Two-dimensional topological fluid dynamics

poisonmammeringMechanics

Oct 24, 2013 (3 years and 11 months ago)

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Two-dimensionaltopologicaluiddynamics
PhilipBoyland
Colloquium
FloridaStateUniversity
February22,2013
FSU,2013p.1
Modelinguids
Therearetwointerconnectedpartsofuidmodeling:
Lagrangian
whichfollowstheuidasitsmovesand
Eulerian
whichsitsata
pointandconsiderslocalquantitieslikethevelocity.
Lagrangian:
Thetrajectorythroughspaceofauidparticlebeginningat
positionxisgivenbyafunctionφ(t)withφ(0)=xandφ(t)
thepositionoftheparticleaftertimet.
IfMistheuiddomain,allthesetrajectoriesarecollected
togetherinasinglefunction,the
uidmotion
,
φ:M×R→M,
usuallywrittenφ
t
(x)=φ(x,t).
FSU,2013p.2
Modelinguids
Eulerian:
Thevelocityeld
is
u(φ
t
(x),t):=
∂φ
t
∂t
(x).(1)
Theequationsofuidmechanicsareusuallywrittenusingthe
velocityeld.Onethensolvesthe
advectionequation(1)
for
thetrajectories.
Forexample,theNavier-Stokesequation
ρ
DX
Dt
=−∇p
t
+
ν
ΔX,
withappropriateboundaryconditions,where
ν
istheviscosity
andp
t
isthepressure.
FSU,2013p.3
Deformationandvorticity
Whenwewatchauidevolve,thereseemtobe(atleast)two
fundamentalthingsgoingon,
stretchingandrotating
.
Theseareexpressedinnitesimallyateachpoint(Eulerian)using
thespacederivativeofthevelocityeld∇u.
FSU,2013p.4
Thedeformation
The
symmetric
partof∇ucanbeorthogonallydiagonalized
yieldingtheinstantaneous,innitesimal
deformation
(∇u)
sym
:=
∇u+(∇u)
T
2





d
1
00
0d
2
0
00d
3




Soagainlocallyandinstantaneously,dx/dt=(∇u)
sym
x
integratestotrajectories
Φ
t
(x)=




e
d
1
t
00
0e
d
2
t
0
00e
d
3
t




x
FSU,2013p.5
Vorticity
The
anti-symmetric
partof∇uyieldstheinstantaneous,
innitesimal
curlorvorticity
,~ω=(ω
1

