Proudman (1956) and Stewartson (1966) analyzed the dynamical properties of a fluid occupying the space between two concentric rotating spheres when the angular velocities of the spheres are slightly different, in other words, when the motion relative to a reference frame rotating with one of the spheres is due to an imposed azimuthal velocity which is symmetric about the equator. The consequences of forcing motion across the equator are explored here. Whereas the flow inside the cylinder circumscribing the inner sphere and having generators parallel to the axis of rotation is similar to that which results when the driving is symmetric, the flow outside is quite different. The Ekman layer on the outer sphere persists outside - its dynamics is modified in the vicinity of the equator - and is instrumental in transferring fluid from one hemisphere to the other. The divergence of this Ekman layer causes slow, axial motion in the inviscid region outside. On two shear layers of thickness O(R⊟2/7) and O(R⊟1/3) (where R is the Reynolds number, assumed large) remove discontinuities in the flow field and return fluid from one hemisphere to the other (rather than one Ekman layer to the other as is the case when the driving is azimuthal).
All Science Journal Classification (ASJC) codes
- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering