There exist two types of stationary instability of the flow over a rotating disc corresponding to the upper, inviscid mode and the lower-branch mode, which has a triple-deck structure, of the neutral stability curve. The linear problem has been investigated by P. Hall ( Proc. R. Soc. Lond. A 406, 93-106 (1986)) and the asymptotic structure of the wavenumber and orientation of these modes has been obtained. Here, a nonlinear investigation of high Reynolds number, stationary instabilities in the three-dimensional boundary layer on a rotating disc is given for the lower branch mode. By considering nonlinear effects and following the framework set up by Hall, asymptotic solutions are obtained that enable the finite amplitude growth of a disturbance close to the neutral location to be described.

Centripetal instabilities in two flows involving time-dependent Rayleigh layers on a rotating circular cylinder are examined. In one case we consider the stability of the flow induced in an infinite expanse of quiescent fluid when the cylinder is impulsively given a constant angular velocity; in the other problem the angular velocity increases as the square root of time so that the undisturbed flow has a constant wall shear. For both situations linear neutral stability curves for vortex motions are calculated by quasi-steady ͑or frozen-time͒ methods, with these results justified, where possible, by Wentzel-Kramers-Brillouin techniques. The topology of the neutral curve for the ramped angular velocity configuration allows a rigorous description of small wavelength, weakly and fully nonlinear vortex structures to be obtained. Our results are compared with the equivalent cases that arise in the study of unsteady thermal Rayleigh layers induced by the sudden heating of a horizontal flat plate.

The stability of the flow produced over an infinite stationary plane in a fluid rotating with uniform angular velocity at an infinite distance from the plane is considered. The basic flow is an exact solution of the Navier-Stokes equations making it amenable to theoretical study. An asymptotic investigation is presented in the limit of large Reynolds number. It is shown that the stationary spiral instabilities observed experimentally can be described by a linear inviscid stability analysis. The prediction obtained for the wave angle of the disturbances is found to agree well with the available experimental and numerical results.