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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
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
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)
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: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
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)
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
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
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