Moffatt eddies in the cone

Abstract

Consider Stokes flow in a cone of half-angle $\alpha$ 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} {\bm f}(\theta;\lambda)\exp{\rm i}m\phi $ where the eigenvalue $\lambda$ satisfies a transcendental equation. It follows, by extending an argument given by Moffatt (1964$a$), that if the eigenvalue $\lambda$ is complex there will exist, associated with the corresponding vector eigenfunction, an infinite sequence of eddies as $r\,{\rightarrow}\, 0$. 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 $74.45^{\circ}$. 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 $r\,{\rightarrow}\, \infty$, 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.