Harry SwinneyHarry SwinneyUniversity of TexasUniversity of Texas

Center for Nonlinear Dynamics and Department of PhysicsCenter for Nonlinear Dynamics and Department of Physics

• Matt Thrasher • Leif Ristroph (now Cornell U)

• Mickey Moore (now Medical Pattern Analysis Co.)

• Eran Sharon (now Hebrew U)

• Olivier Praud (now CNRS-Toulouse)

• Anne Juel (now Manchester)

• Mark Mineev (Los Alamos National Laboratory)

Developments in Experimental Pattern FormationIsaac Newton Institute, Cambridge

12 August 2005

Fractal growth of viscous fingersFractal growth of viscous fingers

Viscous fingering in Hele-Shaw cell

• Velocity of the interface

• Pressure field 2P=0

< 2

b << wFlow

w

LAPLACIAN GROWTH PROBLEM

n)P(b

V 12

2

Saffman-Taylor (1958): finger width → ½ channel width

theoretical assumptions must be re-examined

airoil

air

oil

Previous experiment & theory: steady finger at low flow rates.

U Texas experiment: fluctuating finger as V → 0 :

air

Fluctuations in finger width

gap = 0.051

Capillary number = V/

Ca-2/3

Moore, Juel,Burgess,

McCormick, Swinney,

Phys. Rev. E 65 (2002)

tip splitting

10-110-1

10-2

10-4 10-3 10-2

w

)width( rms

Scaling of finger width fluctuations

10-3

For different gaps b, cell widths w, viscosities

Radial geometry:inject air into center of circular oil layer

60 mm

gap filledwith oil

b=0.127 mm ±0.0002 mm

Pexternal

Silicone oil VISCOSITY

= 0.345 Pa-s

SURFACE TENSION

= 21.0 mN/m

oil

Pin

air

CCD Camera1300 x 1000

1 pixel 2bλMS 3b

288 mm

Instability scale depends on pumping rate

Forcing

airair

airair

pump out oil slowly pump out oil faster

oiloil

AIR AIR

Growth of radial viscous fingering patternstrong forcing

realtime

Viscous fingering pattern

young

old

Praud & SwinneyPhys. Rev. E 72

(2005)

seedparticle

ALGORITHM: ● start with a seed particle

● release random walker particles from far

away, one at a time

Diffusion Limited Aggregation (DLA) Witten and Sander (1981)

young

old

Barra, Davidovitch, and Procaccia, Phys. Rev. E (2002): viscous fingering has D0 > 1.85 and is not in same universality class as DLA

N() number of boxes of size needed to

cover the entire object

N() –D0

Fractal dimension of viscous fingering pattern

10-1

100

101

102

103

101

102

103

104

105

106

N(

)

N()=a.-dim

dim = 1.70 0.01

N() -D0

D0 = 1.70±0.02

Numberof boxes

N()

Fractal dimension D0 of viscous fingering pattern

Fractal dimension of viscous fingering compared to Diffusion Limited Aggregation

Experiments D0 (r/b)max

Present experiments (2005)

Rauseo et al., Phys. Rev. A 35 (1987)

Couder, Kluwer Academic Publ. (1988)

May & Maher, Phys. Rev. A 40 (1989)

1.70 ± 0.021.79 ± 0.07

1.76

1.79 ± 0.04

1200

190

190

DLAWitten & Sander, Phys. Rev. Lett. 47 (1981)

Tolman & Meakin, Phys. Rev. A 40, (1989)

Ossadnik, Physica A 176 (1991)

Davidovitch et al. Phys. Rev. E 62 (2003)

1.70 ± 0.02

1.715 ± 0.004

1.712 ± 0.003

1.713 ± 0.003

square lattice radial

off-lattice radial

off-lattice radial

conformal map theory

Generalized dimensions Dq

)(P)(Z

,log

)(Zlog

qlimD

i

qiq

qq

1

10

Henstchel & Procaccia Physica D 8, 435 (1983)Grassberger, Phys. Lett. A 97, 227 (1983)

Is the radial viscous fingering pattern amultifractal or a monofractal ?

