstable during collapse and the outer surface unstable during expansion.