On a class of steady confined Stokes flows with chaotic streamlines

Abstract

The general incompressible flow u Q ( x ), quadratic in the space coordinates, and satisfying the condition u Q = · 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 u Q ( 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 u 1 ( x ) with closed streamlines, the particle paths are constrained near surfaces defined by an ‘adiabatic invariant’ associated with the perturbation field. When the flow u 1 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.