(i.e., are all Dq the same?)

fractal dim.q = 0

Generalized dimensions

-15 -10 -5 0 5 10 15 201.4

1.5

1.6

1.7

1.8

1.9

2.0

q

Dq

P = 0.50 atmP = 1.25 atmP = 1.75 atm

Generalized Dimension Dq

Dq

q

Conclude:viscous

fingering patternis a

monofractalwith

Dq = 1.70

independentof q

(self-similar)

DLA is also monofractal: Dq = 1.713

Harmonic measure• harmonic measure -- probability measure for

a random walker to hit the cluster.

• Dq for harmonic measure -- difficult to determine because of extreme variation of

probability to hit tips vs hitting deep fjords.

Jensen, Levermann, Mathiesen, Procaccia, Phys. Rev. E 65 (2002):

• iterated mapping technique for DLA –

resolve probabilities as small as: 10-35 :

→ DLA harmonic measure is multifractal

generalized dimensions Dq f() spectrum

r

• Pi(r) ~ r, – singularity strengthwith values min < < max

• f() – probability of value

f() spectrum of singularities

Generalized fractal dimensions Dq

Legendre transform

i

Halsey, Jensen, Kadanoff, Procaccia, Shraiman, Phys. Rev. A 33 (1986)

harmonic measure f(): viscous fingers & DLA

2

1

00 5 10 15 20

1.71

f

DLA viscous fingering clustersof increasing size

Tentative conclusion:DLA and

viscous fingersare in the same

universality class

Mathiesen, Procaccia, Thrasher, Swinney --- preliminary results

Growth dynamics: unscreened angle

largest angle that does not include pre-existing pattern

active region

pre-existing pattern

Distribution of the unscreened angle Θ

50 100 150 200 2500.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(deg)

p(

)P=0.25 atmP=0.50 atmP=1.25 atmP=1.75 atm r/b=200

50 100 150 200 2500.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(deg)

p(

)P=0.25 atmP=0.50 atmP=1.25 atmP=1.75 atm r/b=200

r/b=600

→ P() is independent of forcingbut depends on r/b

P()

Asymptotic screening angle PDF

50 100 150 200 2500.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(deg)

p(

)r/b=160r/b=322r/b=484r/b=644r/b=806

P=1.25 atm

Invariant distribution at large r/b

16032

2484

644

806

r/b

P

Exponential convergence to invariant distribution

0 200 400 600 8000.2

0.3

1.0

3.0

r/b

(r/

b)

P=0.25 atmP=0.50 atmP=1.25 atmP=1.75 atm

/)b/r(asympr ed)](P)(P[)r(

21

221

conver-gencelength=200

(r)

r/b

0.5 atm

p=1.75 atm

1.25 atm

0.25 atm

Asymptotic distribution P(): <> = 145o 36o BUT no indication of a critical angle or 5-fold symmetry

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(deg)

p asym

p()

0 100 200 30010

-4

10-3

10-2

10-1

100

(deg)p as

ymp(

)

Gaussian

Gaussian

Unscreened angle PDF

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(deg)

P(

)

viscous fingeringDLA*

v. f. DLA

<>: 146o 127o

σ: 36o 51o

Skewness: 0.06 0.3Kurtosis: 2.3 3.8

DLA on-lattice algorithm

Kaufman, Dimino, Chaikin, Physica A 157 (1989)

viscous fingering experiment

P(

CoarseningDLA with diffusion & viscous fingering patternsDLA plus diffusion

EXPT

t=0 54 516 4900

t=0 s 115 s 1040 s 10040 s

Lipshtat, Meerson, & Sarasov (2002)

Coarsening: length L1 below which

viscous fingering pattern is smooth

Density-density

correlation

L2: an intermediate length scale -- diluted becausesmall scales thicken while large scales are frozen

L2 defined by minimum in C

C(r)

