Models and Experiments of Active BodiesInteracting with FluidsMichael Shelley – Applied Math Lab, Courant Institute, NYUCollaborators:Silas Alben HarvardJohn Bush MITSteve Childress CIMSPeko Hosoi MITPeter Palffy-Muhoray LCI-KentFabio Piano NYU BiologyNicolas Vandenberghe MarseillesAnna-Karin Tornberg CIMSJon Wilkening CIMS/BerkeleyJun Zhang Physics/CIMS

t = 0 5.0 5.5 6.0 6.5 7.0 9.0 st = 0 5.0 5.5 6.0 6.5 7.0 9.0 s

lightlight

force on waterforce on LCE

light

force on waterforce on LCE

lightlightlight

force on waterforce on LCE

light

force on waterforce on LCE

light

clione antarcticaa shelless Antarctic mollusc

adult

Locomotion by Destabilizing SymmetriesObservation 1: Micro-organisms and birds (and fish) use very

different strategies for locomotionSmall organisms – “slow flow” -- Re << 1Large organisms – “fast flow” – Re >> 1

Observation 2: Some organisms live between these worlds

Flapping WingsReciprocal:Higher Re

Childress and Dudley JFM 2004

juvenile

Non-reciprocal:Low Re

Oscillating cilia

( ) ( ) F=ma0

(incompressib li ) t i y

tRe p∂ + ⋅∇ = −∇ + Δ

∇⋅ =

u u u uu

Reynolds Number: U LRe ρμ

=

Navier-Stokes Eqs:

fluid density= viscosity

characteristic velocitycharacteristic length

UL

ρμ=

==

L

U

Question: Is there some decisive change in the way a fluid and a “free” body interact as Re increases?

Reciprocal flapping is ineffective.-u is a solution of the Stokes Eqs for reversed BCs⇒ reversible flapping gives no net thrust over one period

: viscously dominated Stokes Eqs.

01

0 p

R e−∇ + Δ =

<<∇ ⎭=

⎫⎬

uui

inertially dominated -- vortices Non-reciprocal still produces thrust, but ... Reciprocal flapping + forward motion now gives thrust

Does something happen fo

1:

?(1)r

Re

Re O

>>

∼

Related work for moving bodies:Miller and Peskin, JEB 2004

(flight of small insects)and Z.-J. Wang, 2004

( )( ) sin 2y t A f tπ=

“wing” free to rotate – in either direction

a

frequency Reynolds number

frfA aRe ρμ⋅

=

=

⋅

Rotary Reciprocal Flapper ExperimentVandenberghe, Zhang, and Childress, JFM 2004, VCZ 2005

Rotational speed versus driving frequency

( )3 / 4d aRe

ρμΩ

Ω=

fRe

2 0.26

fStrouhalfrequenc

ReReAfU

y Ω

=

= =

symmetric flow, no rotation

“flight”

hysteresis and bistability

Extrapolating out bearing friction: 20 50critfrRe −∼

Questions from the experiment...• What is the true nature of the bifurcation,

subcritical or supercritical?Friction on axle is a confounding factor.Extrapolation with increasing viscosity:

• What does the work, pressure or viscous forces, as the wing “takes off”?

• Is it really so easy? What is the role of the body mass? Body shape?

20 50critfRe −∼

subcrit. super

Simulate the dynamics of a 2Dflapping elliptical body

(also, Z.-J. Wang, '99, '00)

( )( )

vorticity

&tRe

ω

ω ω ωψ ψ ω⊥

= ⋅ ∇×

∂ + ⋅∇ = Δ⎧⎨

= ∇ Δ = −⎩

k u

uu

Alben & Shelley, PNAS 2005

(Navier-Stokes in vorticity-stream variables)

v

nd

t

n

h

BCs in body frame: (no penetration)

On the body surface (no slip)

In the far-field

Simulated using an 2 order (

==0

=0=

implicit) in time, 4 or

-

der in

Constψψ

ω

⎧⎨⎩⎧⎨⎩−

−

∂

u v

space method

( ) ( ) ( )

