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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Topology, Braids, and Mixing
Jean-Luc Thiffeaultwith
Matthew Finn and Emmanuelle Gouillart
Department of Mathematics
Imperial College London
AMMP Colloquium, 1 November 2005
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
The Taffy Puller
[movie 1]
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
The Four-pronged Taffy Puller
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Experiment of Boyland, Aref, & Stremler
[P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)]
[P. L. Boyland, M. A. Stremler, and H. Aref, Physica D 175, 69 (2003)]
[movie 2] [movie 3]
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
The Connection with Braids
6
t
PSfrag replacements
σ1
PSfrag replacements
σ−2
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Generators of the n-Braid GroupPSfrag replacements
σ−1i
σi
i − 1 i + 1 i + 2i
PSfrag replacements
σ−1i
σi
i − 1 i + 1 i + 2i
A generator of Artin’s braid groupBn on n strands, denoted
σi , i = 1, . . . , n − 1
is the clockwise interchange of thei th and (i + 1)th rod.
Bn is a finitely-generated group,with an infinite number of ele-ments, called words.
These generators are used to characterise the topological motion ofthe rods.
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Presentation of Artin’s Braid Group
The generators obey the presentation
...... ......
... ...... ... ......
=
=
σi+1 σi σi+1 = σi σi+1 σi
σiσj = σjσi , |i−j | > 1
A presentation means that these are the only rules obeyed by thegenerators that are not the consequence of elementary groupproperties.
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
The Two BAS Stirring Protocols
σ1σ2 protocol
σ−11 σ2 protocol
[P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)]
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Train-TracksWhat is the growth rate of an “elastic band” tied to the rods?
Train-tracks give the answer.
PSfrag replacementsa
ab
b
1
2
3
Elastic band has edges (letters) and infinitesimal loops (numbers).As the rods are moved, the edges and loops are mapped as
a 7→ a2b, b 7→ a2b3b, 1 7→ 3, 2 7→ 1, 3 7→ 2.
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Train-TracksWhat is the growth rate of an “elastic band” tied to the rods?
Train-tracks give the answer.
PSfrag replacementsa
ab
b
1
2
3
Elastic band has edges (letters) and infinitesimal loops (numbers).As the rods are moved, the edges and loops are mapped as
a 7→ a2b, b 7→ a2b3b, 1 7→ 3, 2 7→ 1, 3 7→ 2.
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Train-TracksWhat is the growth rate of an “elastic band” tied to the rods?
Train-tracks give the answer.
PSfrag replacementsa
ab
b
1
2
3
Elastic band has edges (letters) and infinitesimal loops (numbers).As the rods are moved, the edges and loops are mapped as
a 7→ a2b, b 7→ a2b3b, 1 7→ 3, 2 7→ 1, 3 7→ 2.
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
The Evolution of Edges: Topological Entropy
The edges and loops are mapped according to
a 7→ a2b, b 7→ a2b3b, 1 7→ 3, 2 7→ 1, 3 7→ 2.
A crucial point is that edges are separated by loops: nocancellations can occur. A transition matrix can be formed:
M =
1 1 0 0 01 2 0 0 0
0 0 0 1 01 1 0 0 10 1 1 0 0
←[ a
←[ b
←[ 1←[ 2←[ 3
The largest eigenvalue gives the asymptotic growth factor of theelastic, 2.6180. The logarithm of this is the topological entropy ofthe braid.
[M. Bestvina and M. Handel, Topology 34, 109 (1995)]
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
The Difference between BAS’s Two Protocols
• Practically speaking, the topological entropy of a braid is alower bound on the line-stretching exponent of the flow!
• The first (bad) stirring protocol has zero topological entropy.
• The second (good) protocol has topological entropylog[(3 +
√5)/2] = 0.96 > 0.
• So for the second protocol the length of a line joining the rodsgrows exponentially!
• That is, material lines have to stretch by at least a factorof 2.6180 each time we execute the protocol σ−1
1 σ2.
• This is guaranteed to hold in some neighbourhood of the rods(Thurston–Nielsen theorem).
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
The Difference between BAS’s Two Protocols
• Practically speaking, the topological entropy of a braid is alower bound on the line-stretching exponent of the flow!
• The first (bad) stirring protocol has zero topological entropy.
• The second (good) protocol has topological entropylog[(3 +
√5)/2] = 0.96 > 0.
• So for the second protocol the length of a line joining the rodsgrows exponentially!