2

3
),
(∇u)
anti
:=
∇u−(∇u)
T
2
=




0−ω
3
ω
2
ω
3
0−ω
1
−ω
2
ω
1
0




,
and(∇u)
anti
x=~ω×x
Soagainlocallyandinstantaneously,dx/dt=(∇u)
sym
x
integratestotrajectorieswhichrotatearoundtheaxis~ωwith
angularvelocity|~ω|.
FSU,2013p.6
Backtomodeling
SolvingtheNavier-Stokesequationisveryhard.Onemustthen
solvetheadvectionequation(1)forthephysicaluidtrajectories.
ItisnotclearatallhowtheEulerianinnitesimaland
instantaneousquantitiesofdeformationandvorticitycontributeto
actualmacroscopicLagrangiandeformationsandrotations.
Thusthereareahostofanalyticandqualitativemethodsfor
gettinginformationabouttheuidevolutionwithoutgoingthrough
alltheselevelsofanalysis.
Oneclassofmethodsgounderthenameof
TopologicalFluid
Mechanics
whichcombinesideasfromTopologyandDynamical
Systemstheory.Today'stalkisaboutsomeofthesemethodsin
two-dimensions
.
FSU,2013p.7
Knotting
Knotsareanessentialingredientof
three-dimensionaltopologyandthusof3D
uiddynamics.
Whatabouttwodimensions?
Co-dimensiontwoisnecessaryforknotting.
Pointsarecodimensiontwointhe
plane.Cantheybeknotted?Yes,if
weconsiderthe
motion
ofthepoints.
Question:
whataretheimplications
ofknottedpointmotions?
Answer:
Exponentialgrowthofmate-
riallines.
FSU,2013p.8
Experimentalillustration
ShowMovie
Veryroughly,ontheleft(pA)thereispuredeformation(the
rotationscancelout)andontheright((fo)thereispurerotation.
Ontheleft(pA)materiallinesaregrowingexponentiallyinlength
andandontheright((fo)thereislineargrowth.
Theleftclearlymixesbetterthantheright.
Foraveryviscousuidthetwoprotocolsrequireveryclosetothe
sameenergy.
Notetheemergentstructureontheleft.
Today'stalkwillfocuson
theexponentialgrowthofmateriallines
anditsimplicationsfordeformation,vorticity,andmixing
.
FSU,2013p.9
Outline
Part1:
KinematicsandTopologicalKinematics.
Part2:
Consequencesforpassivelytransportedscalars(eg.the
creaminyourcoffee).
Part3:
ApplicationstoEulerows.
Part4:
Choosingprotocolstomaximizeatopologicalmeasureof
mixingefciency.
FSU,2013p.10
Mainideas:
Topologicalone-dimensionalgrowthofmateriallinesis
computablefromknottedpointmotions.
Topologicalgrowthislowerboundformetricgrowth.
Metricone-dimensionalgrowthofmateriallinesisapplicableto
two-dimensionaluidmechanics.
Exponentialgrowthofmateriallinesimpliesexponential
growthofgradientsofpassivelytransportedscalars.
Largegradientsenhancediffusionandthusmixing.
Interfacesbetweenmaterialsareone-dimensionalcurvesand
longinterfacesalsoenhancemixing.
Progressionofideas:
Topology→Geometry→Analysis→Fluid
Mechanics.
FSU,2013p.11
Part1
Kinematicsandtopologicalkinematics
FSU,2013p.12
Basicdenitions:Theuidregion
The
uidregion
isasmooth,
one-parameterfamilyof
smooth,multi-connected,
compact,planardomains
M
t
.
Theouterboundaryisheldxedwhiletheinnerdisksmove.
Alwaysassume
time-periodicity
,M
t+1
=M
t
,andthusmodel
stirringbymovingrodsandusetheterminology
stirringprotocol
to
describethemovingregionsM
t
.
Themovingregionsarecalledthe
stirrers
,andtheyareperhaps
permutedeachcycle.
FSU,2013p.13
Basicdenitions:theuidmotion
The
uidmotion
isasmoothone-parameterfamilyof
diffeomorphisms,φ
t
:M
0
→M
t
,withφ
0
=id,justkinematics.
Viewφ
t
asLagrangianuiddisplacementmap:particleatx∈M
0
attime0isatφ
t
(x)∈M
t
attimet.
Iamavoidingtheterminology
ow
becauseindynamical
systemstheorythismeansanR-action,ie.asteadyowinuid
mechanics,whichusuallywon'tbethecasehere.
Theuidmotionis
incompressible
ifitpreservesLebesgue
measureorequivalently,det(Dφ
t
)≡1.
FSU,2013p.14
Basicdenitions:thevelocityeld
Thevelocityeld
is
u(φ
t
(x),t):=
∂φ
t
∂t
(x).
Sinceφ
t
:M
0
→M
t
thevelocityeldsatisestheboundary
conditionsun
i
=
˙
B
i
n
i
,withB
i
themotionofthei
th
boundary.
Initiallystrictly
kinematicsordynamicalsystems
andsothe
velocityeldis
not
yetassumedtosatisfyanyparticularequation.
Thestirringprotocolistime-periodicbutthevelocityeldis
perhaps
not
andsothereisnoPoincarémapingeneral.
Theuidmotionis
incompressible
iffdivu≡0.
FSU,2013p.15
One-dimensionalmetricgrowthrate
A
materialline
intheuidisdescribedbyasmooth
arc
or
simple
closedcurve(scc)
γ.Letℓ
t
(γ)beitslengthwithrespecttosome
smooth,periodicfamilyofRiemannianmetricsontheM
t
.
The
metricgrowthrate
ofγisthegrowthof
L
met
t
(γ):=

t

t
◦γ)

