The effects of axial flow and surface mass-flux on the stability of the rotating-sphere boundary layer

A theoretical investigation is carried out into the linear stability of the boundary-layer flow around a rotating sphere immersed in an incompressible viscous fluid. Two potentially stabilising mechanisms are considered: a forced uniform axial flow in the surrounding fluid, and the introduction of mass suction/injection through the surface of the sphere. The investigation is broadly split into a “local” analysis, where a parallel-flow assumption is made which limits the study to individual latitudinal positions; and a “global” analysis, where the entire streamwise extent of the flow is considered. In the local analysis, both stationary and travelling convective disturbances are considered. For a representative subset of the parameter space, critical Reynolds numbers are presented for the predicted onset of convective and absolute instabilities. Axial flow and surface suction are typically found to postpone the onset of all types of instability by raising the critical Reynolds number, whereas surface injection has the opposite effect. This is further demonstrated by a consideration of the convective and absolute growth rates at various parameter values. The results of the global analysis suggest that the rotating sphere can support a self-sustained, linearly globally-unstable global mode for sufficiently large rotation rates. This is in contrast to the case of the rotating disk, where it is generally accepted that self-sustained linear global modes do not occur.