Moffatt eddies in the cone

Abstract

Consider Stokes flow in a cone of half-angle α 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^λƒ(θ ; λ) exp imΦ where the eigenvalue λ satisfies a transcendental equation. It follows, by extending an argument given by Moffatt (1964a), that if the eigenvalue λ 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 α* 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°. 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 → ∞, 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 α and r₀, 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.