0
(γ)
.
Onsurfaces,themaximalexponentialmetricgrowthrategivesthe
topologicalentropy,connectedtoLyapunovexponents,etc.
FSU,2013p.16
One-dimensionaltopologicalgrowthrate
Forthe
topologicalgrowthrate
,wecomputeleastlengthina
homotopyclass,orequivalently,thelengthofanappropriate
geodesic.
Tomaketheresultstrictlytopologicalrestrictconsiderationtojust
topologicallyessential
curves.
An
essentialarc
isone
thatconnectstwodifferent
boundarycomponents.An
essentialsimpleclosecurve
(scc)
isonethatisneither
null-homotopicnorbound-
aryparallel
FSU,2013p.17
One-dimensionaltopologicalgrowthrate
Thehomotopyclassofanessentialarcallowstheendpointsto
slidealongtheboundaryandforsccusefreehomotopyclasses.
Inbothcasestheclassisdenoted[γ].
The
leastlength
amongcurvesinγ

shomotopyclassis
L
top
(γ):=min{ℓ(σ):σ∈[γ]}
The
topologicalgrowthrateoftheclass
[γ]isthegrowthof
L
top
t
(γ)=
L
top

t
◦γ)
L
top
(γ)
,
Soweevolvecurveforwardfortimetandthenshrinktotheleast
lengthinhomotopyclass.
N.B.
Foranessentialcurveγ,L
met
t
(γ)≥L
top
t
(γ).
FSU,2013p.18
Theleastlengthinahomotopyclass
One-dimensionaltopologicalgrowthrate
Thetopologicalgrowthrateonlydependsontheroughtopology
ofthestirrermotion.
Moreprecisely,recalltwohomeomorphismsf
0
,f
1
:M
0
→M
0
are
isotopic
ifthereisacontinuousfamilyofhomeomorphismsf
t
deformingonetotheother.
Thetopologicalgrowthrateofanessentialcurveγdependsonly
ontheisotopyclassofφ
1
(sincetheprotocolisperiodic,thisisthe
sameisotopyclassasφ
n
foralln∈N).
Thetopologicalgrowthisthesameasthe
growthrateofword
length
fortheinducedmaponthefundamentalgroup(Cayley
graphoffundamentalgroupisquasi-isometrictotheuniversal
cover).
FSU,2013p.20
Thurston-Nielsentheory
Inthelanguageofthistalk,the
Thurston-Nielsentheory
classies
surfacemapsandtheirisotopyclassesintermsoftherateof
topologicalone-dimensionalgrowth,linearorexponential,and
givemethodsforcomputingthegrowthassociatedwithspecic
protocols.
Thefulltheorydealswithisotopy(mappingclasses)onany
surface.
FSU,2013p.21
TheThurston-Nielsentrichotomy
LetM
t
beperiodicstirringprotocolwithuidmotionφ
t
.Theneither
1.
PseudoAnosov(pA):
thereexistconstantsλ>1
(thedilation)
and
0<C
1
<C
2
suchthatforeveryessentialcurveγ,
C
1
λ
t
≤L
top
t
(γ)≤C
2
λ
t
.
2.
Finiteorder(fo):
thereexistsaconstantK>0suchthatforevery
essentialcurveγ,
L
top
t
(γ)<Kn.
3.
Reduciblecase:
(roughlystated)M
0
splitsintoφ
1
-invariant
subsurfacesonwhich(1)or(2)holds.
FSU,2013p.22
Experiment by Mark Stremler, see Boyland, P., Aref, H. and Stremler, M., Topological

fluid mechanics of stirring,
J. Fluid Mech
.,
403
, 277
--
304, 2000.

PseudoAnosov

Finite Order

Initial state

Experiment by Mark Stremler, see Boyland, P., Aref, H. and Stremler, M., Topological

fluid mechanics of stirring,
J. Fluid Mech
.,
403
, 277
--
304, 2000.

PseudoAnosov

Finite Order

1 iterate

Experiment by Mark Stremler, see Boyland, P., Aref, H. and Stremler, M., Topological

fluid mechanics of stirring,
J. Fluid Mech
.,
403
, 277
--
304, 2000.

