Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary * Author to whom correspondence should be addressed. E-mail: crampin@maths.ox.ac. (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth.

We examine the rotor–oscillator flow, a slow viscous flow between long parallel plates driven by the rotation of a slender cylinder (the rotor) and the longitudinal oscillation of one of the plates (the oscillator). For rotor locations of interest to us, this flow exhibits a hyperbolic mixing region, characterized by homoclinic tangling associated with a hyperbolic fixed point, and a degenerate mixing region, characterized by heteroclinic tangling associated with two degenerate fixed points on one of the boundary plates (normally the oscillator). These mixing regions are investigated both theoretically, by applying various dynamical tools to a mathematical model of the flow, and experimentally, by observing the advection of a passive tracer in a specially constructed apparatus. Although degenerate mixing regions have been largely ignored or undervalued in previous research on chaotic mixing, our results demonstrate that more mixing is associated with the degenerate mixing region than the hyperbolic one in many cases. We have also discovered a peculiar phenomenon, which we call Melnikov resonance, involving a rapid fluctuation in the size of the hyperbolic mixing region as the frequency of the oscillator is varied.

An analysis is made of the Stokes flow between parallel planes due to a three-dimensional rotlet whose axis is parallel to the boundary planes. The separation in the plane of symmetry of this flow is compared with that in its two-dimensional analogue, the Stokes flow between parallel planes due to a two-dimensional rotlet. It is found that when the rotlets are midway between the planar walls, both flows exhibit an infinite set of Moffatt eddies. However, when the rotlets are not midway between the walls, the two-dimensional flow has an infinite set of Moffatt eddies, while the three-dimensional flow has at most a finite number of eddies and behaves, far from the rotlet, like the flow due to a two-dimensional source–sink doublet in each of the planes parallel to the boundary planes. An eigenfunction expansion describing a class of asymmetric Stokes flows between parallel planes is also derived and used to show that the far-field behaviour of flows in this class generally resembles the aforementioned flow due to a two-dimensional source–sink doublet in the planes parallel to the walls.

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A study is made of the steady two-dimensional Stokes flow stirred by an infinitesimal rotating cylinder (a line rotlet) in the annular region between two fixed concentric cylindrical walls. It is shown that this simple flow exhibits a rich diversity of possible flow structures. Generally, one component of the flow is associated with a net flux through the annular region in either a clockwise or anticlockwise direction depending on whether the radial distance between the rotlet and the center axis of the bounding cylinders is greater or less than a unique value at which this flow component vanishes. The remaining components of the flow are either eddies attached to the boundary or free eddies in the interior of the flow. Owing to some of the dynamical features of this flow, the results obtained here may be applicable in engineering settings where mixing by the chaotic advection of fluid particles is desirable.