On a class of three-dimensional corner eddies

Consider Stokes flow in a viscous fluid filling a corner, of angle 2 alpha, bounded by two infinite plane walls. Assume that the flow is symmetrical about some plane which is normal to the walls bounding the corner. Since superposition is valid we may consider flows that are symmetrical about the plane bisecting the corner and those that are antisymmetrical about this plane. In either case it is shown that for a class of corner eddies, the corner flow is made up of an infinite sequence of eddies as r --> 0, where r is the radial distance from the corner. Moreover, the eigenvalues lambda which determine the structure of the corner eddy fields satisfy the same equation, sin lambda alpha = +/-lambda sin 2 alpha, that arises in the corresponding plane case. The three-dimensional velocity fields are, however, quite different from those seen in the plane case. In particular, in the symmetric case the streamlines are not closed and foci, rather than elliptic stagnation points, are the centres of the eddies in the plane of symmetry. These results represent, in this special context, a generalization to three-dimensions of Moffatt's classical result for planar corner eddies.