PseudoAnosov

Finite Order

2 iterates

Experiment by Mark Stremler, see Boyland, P., Aref, H. and Stremler, M., Topological

fluid mechanics of stirring,
J. Fluid Mech
.,
403
, 277
--
304, 2000.

PseudoAnosov

Finite Order

9 iterates

RemarksonTNtrichomtomy
Wecallthestirringprotocolniteorder,pseudoAnosov,or
reducibleaccordingtotheTN-type.
WefocusherejustonthepseudoAnosovcase,where
every
essentialcurvehasthesame
topological
exponentialgrowthrate,
namely,λ.
Thisis
independentofthedetailsoftheuid
motion,butjust
dependsonthetopologyofthestirrermotionasdescribedshortly.
Thetopologicalgrowthisjustalowerbound,themetricgrowth
couldbemuchlarger.
Inthe
pseudoAnosov
casethetheorygivesmuchmore
informationaboutthedynamics:waystocomputethe
dilatation
λ,
periodicorbitsandinvariantmeasuresthatmustbepresent,a
lowerboundoflog(λ)forthetopologicalentropy,etc.
FSU,2013p.23
Braids,stirringprotocolsandisotopyclasses
TNtypeandthetopologicalone-dimensionalgrowthdependjust
ontheistotopyclassofφ
1
.
Theisotopyclassjustdependsonthetopologyofthemotionof
thestirrersandthisinturncanbevisualizedandcharacterizedby
theirspace-timetraceor
braid
.
ThealgebraofthebraidcanbeusedtocomputetheTN-type.
Thetwoprotocolsoftheex-
perimenthaveinequivalent
braids;oneisniteorder
(lineargrowth)andtheother
pA(exponentialgrowth).
PseudoAnosov
Finite Order
FSU,2013p.24
Braidsandstirringprotocols
Takenfrom
FinnandThiffeault
.
FSU,2013p.25
Part2
Passivelyadvectedscalars
FSU,2013p.26
Basicdenitions
Givenauidmotionφ
t
,afunctionα:M
t
×R→Riscalleda
passivelyadvectedscalar
ifitisconstantontrajectories,
α
t

t
(x))=α
0
(x),
orequivalently,
i
∂α
t

t
(x))
∂t
=0,
wherewehavewrittenα
t
(x)forα(x,t).
Examples,dyeinuid,sugarinchocolate,orcreamincoffee
ignoringdiffusion
.
Inthelanguageofglobalanalysisonesaysthatα
t
isthe
push
forward
ofα
0
andwrites(φ
t
)


0
)=α
t
,with

t
)


0
)=α
0
◦(φ
t
)
−1
.
FSU,2013p.27
Twofundamentaltypesofadvectedscalars
Foranyfunctionf:M
0
→Rweobtainapassivelyadvected
scalarjustbydeningα
t
:=(φ
t
)

(f),andsoonlytheinitial
congurationandtheuidmotionmatter.
Forexample
,ifthedensityofadyetracerisinitiallygivenbyα
0
aftertimetthedensityisgivenbyα
t
:=(φ
t
)


0
)
However
,sometimesinaphysicaluidα
t
mayrepresentascalar
ofinterestthatiscomputedateachtime
fromthevelocityeld
.
Thusisrepresentsaconservedquantity.
Forexample
,intwodimensionsthecurl,ω
t
=∇×u,isa
passivelyadvectedscalarforanEulerow.
Therstcaseisrelevanttomixingwhilethesecondto
understandingdynamicsofEuleruids.
FSU,2013p.28
HeuristicconsequencesofapseudoAnosovprotocol
ForpseudoAnosovstirringprotocolthe
metriclengthofessential
curves
isgrowingexponentiallyfast.
Thus
tangentvectors
tothesecurves(materiallineelements)
mustbegrowinginlengthexponentiallyundertheactionofthe
spacederivativeoftheuidmotion,Dφ
t
.
Forapassivelyadvectedscalarα
t
,sinceα
t