Non-self-similar coarsening of pattern:described by two lengths L1 and L2

Non-self-similar coarsening:lengths L1 and L2

power law exponents and

• Viscous fingers — = 0.22 ± 0.02, = 0.31 ± 0.02

• DLA cluster with diffusion — = 0.22 ± 0.02 (at intermediate times), = 1/3

Lipshtat, Meerson, & Sarasov, Phys. Rev. E (2002)Conti, Lipshtat, & Meerson, Phys. Rev. E (2004)

Sharon, Moore, McCormick, SwinneyPhys. Rev. Lett. 91 (2003)

Fjords between viscous fingerssector geometry

Lajeunesse & CouderJ. Fluid Mech.

419 (2000)

“A fjord center line follows approximately a curve normal to the successive profiles of

stable fingers.”

FJORD

Can ramified finger be fit to theory for inviscid fingering?

Exact non-singular solutions for Laplacian growth with zero surface tension

The motion in time t of a point (x,y) on a moving interface is given by (with z = x + iy)

)ln(iitz k

N

kk

1

1

)ln(iit mk

N

mmkk

1

1

where k and k are complex constants of motion.

Mineev & Dawson, Phys. Rev. E 50 (1994)

A fit with 43 sets of complex constants k and k

Evolve solution forward in timepreliminary

Moore, Thrasher, Mineev, Swinney

which have different:

– lengths– widths– propagation directions

(relative to channel axis or radial line)

– forcing levels (tip velocity V)– geometries

• circular• rectangular (and vary aspect ratio w/b )

w

Search for selection rules for fjords

Fjord dependence on forcing

0010.V

Ca

0010.V

Ca

Ca = 0.040

Predict fjord width W

original interface

emergent fjord

emergent finger

emergent finger

Conclude

W = (1/2)c

V

Wavelength of instability of an interface

Vb

Ca

bc

Chuoke, van Meurs, & van der Pol, Petrol. Trans. AIME 216 (1959) (fluid)

Mullins & Sekerka, J. Appl. Phys. 35 (1964) (solidification front)

surface tension

interface velocityviscosity

Tip splits and forms a fjord

tipcurvature=0t=0

time dependence

-5 0 5 10 15time (s)

t=0

(cm-1)

V(cm/s)

curvature

tip velocity

Channel base width: W0 = W(ℓ=0, t=0)

0 5 10 15fjord length ℓ (cm)

W4

fjordwidth(cm)

Ca

bc

221

Compare theory and experiment

theory

Theory predicts parallel walls of fjord:

channel wall

channel wall

Measure fjord opening angle

Mineev, Phys. Rev. Lett. 80 (1998)Pereira & Elezgaray, Phys. Rev. E 69 (2004)

FJORDstagnation

point

sequence of snapshots of interface, t = 50 sec

7.5o

fjord length ℓ (cm)

Opening angle of a fjordrectangular cell

Ristroph, Thrasher,Mineev, Swinney

2005

(deg)

rectangular cell< > = 7.90.8 deg

circular cell<>= 8.21.1 deg

p()

(degrees)

Opening angle probability distributionRESULT: < > = 8.0 1.0 deg

Invariant with fjord• width• length

• direction• forcing

• geometry

Fractal growth phenomena: same universality class ?

Dielectric breakdown

Niemeyer et al. PRL (1984)

U Texas (2003)

DLA

Witten & Sander (1981)

Bacterial growth

Matsushita (2003)

Diffusion Limited AggregationViscous fingers

Electrodeposition

Brady & Ball, Nature (1983)

andmetal corrosion,brittle fracture,

…

•Viscous fingers and DLA: same universality class

• pattern: monofractal with Dq = 1.70 for all q

• harmonic measure: same multi-fractal f() curve

• Fjord selection rules for viscous fingers: for all lengths, widths, directions, and forcings

in both circular and rectangular geometries: • width: W = (1/2)c

• opening angle: 8 1 deg

• Viscous finger width fluctuations:(width)rms Ca-2/3 (for small Ca)

Conclusions