The problem of the linear stability of a time-dependent shear flow is investigated and the effects of contraction and expansion of the layer are discussed. Models of flows observed prior to breakdown are constructed and used to investigate the resulting secondary instabilities which are shown to be extremely violent and in essential agreement with recent experiments. In relatively short time, one half period of the primary oscillation, the energy in the secondary disturbance increases by two orders of magnitude; the wave-number corresponding to maximum amplification is five times that of the primary wave.

An investigation is made into the possible types of similarity solutions that can describe the symmetric flow of a fluid into an empty spherical cavity. The flow is homentropic, and the fluid obeys a perfect gas law $p\; =\; k \rho ^{\gamma}$. Values of γ in the range 7 ≥ γ > 1 are discussed. In this range, we find that similarity solutions in which the flow accelerates into the cavity exist for $\gamma \textgreater {\frac{3}{2}$. For these solutions, the radius R of the cavity decreases as the nth power of time measured from the instant at which the cavity disappears. This power n increases monotonically as γ decreases, and attains the value 1 for $\gamma = {\frac{3}{2}$ Similarity solutions in which the cavity collapses with constant velocity are given by the value n = 1, and such solutions are possible for all values of γ in the range considered.

In an earlier paper Hall (1961) proposed a simplified model for the vortex core formed over a slender delta wing at incidence by the rolling-up of the shear layer that separates from a leading edge. This model enabled an outer inviscid solution and an inner viscous solution for the core to be obtained from the equations of motion. However the procedure used for the inner solution led to a number of defects: in particular, the matching of the inner and outer solutions seemed unsatisfactory. In the present paper the defects are avoided by using a different procedure. The first approximation, in the sense of boundary-layer theory, is sought. A solution, in special variables, is obtained which is in the form of an asymptotic expansion containing inverse powers of the logarithm of a Reynolds number. The leading terms of the expansion are computed, and the results confirm that the inner and outer solutions are properly matched.