[ ]

nd

, 2 cos 2

ˆ

Determining :

Find by Newton's 2 Law:

with

Invariant:

the total horizontal momentum Paramet

2

er

s

:

x y y

x

xfr fluid fluid body

x

v v v fA ft

vdvM Re p dsdt

M v dA

Re

π π= =

= ⋅ = − +

•

•

+ ⋅

∫

∫

v

x F F I E n

x u

2

aspect ratio of body

chord-to-amplitude ratio (set to )

= effective

1/

m

2

ass

fr

body

fluid

fA L LW

Area AML L

ρμ

ρρ

⋅ ⋅=

=

2A

L

W

Re=7, M=1,

Re 7, 1, / 3fr M L W= = =

Symmetric fluid response

“low” Reynolds number flapping

Re 35, / 1, / 5f b L Wρ ρ= = =

A faster body …

Swimming? Chaotically perhaps

-5 0 5 10 15 20-5

0

5

Re 35, / 32, / 5fr b L Wρ ρ= = =

let’s make the “swimmer” a little heavier …

-40 -35 -30 -25 -20 -15 -10 -5 0-4-2024

t = 16

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

B C

F

t = 0.4 t = 4.4 t = 6.2

0 5 10 15-6

-4

-2

0

2

t

A

E

D

t = 6.20 1 2 3 4s

-40 -35 -30 -25 -20 -15 -10 -5 0-4-2024

t = 16

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

B C

F

t = 0.4 t = 4.4 t = 6.2

0 5 10 15-6

-4

-2

0

2

t

A

E

D

t = 6.20 1 2 3 4s

D

t = 6.20 1 2 3 4s

vortex collision yields dipole initiates locomotion

pressure forces> viscous forceslocomotion decreases input power

Find that txv eα∼

Critical 7frRe ≈

0 5 10 150

0.5

1

1.5

2

α

3.3:15:110:120:1

0 10 20 30 40 50 60 70 80 90 100 1100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 20 40 60 80 1000

400

800

1200

2; fr

UU

ReULRe StRe

ρμ

= =

Critical 10frRe ≈

0.18St →

First bifurcation is von Karman instability of a symmetric wakeSecond is to unidirectional locomotion – looks supercriticalSt=0.2 - 0.4 typically observed for animal locomotion

frRefrRe

UReSt

The effect of shape? a flatter body

Thinner body move more directly into locomotion

Re 35, / 32, / 10fr b L Wρ ρ= = =

sym

met

ric

perio

dic

osci

llatio

n

nonperiodic oscillation

Strouhal number

● Unidirectional locomotion can be a strongly attracting statefor even a fore-aft symmetric body

● The transition occurs at a finite Reynolds number, markinga boundary between high and low Reynolds number behavior

● We can relate loss of a symmetric flow responseto wake transitions at moderate Re for steadilytranslating bodies (formation of eddies via von Karman shedding)

• The transition appears to be supercritical and notsubcritical – (effect of bearing friction?)

Different start-up gives different wake structure

Liquid crystal elastomersSample

ArLaser

Sample

ArLaser

sample: nematic elastomer EC4OCH3+ 0.1% dissolvedDisperse Orange 1 azo dye

5 mm (x 0.3 mm)

Dynamic ResponseDynamic Response

0 100 200 300 400 5000.0

0.5

1.0

1.5

2.0

2.5

time (ms)

forc

e (m

N)

50 100 150 200 250 300-3

-2

-1

0

1

time (ms)

log

(forc

e)

τ=75ms

0 100 200 300 400 5000.0

0.5

1.0

1.5

2.0

2.5

time (ms)

forc

e (m

N)

50 100 150 200 250 300-3

-2

-1

0

1

time (ms)

log

(forc

e)

