Topological chaos in spatially periodic mixers

1
Topological chaos in spatially periodic mixers
Matthew Finn and Jean-Luc Thiffeault
Department of Mathematics, South Kensington
Campus, Imperial College London, London SW7 2AZ,
United Kingdom

• Figure 6 Stirring in a doubly-periodic domain.
  The flow is periodic in both directions, so the
  left and right edges and the top and bottom edges
  of the box are identified. The images illustrate
  three complete applications of the braid t1r1.
• Since the hole acts as a stirrer, only two
  stirring rods are required to guarantee
  topological chaos! We illustrate this in Figure 5
  for the braid s1 t1, which is equivalent to the
  braid in the plane S1S1S2S2S1 for which the Burau
  and train-track entropy bounds are both 1.32.
  Figure 2 shows the interface length. We calculate
  the entropy of our example flow to be 1.38, so in
  this case the lower bound based on the braid
  alone is an impressive 96 sharp! This suggests
  that most of the chaotic behaviour in this flow
  is directly due to the motion of the rods, and
  not to the creation of secondary periodic orbits.
• Periodicity in both directions
• In the doubly-periodic domain we introduce braid
  elements ri corresponding to touring the second
  periodic direction. Unfortunately, we cannot
  associate this new torus braid group with the
  braid group for bounded flow as no conformal map
  exists from the torus to plane.
• Although topological chaos can be guaranteed with
  three rods in the plane, and just two rods on the
  cylinder, it does not follow that TC can be
  produced with only one rod on the torus. The
  torus braid group with one stirrer is t1, r1.
  Both of these elements commute with each other
  and so it is not possible to generate any
  non-trivial braids. Flows with one stirrer can of
  course still be chaotic, with a finite TE, but
  topological considerations based on the rod
  motion tells us nothing.
• We illustrate the poor quality of stirring
  achieved with a single rod for the braid t1 r1 in
  Figure 6. This flow has a low TE of 0.55 and
  produces visibly poorer stirring than can be seen
  in Figures 1 and 5.
• Our current work is concerned with matrix
  representations for the torus braid group that
  will enable us to compute entropy lower bounds.
  We are also investigating train-tracks for the
  torus.
• References

