GFD Program 2010
atWoods Hole Oceanographic Ins<tu<on

19th/January/2011 WTK オンラインセミナー
小布施 祈織 （京都大学数理解析研究所）

Fellows

AmericanDutch
RussianJapanese
KoreanBrit(English)

IndianBrit(ScoJsh)Woods HoleTHE LIFE AND THE LECTURES

WHOI

Woods HoleVillage facili<esWoods Hole

Woods Hole

Accomoda<onWalsh CoPageLibraryLecturers

Jean‐Luc Thiﬀeault,
University of
Wisconsin, Madison+ Neil
Balmforth,
University ofBri<sh Columbia

Glenn Flierl, MIT

Antonello Provenzale,
Is<tuto di Scienze
Dell'Atmosfera

Lecturers
Lecture 1: S<rring and Mixing (Jean‐Luc Thiﬀeault)
Lecture 2 : Introduc<on to Biological models (Glenn Flierl)
Lecture 3 : Eﬀec<ve Diﬀusivity and Swimming Organisms (Jean‐Luc Thiﬀeault)
Lecture 4 : Local Stretching Theories (Jean‐Luc Thiﬀeault)
Lecture 5 : Social Behaviour, Mixing, and the Evolu<on of Schooling (Glenn Flierl)
Lecture 6 : Mixing in the Presence of Sources and Sinks (Jean‐Luc Thiﬀeault)
Lecture 7 : Examples at the Mesoscale (Antonello Provenzale)
Lecture 8: Dynamics of Heavy Impuri<es with Finite Size (Antonello Provenzale)
Lecture 9: Plankton Sinking and the Role of Turbulence (Antonello Provenzale)
Lecture 10: Evolu<onary Models: Movement and Mixing in Trait and PhysicalSpace Glenn Flierl)

Lecturer notes

BBQ

Beer, beer, …, beer, vodka, vodka, wine, cocktail

BeachSohballGeorge Veronis,Yale University

SohballJapanese PartyOthers

THE PROJECTTrajectories of a treadmilling swimmer
Can it escape from its image?

Kiori Obuse (RIMS, Kyoto University)
Supervised by
Jean‐Luc Thiﬀeault (University of Wisconsin)

Microorganism
Bacillus

Chlamydomonas

VolvoxhPp://blackmonsbacillus.pbworks.com/
hPp://en.wikipedia.org/wiki/Chlamydomonas
hPp://www.freewebs.com/jennalb03/

Tetrahymena

hPp://en.wikipedia.org/wiki/Tetrahymena

hPp://www.damtp.cam.ac.uk/user/gold/movies.html

Escherichia coli

hPp://ja.wikipedia.org/wiki/%E5%A4%A7%E8%85%B8%E8%8F%8C

hPp://www.marlerblog.com/tags/e‐coli/

Microorganism and the Stokes ﬂow
cilia or ﬂagella

Small speed(Smalllength scale(1 – 200 μm )Low Reynolds number(ex. Escherichia coli:)→ viscous eﬀect becomes dominantμm/s )Treadmilling in a free space

Nutri<onhPp://gizmodo.com/338598/ﬁt‐fur‐life‐doggy‐treadmill‐walks‐that‐muP‐so‐you‐wont‐have‐toMicroorganisms near a boundary
E. Coli swimming in circles above
a ﬂat glass surface, (Lauga et al. 2006)

Accumula<on of E. Coli near boundaries(measurement and force dipole
singularity model), (Berke et al. 2008)“waltzing” mo<on of pair of Volvox ,(Drescher et al. 2009)Preceding studies： bouncing above a no‐slip wall

Trajectory of the two‐sphere swimmer near wallTwo‐rota<ng‐sphere model(Or and Murray, 2009)

Trajectory of the swimmer
Rota<ng two cylinders to generatemacroscale robo<c prototype swimming(Zhang et al. , 2010)

Stokeslet, stresslet, rotelet …
Stokes equa<onStress tensor
Dyadic Green’s func<on\AtAmbient
ﬂow ﬁeld:

Rigid par<cle, take the Taylor series in

Stokeslet(point force)

about

Symmetric partAsymmetric partStresslet (force‐free,torque‐free )rotlet(point torque)Preceding studies(singularity model):Near an inﬁnite no‐slip wall (Crowdy and Or , 2010)
Simpliﬁca<on to the synchronised
ac<on of cilia.

