Moffatt eddies in the cone

Consider Stokes flow in a cone of half-angle a filled with a viscous liquid. It is shown that in spherical polar coordinates there exist similarity solutions for the velocity field of the type r(lambda)f(theta;lambda) exp im phi where the eigenvalue lambda satisfies a transcendental equation. It follows, by extending an argument given by Moffatt (1964a), that if the eigenvalue is complex there will exist, associated with the corresponding vector eigenfunction, an infinite sequence of eddies as r -> 0. Consequently, provided the principal eigenvalue is complex and the driving field is appropriate, such eddy sequences will exist. It is also shown that for each wavenumber m there exists a critical angle alpha(*) below which the principal eigenvalue is complex and above which it is real. For example, for m = 1 the critical angle is about 74.45 degrees. The full set of real and complex eigenfunctions, the inner eigenfunctions, can be used to compute the flow in a cone given data on the lid. There also exist outer eigenfunctions, those that decay for r -> infinity, and these can be generated from the inner ones. The two sets together can be used to calculate the flow in a conical container whose base and lid are spherical surfaces. Examples are given of flows in cones and in conical containers which illustrate how alpha and r(0), a length scale, affect the flow fields. The fields in conical containers exhibit toroidal corner vortices whose structure is different from those at a conical vertex; their growth and evolution to primary vortices is briefly examined.