On a class of steady confined Stokes flows with chaotic streamlines

The general incompressible flow uQ(x), quadratic in the space coordinates, and satisfying the condition uQ = · n = 0 on a sphere r = 1, is considered. It is shown that this flow may be decomposed into the sum of three ingredients – a poloidal flow of Hill's vortex structure, a quasi-rigid rotation, and a twist ingredient involving two parameters, the complete flow uQ(x) then involving essentially seven independent parameters. The flow, being quadratic, is a Stokes flow in the sphere. The streamline structure of the general flow is investigated, and the results illustrated with reference to a particular sub-family of ‘stretch–twist–fold’ (STF) flows that arise naturally in dynamo theory. When the flow is a small perturbation of a flow u1(x) with closed streamlines, the particle paths are constrained near surfaces defined by an ‘adiabatic invariant’ associated with the perturbation field. When the flow u1 is dominated by its twist ingredient, the particles can migrate from one such surface to another, a phenomenon that is clearly evident in the computation of Poincaré sections for the STF flow, and that we describe as ‘trans-adiabatic drift’. The migration occurs when the particles pass a neighbourhood of saddle points of the flow on r = 1, and leads to chaos in the streamline pattern in much the same way as the chaos that occurs near heteroclinic orbits of low-order dynamical systems. The flow is believed to be the first example ofa steady Stokes flow in a bounded region exhibiting chaotic streamlines.