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Rob Sturman |
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Mixing in microfluidics
I am currently working with Professor Stephen Wiggins at the
University of Bristol on an EPSRC funded project entitled Chaotic
Advection, Stirring, Mixing and Optimal Design Principles for
Micromixers. Our work is based on applying the ideas of linked twist
maps to the mixing of fluids at the microscale. This involves extending
and generalising results from classical ergodic theory, producing
numerical computations which demonstrate the breakdown of mixing,
studying the rate of mixing via the decay of correlations, and
ultimately producing a set of design criteria for producing optimal
mixing results. Chaotic mixing in fluids has for a long time been
studied from the topological viewpoint, with the existence of a
horseshoe frequently pointing the way towards chaotic dynamics, and
hence mixing. However, horseshoes are objects of zero volume, and as
such do not guarantee that islands of unmixed fluid do not exist.
Ergodic theory, and in particular linked twist maps, gives a framework
in which mixing results on sets of positive (and even full) volume can
be formulated.
We have produced a book, co-authored with J. M. Ottino, entitled
The
Mathematical Foundations of Mixing:
The Linked Twist Map as a
Paradigm in Applications, Micro to Macro, Fluids to Solids,
which contains a review of a diverse range of mixing devices which can
be studied via the linked twist map framework, an original exposition of
the mathematical topics of ergodic theory and hyperbolicity, aimed at
the non-specialist reader, followed by a synthesis and detailed
explanation of existing results for linked twist maps, which until now
exist only in obscure specialized conference proceedings. The book also
contains a large chapter of original open problems and future directions
for this field of research to take.
For example we have modelled existing mixing devices which are used in industry to increase the hybridization of DNA in microarrays. Using fundamental ideas from ergodic theory of hyperbolic dynamical systems we are able to suggest a set of design principles which dramatically improves the quality of chaotic mixing in such microarrays. These results may have significant implications for both reliability and speed in DNA hybridization experiments.
Piecewise isometries and granular tumbler mixers
Recent
research in collaboration with Steve Meier and J. M. Ottino (Northwestern)
has revealed some rich dynamical structure in the behaviour of
three-dimensional granular tumbler mixers. In particular efficient
mixing behaviour seems possible in the absence of the stretching and
folding usually associated with chaotic mixing. This complex dynamics
stems from piecewise isometries, and is the subject of much
current theoretical and applicable research.
Musical tuning and dynamical systems I am also interested in musical tuning, and have used ideas from dynamical systems, and in particular frequency locking in the Arnold Sine Circle Map, to devise a method of producing systematically out-of-tune nonlinear musical scales. This technique may have uses in psychoacoustics, and is also currently being used by a young composer who writes music for a microtonally tuned harp. Intermittency and bursting thourgh symmetry-breaking
As a post-doctoral research assistant I worked with Dr Alastair
Rucklidge (Leeds) and Dr Peter Ashwin (Exeter) on intermittent dynamics
in systems with invariant subspaces forced by the presence of
symmetries. In particular, robust stable heteroclinic cycles can appear
in such systems, and be robust with respect to perturbations which
preserve the symmetry. Cycles between more complicated invariant sets
are possible; cycles between chaotic sets are termed cycling chaos. We
characterised two different types of dynamics that can result when
cycling chaos loses stability in a resonance bifurcation. We extended
these results from maps to flows, and also to a coupled map lattice,
where the cycling can be used to construct non-ergodic behaviour - that
is, trajectories for which long-term averages do not converge, but
instead give a system of decelerating defects in a pattern. We applied
the ideas of intermittent chaotic dynamics to a model of the solar
dynamo, using a Markov chain model that reproduces the time spent close
to the invariant subspaces and the switching between the different
possible invariant subspaces.
Strange nonchaotic attractors in quasiperiodically
forced systems
My PhD studies, supervised by Dr Jaroslav Stark (UCL),
concentrated on skew-product maps with quasiperiodic forcing. A common
feature of such systems appears to be the presence of attractors with a
complicated (strange) geometric structure, but for whom typical
trajectories are not chaotic. We formulated semi-uniform versions of the
Birkhoff and sub-additive ergodic theorems, and applied them to show
that any strange compact nonchaotic invariant set in such a system must
support an invariant measure with a non-negative maximal normal Lyapunov
exponent. This remains of the few rigorous mathematical results known
about SNAs. We investigated numerically a paradigm example which
highlights the structural complexity of SNAs, and used rational
approximations to the irrational forcing to understand a type of
intermittency present in the SNA, with similar scaling properties to the
intermittency in an attractor-merging crisis. To reconcile the numerical
and analytic work we used renormalisation techniques to discuss
fine-scale structure with some mathematical rigour.
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