τ=75ms

0.6P W=3d mm=

5 5mm mm×

Very fast, O(50ms) Applications: artificial muscle, actuators, propulsion, pumping, …

bend along director field

LCEs‘SOLID LIQUID CRYSTALS’ combining features of

LIQUID CRYSTALS & ELASTIC SOLIDS

ilil

n 0.8S =

NematicNematic liquid crystalsliquid crystalscan have can have orientationalorientationalorder:order:

director

LCE Structure

If orientational order increases,

expansion contraction

If orientational order decreases,

expansioncontraction

see Warner & TerentjevLiquid Crystal Elastomers2003

Conceptualized by de Gennes ’75Synthesized by Finkelmann et al ‘81

Laser beam

container

waterElastomer

Swimming into the dark

A liquid crystal elastomer swims away from the lightPalffy-Muhoray, Comacho-Lopez, Finkelmann, Shelley, Nature Mat. 2004

Re ~ 200

A Model of Swimming – a wave of fixed shape moving through an inextensible membrane

( ) ( ) ( )( )( ) ( )( )

( )1

0 0

, ( ) , , ,

ˆ, , ,

,

t t

Ltt fluid

t q t x t y t

x t y t V

mq m d x t ds

α α α

α α

α α

= +

=

+ =∫ ∫

X

s

F xi

momentum balance on plate:

Wilkening & Shelley 2005

“steady-state” solution: Re=100 (based on depth)

Poor swimmer – 1/10 length displacement/cycle

large-amplitude waves on thin layers --Newtonian Snail

Pressure

highlow

see Chen, Balmforth, Hosoi ’05 for long-wave analysisand Robo-Snail(s)

Dynamics and interactions of elastic micro-structure in complex fluids

Dynamics of non-Newtonian fluids Reinforced composite materialsBiological locomotionPolymer solutionsElastic “turbulence” & low Re mixingGroisman, Enzelberger, & Quake ‘03

Microfluidic switchesGroisman & Sternberg ’00, ’01

How does one simulate the dynamics of large aspect ratio ( ε = L/r << 1 ) flexible filaments moving in a Stokesian fluid?

Non-local hydrodynamics of slender bodies:Place fundamental solutions (Stokeslets and dipoles) on filament centerline.Technique of matched asymptotics used to derive approximate equation (accuracy O(ε2ln ε) ). Formulation closed by enforcing no-slip condition on filament surface.

Keller and Rubinow (1976), Johnson (1980), Gotz (2000).

In the fluid:0

0p μ−∇ + Δ =∇ ⋅ =

uu

f

length L

radius r

( ) ( )^( ) ( )| ( )

( )s s

ds ss>>

+ − −−∫

^

Y X

I Y X Y Xu f

Y X∼

X(s)

Y

[ ] ,; ; [ ];i i sssi i i kk

si

kT≠

= + −⎡ ⎤⎣ ⎦ ∑Λ R XfXXX R X

iX

( ) [ ] [ ]0 , ; ;ii i i k i k

k i

tt ≠

∂= + + −

∂ ∑X U X K f X K f X X

Form of Dynamics:

kX

A system of nonlocal and coupled fourth-order PDEs(3 per filament) evolving under constraints.

The nonlocal hydrodynamics

of many filamentsin an oscillating shear flow

Tornberg & ShelleyJCP ‘04

The nonlocal hydrodynamics of many filamentsin an oscillating shear flow

Tornberg & Shelley, JCP ‘04

1N

2,el ii fils

E dsκ= ∑ ∫

Not yet equilibrated …

Self-locomotion -- an elastic body chasing its desired shape

( )( ) ( )

time-dependent curvature of preferred shape (traveling wave)

Single nondimensional parameter:

Zero n

( );

8 ³ / et force, zero net torque

and

( ) 0

ss s ssE T

s Vt

µ µVL cE

ds s

κ κ

κ κ

π

= − −

= −

=

=∫

f n x

f ( )( ) ( ) 0ds s s× =∫ x f

see Goldstein & Langer ‘95for energetic formulation

1µ = 1000µ =

“kinematic” locomotion

Taylor, Lighthill, Purcell,and many others…

challenging

challenged

w. Fabio Piano (NYU Biology; C. elegans development), John Bush (MIT)

N1=.125 N1=.146

N1=.137

“active” particles