Introduction Topological chaos (TC) may be used
to design fluid mixers with a robust stirring
quality that depends only on the relative motions
of stirring rods and is independent of flow
properties such as compressibility or viscosity
1. This mixing quality can be guaranteed
because rods are topological obstacles to flow,
and if rod trajectories form an intertwined
braiding pattern, then by continuity the fluid
must be braided too. Thurston-Nielson theory
tells us that under the action of these
non-trivial braided stirrer motions the
displacement of the fluid is isotopic to a
pseudo-Anosov map, and that the topological
entropy (a popular mixing measure) of the flow is
at least as large as that in this associated map.
In two-dimensional flows, the topological
entropy is the same as the growth rate of the
length of material lines. To guarantee a positive
topological entropy it can be shown that at least
three trajectories must be involved in a braid.
However, this does not mean mixers must have
three or more stirrers to produce TC 2. This is
because we can consider trajectories of anything
that acts as a topological obstacle in a
two-dimensional flow this could be a fluid
particle itself, or an entire periodic island
from a Poincaré section. Thus, all good mixers
generally produce TC in some form, since it is
easy to find sets of periodic trajectories that
braid. This means that TC can used to diagnose
mixing, even for flows with fewer than three
stirrers 2. On this poster we study TC in
regimes where flow is driven by moving stirring
rods. We consider the effect of extra rod motions
that are possible when the fluid domain has
spatial periodicity. Even though physical
realisations of periodic mixers are lacking, we
study topological chaos in such conditions
because many prototypical chaotic flows are
spatially periodic. Stirring in a bounded
domain In bounded flows, braiding is
characterised by recording exchanges of position
of adjacent stirring rods. Rod positions are
projected onto an axes and labelled 1 to n from
left to right. If the ith and i1th rods exchange
order in a clockwise sense we assign this the
braid element si an anti-clockwise crossing is
labelled si-1. Stirring protocols correspond to
elements of the braid group, which is the set of
all braids generated by sequences of s1 to sn-1
and their inverses. The braids encode much
information about time-periodic fluid flow. One
such piece of information is a lower bound on the
flows topological entropy. Two ways of
calculating the lower bound have been considered
matrix representations of the braid group 3,
and the Bestvina-Handel train-track algorithm
4. In the former case, the logarithm of the
spectral radius of a matrix representation of
each braid is used as an estimate of the entropy
the most popular is the one-parameter Burau
representation 3, though there are others. For
n stirring rods the Burau representation consists
of square matrices of size n-1. The matrices for
each si have unit eigenvalues to represent the
fact that the repeated action of interchanging
any one pair of stirrers forms a trivial braid
that should produce zero entropy. Only
non-trivial braids which involve products of
different Burau matrices can produce a positive
topological entropy. A more sophisticated method
for computing entropy lower bounds is via the
Bestvina-Handel train-track algorithm 4. This
more computationally intensive algorithm requires
the construction of a special directed graph for
each braid.
0
½
3
0
½
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Figure 3 Additional braid group elements
for spatially periodic flows. In addition to the
usual si, corresponding to interchanging adjacent
stirring rods, there are elements corresponding
to stirrers making a tour around one or both
periodic directions. For flows that are periodic
in one direction we allow the elements ti. For
doubly-periodic flows we also consider the
elements ri. Figure 4 The conformal
transformation maps a
singly-periodic domain with n stirrers onto a
bounded annular domain. The central region acts
as an n1th stirrer. The braid element ti
corresponds to the ith stirrer making a clockwise
tour of all the stirring rods i1, , n1, so
that ti ? Si Sn Sn Si . In Figure 4 we show
the result of conformally mapping the square in
the z-plane in Figure 3 (with )
onto an annulus in the w-plane according to
. A key feature in this bounded
w-domain is that there is a hole which acts as a
fixed n1th stirrer. Braid elements si dont
involve the hole, but the periodic motions ti
make a clockwise tour of the hole. Thus the braid
group on a singly-periodic domain with n strings
si, ti can be associated with the braid group
in the plane with n1 strings Si if we assign
si ? Si and ti ? Si Sn Sn Si. Figur
e 5 Stirring in a singly-periodic domain. The
top and bottom of the box are no-slip boundaries.
The flow is periodic in the horizontal direction,
so that the left and right edges of the box are
identified. The images illustrate three complete
applications of the braid s1t1. In the bounded
conformally mapped domain w the corresponding
braid is S1S1S2S2S1, with a topological entropy
lower bound of 1.32.
Figure 1 Stirring in a bounded square
box under the repeated action of the braid s1
s2-1. Box zero shows the initial condition of
blue and orange dye. Arrows in each box indicate
stirrer motions leading to the next image. Three
complete applications of the braid are shown. For
n 3 stirrers, train-tracks give exactly the
same entropy bound as the Burau representation.
For n gt 3 train-tracks give a sharper bound. We
illustrate some bound calculations for a
Stokes-flow mixing device. Figure 1 illustrates a
braiding motion of three circular stirring rods
mixing blue and orange fluid inside a square box
with no-slip walls. Motion of the stirring rods
corresponds to the braid s1 s2-1. Since there
are three stirrers, the Burau representation and
train-track algorithm both give the same entropy
bound of 0.96. The actual topological entropy of
this flow was found by computing the exponential
rate at which the boundary between the orange and
blue is stretched. This is plotted in Figure 2.
The computed value of the entropy is 1.27, and so
the estimate based on the braid alone is a
reasonable 76 sharp. Periodicity in one
direction For a periodic domain, in addition to
the braid elements si, we introduce stirrer
motions ti corresponding stirrers making a tour
in the periodic direction, as illustrated in
Figure 3. However, the dynamics on a
singly-periodic domain can be reduced to the
dynamics in a bounded domain, as long as we allow
for an extra fixed rod in the bounded domain.
This is because there exists a conformal mapping
from the periodic strip to a bounded annulus, and
the topological entropy of the flow is preserved
under this conformal mapping. Figure
2 Plot of orange-blue interface length versus
time for braids in Figures 1, 5 and 6. The length
is shown on a log scale. The slope of each curve
for large t gives the topological entropy of each
flow. Corresponding entropy lower bounds
according to the Burau representation and
train-track algorithm are shown in the table.
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• Thanks to Toby Hall for help with the train-track
  algorithm. This work was funded by the EPSRC.