0
◦(φ
t
)
−1
,
∇α
t
=∇α
0
(Dφ
t
)
−1
.
Ifuidmotionisincompressible,det(Dφ
t
)=1,andso(Dφ
t
)
−1
alsohasaneigenvaluegrowingexponentially.
Thus
|∇α
t
|isgrowingexponentially.
FSU,2013p.29
Issueswiththeargument
Thereare(atleast)three
problems
withmakingthisrigourous.
Thepositionalongthemateriallineatwhichwehavegrowth
oftangentvectorsismovingintime
Therecouldbeanunfortunatecoincidencewhere∇α
0
stays
alignedwiththestableeigen-directionof(Dφ
t
)
−1
.
Thescalarcouldhavepatcheswhere∇α
0
≡0andthese
couldbejustwherethemateriallinesarestretching.
Theseissuesaredealtwithbyusingamore
global
argumentand
assumingtheinitialcongurationofthescalaris
generic
ina
precisesense.
FSU,2013p.30
Theoremonpassivelyadvectedscalars
Theorem:
M
t
isatime-periodicstirringprotocolof
pA
typewith
incompressibleuidmotionφ
t
.Ifα
t
isapassivelyadvectedscalar
suchthatitsinitialstateα
0
isagenericC
2
-function,thenthereare
positiveconstantsc,c

sothat
sup
x∈M
0
|∇α
t
(x)|≥cλ
t
and
Z
M
t
|∇α
t
(x)|≥c

λ
t
forallt∈R,whereλ>1isthedilationofthepseudoAnosovprotocol.
Thuskα
t
k
C
1
andkα
t
k
1,1
bothgotoinnityexponentiallyfast.
FSU,2013p.31
Ideaofproof
FindaC
2
-open,densesetGinsidetheMorsefunctionsonM
0
so
thatα
0
∈Gimpliesthatα
0
hasabandofregularinverseimages
ofessentialarcsorcircles.
ThepAprotocolforcesstretchinlengthbyλ
t
.Thiscoupledwith
areapreservationandthepassivetransportforcelevelsetsof
α
t
=(ϕ
t
)


0
)tobunchupintransversedirection,whichcauses
k∇α
t
k

→∞likeλ
t
.
FSU,2013p.32
Noteontime-periodicuidmotions
Ifthe
velocityeldistime-periodic
withperiodone,thenthereisa
Poincarémapφ
1
whichsatisesφ
n
=(φ
n
)
whereasuperscriptis
repeatedcomposition.
Ifthereisapassivelyadvectedscalarα
t
thatdependsonthe
velocityeld,thenα
t+1

t
.
Thismeansthatthescalarisan
integralofmotion
α
0
(x)=α
1

1
(x))=α
0

1
(x)).
Ifα
0
isnon-degenerateenough(eg.C
2
-genericagain),this
impliesthatthe
dynamics
ofφ
t
are
verysimple
,inparticular,ithas
zerotopologicalentropyandatmostlinearone-dimensional
metricgrowthofarcs.
FSU,2013p.33
Part3
ApplicationstoEulerFluidMotions
FSU,2013p.34
Euleruidmotions
Nowassumethevelocityeldu(x,t)oftheuidmotionφ
t
satisestheincompressible,constantdensity(ρ≡1),Euler
equation
Du
Dt
=−∇p
t
,div(u)=0,
withslipboundaryconditionsonthemovingboundary.
Thenφ
t
iscalledan
Euleruidmotion
.
Recallthatfor
two-dimensional,divergence-free
velocityeldsa
classicalresultsaysthatthe
curl
coupledwiththe
circulations
aroundtheboundarycomponentsandthe
boundaryconditions
un
i
=
˙
B
i
n
i
determinetheeldcompletely.
FSU,2013p.35
Existenceofsolutions:2DEuler
OnehasglobalclassicalsolutionsoftheincompressibleEuler
equationsinthiscase:
Theorem(Kozonoi1985)Givenasmoothfamilyofsmooth
compactplanarregionsM
t
andanysmoothdivergence-free
vectoreldu
0
withslipboundaryconditionsu
0
n
i
=
˙
B
i
n
i
(or
equivalentlyinitialcurlandcirculations)
onM
0
,thereisaunique,
smoothEuleruidmotionwiththatinitialdata.
Ingeneral,weassumeaglobalsolutionwiththeregularityofthe
initialdataandanalyzeitsdynamics.
FSU,2013p.36
Helmholtz-Kelvintheorem:Euleruidmotions
TheHelmholtz-KelvinTheorem(1890's)allowsonetousethe
methodsofdynamics/globalanalysisonEuleruidmotions.
Helmholtz-KelvinTheorem:
Atwo-dimensionalarea-preserving
uidmotion(M
t

