© 1968 by Oxford University Press
ON STOKES AND RAYLEIGH LAYERS IN A ROTATING SYSTEM
( Department of Mathematics, Imperial College, University of London )
We consider the flow generated in a semi-infinite expanse of fluid bounded by an infinite disk, when both the fluid and the disk are in solid-body rotation and, additionally, the disk performs non-torsional oscillations in its own plane. Periodic solutions are first sought and we find that a modified Stokes layer is formed on the disk for all frequencies except for the resonant frequency, which is twice the angular velocity of rotation. In the latter case there is no oscillatory solution which satisfies the boundary conditions.
In order to seek a resolution of the difficulty associated with the resonant case, an initial-value problem is posed; in most cases the oscillatory solutions are reached at large times. In the resonant case, however, we find that the flow is a linear combination of a modified Rayleigh layer, which penetrates outwards perpetually from the disk in the standard diffusive manner, and a layer confined to the disk which, at large times, becomes a modified Stokes layer. The shear oscillations continue to penetrate outwards indefinitely, unless the imposed oscillations are chosen so that the velocity vector of the disk rotates with constant magnitude in the same direction as the basic angular rotation, but with twice its angular speed; then the Rayleigh layer is absent. On the other hand, the presence of a second disk produces, at large times, in the resonant case, a modified plane Couette flow of oscillatory amplitude superimposed on the modified Stokes layer.
Initial-value problems which lead to steady boundary layers are considered. Also the linearized problem for torsional oscillations can be studied by the present analysis if only one disk is present.