(xd, ‐yd )Preceding studies(singularity model):Near an inﬁnite no‐slip wall (Crowdy and Or , 2010)

Preceding studies(singularity model):Near an inﬁnite no‐slip wall with a gap (Crowdy and Ophir, 2010)
y(xd, yd)2ε‐101xProject: Near an half‐inﬁnite no‐slip wall ??
y(xd, yd )εNo‐slip wall ofinﬁnitesimal thickness

0tangen<al velocity proﬁle::const. , sets the <me scale for the treadmilling ac<on byx

Biharmonic equa<on and Goursat func<ons()(Biharmonic equa<on )()Singularity model for 2D Stokes ﬂow()()()

Treadmilling swimmer and a stresslet
Seek solu<ons for Goursat func<ons of the form;

No Stokeslet: Force‐free
No rotlet: Torque‐freeBoundary condi<on on the wall:Image of the swimmer
Goursat func<ons

Im(z)=yzd0
No‐slip wall ofinﬁnitesimal thickness

0

Where should theimage be placed ?
Re(z)=xConformal mapping (change of variables)

AB

0

A0

B

Analy<cal single‐valued func<onsandAssume

to have the form (image system method)

is determined by the boundary condi<on on the wall :0,,, andnecessary boundary condi<on on the treadmiller’s body :Nearhave to have singulari<es wriPen as
and,,,Governing equa<ons= velocity at
induced by an image(regular part of the velocityat)=vor<city at
induced by an image(regular part of the vor<cityat)

Example of the velocity ﬁeld:yyxx

y = 2.0xx

Ini<al condi<ons and parametersy20Θ0zd0 = xd0 + iyd0
ε=11‐200120x

Time integra<on: ode45 solver,dx = 0.5, dy = 0.5, dΘ0 = π/100, Maximum <me: tmax = 1500,ε = 1, μ = exp(2iΘ(t))

Examples of trajectories
x d0 = 10, yd0 = 2x d0 = ‐1, y d0 = 2x d0 = 0, yd0 = 2x d0 = ‐15, yd0 = 2

Examples of the trajectories: Escaping from the wall

Examples of the trajectories: Escaping from the wall

Examples of the trajectories: Being above the wall

Examples of the trajectories: Going beneath the wall

Examples of the trajectories: Crashing into the wall

The posi<on at a suﬃciently large <meyy②①(xd0, yd0)

δ=0.2ε⑤0x

δ=0.2εδ=0.2ε③④x

From an ini<al point xd0 > 0, yd0 > 0Θ0①②④⑤Θ0

From an ini<al point xd0 < 0, yd0 > 1

Θ0①②⑤

Θ0

From diﬀerent ini<al points

From diﬀerent ini<al points: Several rules

y20①②⑤①②④⑤
1‐200①②③④
120x

Escaping probability

for each ini<al point,Temporal varia<on of the speed: Escaping from the wall

Trajectory of a swimmer

ySpeed of a swimmer

xy‐posi<on of a swimmertConclusions
・ Treadmilling swimmer feels the presence of the wall
escaping probability < 1
・ The speed of a swimmer slows down as the swimmer goes
further from the wall since the image should remain on the wall

Further works??

・ half a wall 〜 size of the swimmer << width of the gap
yThe original problem
・near a cornerx

Acknowledgements

林先生をはじめ, 北大・神戸大
GCOE/CPS関係者のみなさまに深く感謝致します.

Many thanks to everyone
Jean‐Luc, MaP, and the fellows

AppendixesLecturer note 1
Dynamics of swimming organism ( posi<on, velocity

Water
Preferred Randomvelocity swimming accelera<onvelocity

・ taxis (describes a large‐scale preferredvelocity that the group tends to. ):depends on gradient of cue ﬁeldcue
may beenvironmental (food, light, depth, etc.) or social(posi<ons of neighbors, etc.).
Here, we deﬁne taxis as a preference formoving up the gradient of the cue ﬁeld,):

Lecturer note 1・ kinesis (describes an individual’s tendency to move
randomly):
depends on the cue ﬁeld:

・schooling (describes the behavior of the organismsthat tend to swim similarly to their neighbours.):
depends on neighbor’s :The preferred direc<on of the swimming organisms
results from a combina<on of aPrac<on and alignment
tendencies, and so schooling can be represented as

Singulari<es in Goursat func<ons
Example:Assumeof the form()

Then the complex velocity ﬁeld is

Insist that the velocity ﬁled should be both
single‐valued and, at least, logarithmically singularshould have concomitant singulari<es to those in, but not but notconversely (can have singulari<es which is independent of thoseof.).

necessary boundary condi<onon the treadmiller’s body(Near,andhave to have singulari<es wriPen as)Goursat func<ons, andExample of the velocity stream func<on:yyxx

From diﬀerent ini<al points
52x

0‐5‐202