t
)isEuler
ifandonly
ifitsvorticityispassively
transported,
∂ω
t

t
(x))
∂t
=0
andcirculationsaroundallsmoothsimpleclosedcurvesCare
preserved,
d
dt
I
φ
t
(C)
udr=0.
Thepreservationofboundarycirculationisafeatureof
multi-connecteddomains.
FSU,2013p.37
Transportofvorticity
FSU,2013p.38
Transportofvorticity
FSU,2013p.39
Anexponentialgrowththeorem
Helmholtz-Kelvinsays
vorticityisapassivelyadvectedscalar
for
anEuleruidmotionandsowemayusethetheoremabove:
Theorem:
LetM
t
beatime-periodicstirringprotocolof
pA
type
withEuleruidmotionφ
t
.Iftheinitialvorticityω
0
isageneric
C
2
-function,therearepositiveconstantsc,c

sothat
sup
x∈M
0
k∇ω
t
(x)k≥cλ
t
and
Z
M
t
k∇ω
t
(x)k≥c

λ
t
forallt∈Rwhereλ>1isthedilationofthepAprotocol.
ThuskΔu(x,t)k

=k∇ω
t
k

→∞,ku
t
k
C
2
→∞,and
ku
t
k
2,1
→∞alllikeλ
t
souis
nottime-periodic
.
FSU,2013p.40
Remarksonexponentialgrowththeorem
Yudovich(1974,2000)andothersshowedlineargrowthofk∇ω
t
k
forperturbationsofmanytwo-dimensionalsteadyEuleruid
motions.Theseresultsarecentraltostabilityanalysis.
Arnol'd(1972),FriedlanderandVishik(1992)andothershave
showntheimportanceofexponentialgrowthofdistortionfor
stabilityanalysis.
Thebasicmechanisminplayhereisthesame:unbounded
distortionast→∞.Sinceω
t

0
◦φ
−1
t
,
∇ω
t
=∇ω
0
(D
x
φ
t
)
−1
Herethegrowthofk∇ω
t
kisexponentialandforcedbythe
topology
ofthepAstirringprotocolwhichforcesthemaximal
spectralradiusofD
x
φ
t
togrowlikeλ
t
.
FSU,2013p.41
TheenergyofastirredEuleruidmotion
TotalenergyisconservedforEuleruidmotionsinstationary
boundeddomains.Whathappenswithmovingboundary?Since
k∇ωk→∞,perhapstheenergyisalsounbounded?
Usualargumentforasteadydomainyields
dE
dt
=−
X
I
φ
t
(C
i
)
p
˙
B
i
dn
i
.
sinceuidcandoworkonthestirrersandviceversa(withthe
sumovertheboundarycircles).
However,fairlystandardargumentsyieldthatfor
periodic
boundarymotiontheenergyisuniformlyboundedintime.
Question:
Doestheenergyoscillate,gotoanasymptote,stay
constant,etc.?
FSU,2013p.42
SpeculationsonapplicationstomoregeneralEulerows
Mainidea:
Usepointsintheuidasvirtualstirrersorghost
rods(Bowen1978,ThiffeaultandFinn2006).
ExtendTN-theorytogetexponentialtopologicalgrowthfromaset
ofpointsfornon-periodicuidmotions(probablydoable).
Showthatfortypicalinitialvorticity,atwo-dimensionalEuleruid
motionalwayshassuchorbits(??).
Providesanewperspectiveonaversionofthe
Yudovich
Hypothesis/Conjecture
:Forgenericinitialvorticitya
two-dimensionalEuleruidmotionsatises
kΔu(x,t)k