• That is, material lines have to stretch by at least a factorof 2.6180 each time we execute the protocol σ−1
1 σ2.
• This is guaranteed to hold in some neighbourhood of the rods(Thurston–Nielsen theorem).
![Page 15: Topology, Braids, and Mixingjeanluc/talks/imperial_ammp-colloq2005.pdf · Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions References T{G{F, \The Size of](https://reader034.vdocuments.us/reader034/viewer/2022042708/5f3828bcfde1260c7a57d284/html5/thumbnails/15.jpg)
Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
The Difference between BAS’s Two Protocols
• Practically speaking, the topological entropy of a braid is alower bound on the line-stretching exponent of the flow!
• The first (bad) stirring protocol has zero topological entropy.
• The second (good) protocol has topological entropylog[(3 +
√5)/2] = 0.96 > 0.
• So for the second protocol the length of a line joining the rodsgrows exponentially!
• That is, material lines have to stretch by at least a factorof 2.6180 each time we execute the protocol σ−1
1 σ2.
• This is guaranteed to hold in some neighbourhood of the rods(Thurston–Nielsen theorem).
![Page 16: Topology, Braids, and Mixingjeanluc/talks/imperial_ammp-colloq2005.pdf · Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions References T{G{F, \The Size of](https://reader034.vdocuments.us/reader034/viewer/2022042708/5f3828bcfde1260c7a57d284/html5/thumbnails/16.jpg)
Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
The Difference between BAS’s Two Protocols
• Practically speaking, the topological entropy of a braid is alower bound on the line-stretching exponent of the flow!
• The first (bad) stirring protocol has zero topological entropy.
• The second (good) protocol has topological entropylog[(3 +
√5)/2] = 0.96 > 0.
• So for the second protocol the length of a line joining the rodsgrows exponentially!
• That is, material lines have to stretch by at least a factorof 2.6180 each time we execute the protocol σ−1
1 σ2.
• This is guaranteed to hold in some neighbourhood of the rods(Thurston–Nielsen theorem).
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
One Rod Mixer: The Kenwood Chef
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Poincare Section
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Stretching of Lines: A Ghostly Rod?
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Stretching of Lines: A Ghostly Rod?
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Particle Orbits are Topological ObstaclesChoose any fluid particle orbit (blue dot).
Material lines must bend around the orbit: it acts just like a “rod”!
[J-LT, Phys. Rev. Lett. 94, 084502 (2005)]
Today: focus on periodic orbits.
How do they braid around each other?
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Motion of Islands
Make a braid from the mo-tion of the rod and the peri-odic islands.
Most (74%) of the line-stretching is accounted forby this braid.
[G–T–F, “Topological Mixing with Ghost Rods,” preprint, 2005.]
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Motion of Islands and Unstable Periodic Orbits
Now we also include unstableperiods orbits as well as the sta-ble ones (islands).
Almost all (99%) of the line-stretching is accounted for bythis braid.
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Conclusions
• Topological chaos involves moving obstacles in a 2D flow,which create nontrivial braids.
• A braid with positive topological entropy guarantees chaos insome region.
• Periodic orbits make great obstacles (in periodic flows),especially islands.
• This is a good way to “explain” the chaos in a flow —accounts for stretching of material lines.
• Other studies:• braids on the torus and sphere;• random braids;• optimisation via braids;• applications to open flows. . .
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
Conclusions
• Topological chaos involves moving obstacles in a 2D flow,which create nontrivial braids.
• A braid with positive topological entropy guarantees chaos insome region.
• Periodic orbits make great obstacles (in periodic flows),especially islands.
• This is a good way to “explain” the chaos in a flow —accounts for stretching of material lines.
• Other studies:• braids on the torus and sphere;• random braids;• optimisation via braids;• applications to open flows. . .
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Topological Mixers Braids Stretching of Lines The Kenwood Chef Conclusions
References
• T–G–F, “The Size of Ghost Rods,” in Proceedings of the
Workshop on Analysis and Control of Mixing
(Springer-Verlag, 2006, in press).
• G–T–F, “Topological Mixing with Ghost Rods,” preprint,2005.
• F–T–G, “Topological Chaos in Spatially Periodic Mixers,” insubmission, 2005.
• T, “Measuring Topological Chaos,” Physical Review Letters
94 (8), 084502, March 2005.
Preprints and slides available at www.ma.imperial.ac.uk/˜jeanluc