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 ul(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 Poincar6 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 of a steady Stokes flow in a bounded region exhibiting chaotic streamlines.
[1]
G. Batchelor,et al.
An Introduction to Fluid Dynamics
,
1968
.
[2]
H. K. Moffatt,et al.
The degree of knottedness of tangled vortex lines
,
1969,
Journal of Fluid Mechanics.
[3]
Y. Zel’dovich,et al.
Origin of Magnetic Fields in Astrophysics (Turbulent "Dynamo" Mechanisms)
,
1972
.
[4]
V. Arnold,et al.
Ordinary Differential Equations
,
1973
.
[5]
V. Arnold.
Mathematical Methods of Classical Mechanics
,
1974
.
[6]
H. K. Moffatt.
Magnetic Field Generation in Electrically Conducting Fluids
,
1978
.
[7]
A. Lichtenberg,et al.
Regular and Stochastic Motion
,
1982
.
[8]
H. Aref.
Stirring by chaotic advection
,
1984,
Journal of Fluid Mechanics.
[9]
Mark R. Proctor,et al.
Topological constraints associated with fast dynamo action
,
1985,
Journal of Fluid Mechanics.
[10]
M. Tabor,et al.
Experimental study of Lagrangian turbulence in a Stokes flow
,
1986,
Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.
[11]
Hassan Aref,et al.
Chaotic advection in a Stokes flow
,
1986
.
[12]
Uriel Frisch,et al.
Chaotic streamlines in the ABC flows
,
1986,
Journal of Fluid Mechanics.
[13]
C. K. Chu,et al.
Lagrangian turbulence and spatial complexity in a Stokes flow
,
1987
.