=k∇ω
t
k

→∞
k∇ω
t
k
L
1
→∞
exponentiallyfast.
FSU,2013p.43
Part4
Optimizingatopologicalmeasureofmixingefciency
withJasonHarrington
FSU,2013p.44
Mixing
Mixing
referstotheprocessbywhichdifferentmaterialsare
intermingledbystirringauid.Examplesincludeplastics,
cosmetics,rubber,candy,paint,creamincoffee,....
Inthemostbasicmodelsoneusuallyconsidersjusttheuid
evolutionwithnodiffusionorchemicalreactions.Thisis
sometimescalled
stirring
todistinguishitfrommoredetailed
models.
Turbulentuidsmixwell,butapplicationsdemandusingminimal
energyandavoidingtearing,bubbles,etc.
FSU,2013p.45
Mixing
Itisclearthat
stretching
(andthusfolding)ofmateriallinesis
necessaryforgoodmixing(butmaybe
notsufcient
).
Wehaveseenthatexponentialgrowthofmateriallinescauses
exponentialgrowthofgradientsoftransportedscalars.
Also,intwodimensionsinterfacesbetweenmaterialsare
one-dimensional,thisgivesrisetothepossibilityofenhanced
diffusionacrosstheinterface.
Thusitisreasonabletousetheglobalratesof
stretchingof
materiallines
asanapproximate
measureofgoodmixing
.
Aswehaveseen,apAstirringprotocolalwayscauses
exponentialstretchingofmateriallineswithaspeciedrateofλ
t
soweuseλasonemeasureofmixing.
FSU,2013p.46
Entropyefciency
Thusweseekprotocolswhich
maximizetheλ
while
minimizing
theamountofstirrermotion
.
Wemeasuretheamountofstirrermotiontopologically,butother
measuresmightbemorerealistic.
The
entropyefciency
isthusthestretchfactorλnormalizedby
takingthek
th
rootwheretheprotocoluseskseparatestirrer
motions(theunitstirrermotionwillbedenedshortly).
Noteonterminology:
Itiscommontakealogarithmandthe
entropyoftheprotocolisdenedtobelog(λ)andthenthe
efciencywouldbelog(λ)/k.
Wearethusfacedwith
nonlinear,topologicaloptimization
problem
ofmaximizingtheentropyefciencyamongsomeclass
ofprotocols.
FSU,2013p.47
π
i
-stirringprotocols
We
restrict
toaspecialclassofstirring
protocols.inwhichasinglestirrerSmoves
aroundNxedobstacles.
Eachsuchprotocolisuniquelydescribedbyaclosedpathstarting
andendingatS.
Thusthecollectionofsuchprotocolsisnaturallyisomorphicto
π
1
(diskminusNpoints),thefreegrouponNletters(moreonthe
fundamentalgrouplater).
FSU,2013p.48
Entropyefciencyofaπ
1
-protocol
Thegeneratorα
4
.
Wemaywriteeach
π
1
-stirringprotocol
uniquelyas
η=α
ǫ
i
1
i
1
...α
ǫ
i
1
i
k
where
α
j
isgoingaroundthe
j
th
holeonceclockwise
andǫ
i
j
=±1.
Usingλ(η)todenotethetopologicalstretchrateassociatedwith
theprotocolηits
entropyefciency
is
eff(η)=λ(η)
1
#(η)
,
where#(η)isthenumberofα
j
usedinη.
FSU,2013p.49
Maximumentropyefciency
ForeachN,letPP(N)bethegroupofallπ
1
-protocolswithN
xedobstacles.
Inthiscontextthe
maximalentropyefciency
foragivenNis
Eff(N):=sup{eff(η):η∈PP(N)}
Numerically,Eff(N)appearstobeachievedby
HSP
N
:=α
1
α
2
...α
N
.
Whatcanyouprove?
FSU,2013p.50
Maximumefciency;N=2
Thecaseof
two
obstaclesusesspecialmethods.
Inthiscaseastandardtrick(hyperellipticinvolution)liftsthe
problemtolinearautomorphismsofthetwotorus.
Thensolvetheoptimizationproblemtherebyhand:themaximal
entropyefciencyforN=2isrealizedbytheprotocolα
1
α
−1
2
and
hasvalueEff(2)=1+

