Buoyancy-induced Stokes flow in a wedge-shaped enclosure

The problem of buoyancy-induced Stokes flow in a sectorial region is addressed. Skew-symmetric flows are considered for wedge or opening angles of the sector in the range 0 < α ≤ π. The basic structure and character of the motion are found to depend critically upon the relative dominance, near the sector vertex, of the particular solution of the system with respect to the leading eigenfunction. A simple criterion is developed for the appearance of eddies, such as those observed by Moffatt (1964), in the neighbourhood of the sector vertex. A calculation is carried out for the specific case of motion induced by different temperatures on the radial boundaries of the enclosure. It is found that corner eddies may be present in this circumstance for wedge angles in the range 126^º ≲ α ≲146º. The eddying motion near the vertex is examined, in some detail, for the wedge angle α = 135^º. In the limiting case of α = π, corresponding to a semicircular-shaped sector, the particular solution is found to exhibit singular behaviour. However, this singular nature is found to be spurious, as a bounded particular solution can be constructed with the aid of one of the eigensolutions. Results are given for no-slip and shear-free conditions on the circular boundary of the sector for the purpose of comparison.