2.
Thepathofthestirrerisagureeightwith
anobstacleinsideeachloopandthispro-
tocolisoftencalledthe
taffypuller
FSU,2013p.51
TheoremonmaximumefciencywhenN>2
Roughly,Eff(N)is
asymptotically3
.
Theorem:
ThereareexplicitlydenedmatricesH
(N)
and
ˆ
H
(N)
with

3
N
−3N−1
N

1/N
≤ρ(H
(N)
)
1/N
≤Eff(N)
≤ρ(
ˆ
H
(N)
)
1/N
≤(3
N
−2)
1/N
,
whereρ(M)isthespectralradiusofamatrixM.
ThusEff(N)→3asN→∞.
Heuristically,thebestyoucandoistotriplelengthswitheach
stirrerloop.
FSU,2013p.52
Theoremonmaximumefciency
Intuition:
Eachloopingaroundaholeaddsprevioustimesthree,
yielding1+3++3
N−1
=(3
N
−1)/2forNloops.
FSU,2013p.53
Remarksontheorem
Numericalobservation:
ρ(H
(N)
)=3
N
−L(N)and
ρ(
ˆ
H
(N)
)=3
N

ˆ
L(N)tohighaccuracyforlinearfunctionsLand
ˆ
L.
ρ(H
N
)areallSalemnumbersandρ(
ˆ
H
N
)areallPisotnumbers
PlotofNvslog(Eff(N)).
FSU,2013p.54
PisotandSalemnumbers
A
Pisotnumber
isarealalgebraicintegerα>1suchthatallits
Galoisconjugatesarelessthan1inmodulus.
A
Salemnumber
isarealalgebraicintegerα>1suchthatallits
Galoisconjugatesarelessthanorequalto1inmodulusandat
leastoneconjugateisontheunitcircle.Thisimpliesthat1/aisa
Galoisconjugateandallotherconjugatesareontheunitcircle.
FSU,2013p.55
Stepsinproofoftheorem
The
lowerbound
Sincetheentropyefciencyisamaxoverprotocols,anyprotocol
canbeusedasalowerbound.
ComputetheentropyefciencyofthenumericalwinnerHSP
N
.
Thiscomputationalsoinvolvesalinearization,butthistimeusing
homologyinaspecialcoveringspace.
Finally,estimatethespectralradiusofamatrix
ˆ
H
(N)
.
FSU,2013p.56
Stepsinproofoftheorem
The
upperbound
Transformthetopologicaloptimizationproblemtoanonlinear
algebraiconeusingalgebraictopology,specically,the
fundamentalgroup.
Showthatthesolutiontothisproblemisboundedabovebythe
solutiontoitslinearanalog(thejointspectralradius)
Provetheneededjointspectralradiusisachievedbythematrix
H
(N)
.
FSU,2013p.57
Entropyefciencyusingneighborswaps
Thenaturalrstquestionistoconsiderthemaximalstretchrate
perunitswapofadjacentstirrers(thesearetheusualgenerators
ofthebraidgroup).
FinnandThiffeault(2010)usingtheargumentdevelopedhere
showthatthemaximalentropyefciencywiththesegeneratorsis
boundedaboveby(1+

5)/2.
Theboundisachievedfor3rodsandforn>3rodsthemaximal
entropyefciency
decreases
forincreasingn.
Theyalsoconsideraclassofprotocolswhereawholecollection
ofrodsmoveatonce.
Weconsiderhereprotocolswhereasinglerodmovesinwhich
casethemaximalentropyefciency
increases
forincreasingn.
FSU,2013p.58
Conclusions
Intwodimensionstheproperbraidingorknottingofuid
trajectoriesgivesrisetothe
exponentialstretching
oftopologically
essentialmateriallines.
This,inturn,impliestheexponentialgrowthofthemaximum
deformationandthusofthe
gradientsofanytransportedscalar
.
Applicationsto
Euler
uidmotionsthenfollowfromthe
Helmholtz-KelvinTheorem.
Onemayformulateandinsomecasessolvethe
topological
optimizationproblem
ofmaximizingthestretchwhileminimizing
thestirrermotion.
FSU,2013p.59