Fractal growth of viscous fingers

1
Fractal growth of viscous fingers
Developments in Experimental Pattern
Formation Isaac Newton Institute, Cambridge 12
August 2005
Harry SwinneyUniversity of TexasCenter 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)

2
Viscous fingering in Hele-Shaw cell
m1 lt m2
b ltlt w
Flow
w
LAPLACIAN GROWTH PROBLEM
Saffman-Taylor (1958) finger width ? ½ channel
width
3
Fluctuations in finger width
Previous experiment theory steady finger at
low flow rates.
air
U Texas experiment fluctuating finger as V ? 0
gap 0.051
? theoretical assumptions must be re-examined
4
Scaling of finger width fluctuations
For different gaps b, cell widths w, viscosities m
10-1
10-1
Ca-2/3
tip splitting
10-2
Moore, Juel, Burgess, McCormick, Swinney, Phys.
Rev. E 65 (2002)
10-3
10-4
10-3
10-2
Capillary number mV/s
5
Radial geometry inject air into center of
circular oil layer
CCD Camera 1300 x 1000 1 pixel ? 2b ?MS ? 3b
288 mm
gap filled with oil b0.127 mm 0.0002 mm
60 mm
6
Instability scale depends on pumping rate
pump out oil slowly
pump out oil faster
oil
oil
AIR
AIR
Forcing
7
Growth of radial viscous fingering pattern
strong forcing
real time
8
Viscous fingering pattern
Praud Swinney Phys. Rev. E 72 (2005)
young
old
9
Diffusion Limited Aggregation (DLA)
Witten and Sander (1981)
ALGORITHM ? start with a seed particle ?
release random walker particles from far away,
one at a time
young
old
seed particle
Barra, Davidovitch, and Procaccia, Phys. Rev. E
(2002) viscous fingering has D0 gt 1.85 and is
not in same universality class as DLA
10
Fractal dimension of viscous fingering pattern
N(e) number of boxes of size e needed to cover
the entire object
11
Fractal dimension D0 of viscous fingering
pattern
Number of boxes N(e)
e
12
Fractal dimension of viscous fingering compared
to Diffusion Limited Aggregation
13
Generalized dimensions Dq
Henstchel Procaccia Physica D 8, 435
(1983) Grassberger, Phys. Lett. A 97, 227 (1983)
fractal dim. q 0
Is the radial viscous fingering pattern
a multifractal or a monofractal ? (i.e., are
all Dq the same?)
14
Generalized Dimension Dq
Generalized dimensions
Dq
Conclude viscous fingering pattern is
a monofractal with Dq 1.70 independent of
q (self-similar)
q
DLA is also monofractal Dq 1.713
15
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

16
generalized dimensions Dq ? f(a) spectrum
Halsey, Jensen, Kadanoff, Procaccia, Shraiman,
Phys. Rev. A 33 (1986)

• Pi(r) ra, a singularity strength
• with values amin lt a lt amax
• f(a) probability of value a

i
f(a) spectrum of singularities
Legendre transform
Generalized fractal dimensions Dq
17
harmonic measure f(a) viscous fingers DLA
Mathiesen, Procaccia, Thrasher, Swinney ---
preliminary results
2
Tentative conclusion DLA and viscous fingers are
in the same universality class
1.71
f(a)
1
DLA
viscous fingering clusters of increasing size
0
20
0
5
10
15
a
18
Growth dynamics unscreened angle ?
active region
largest angle that does not include
pre-existing pattern
pre-existing pattern
19
Distribution of the unscreened angle T
P(Q)
? P(Q) is independent of forcing but depends on
r/b
20
Asymptotic screening angle PDF
P(Q)
484
322
644
160
r/b
806
Invariant distribution at large r/b
21
Exponential convergence to invariant distribution
Dp1.75 atm
G(r)
1.25 atm
conver- gence length x200
0.5 atm
0.25 atm
r/b
22
Asymptotic distribution P(Q) ltQgt 145o ? 36o
BUT no indication of a critical angle or 5-fold
symmetry
Gaussian
Gaussian
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Unscreened angle PDF
P(Q
DLA on-lattice algorithm
Kaufman, Dimino, Chaikin, Physica A 157 (1989)
viscous fingering experiment
v. f. DLA lt?gt 146o 127o s
36o 51o Skewness 0.06 0.3 Kurtosis
2.3 3.8
24
CoarseningDLA with diffusion viscous fingering
patterns
DLA plus diffusion
Lipshtat, Meerson, Sarasov (2002)
t0
54
516
4900
EXPT
115 s
t0 s
1040 s
10040 s
25
Coarsening length L1 below whichviscous
fingering pattern is smooth
Density- density correlation
26
L2 an intermediate length scale -- diluted
because small scales thicken while large scales
are frozen
DC(r)
L2 defined by minimum in DC
27
Non-self-similar coarsening of patterndescribed
by two lengths L1 and L2
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Non-self-similar coarseninglengths L1 and L2
power law exponents a and b

• Viscous fingers
• a 0.22 0.02, b 0.31 0.02
• DLA cluster with diffusion
• a 0.22 0.02 (at intermediate times), b 1/3

Sharon, Moore, McCormick, Swinney Phys. Rev.
Lett. 91 (2003)
Lipshtat, Meerson, Sarasov, Phys. Rev. E
(2002) Conti, Lipshtat, Meerson, Phys. Rev. E
(2004)
29
Fjords between viscous fingerssector geometry
Lajeunesse Couder J. Fluid Mech. 419 (2000)
FJORD
A fjord center line follows approximately a
curve normal to the successive profiles of stable
fingers.
30
Can ramified finger be fit to theory for inviscid
fingering?
31
Exact non-singular solutions for Laplacian growth
with zero surface tension
Mineev Dawson, Phys. Rev. E 50 (1994)

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

where ak and bk are complex constants of motion.
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A fit with 43 sets of complex constants ak and bk
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Evolve solution forward in time
preliminary Moore, Thrasher, Mineev, Swinney
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Search for selection rules for fjords

• 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
35
Fjord dependence on forcing
Ca 0.040
36
Predict fjord width W
V
emergent finger
original interface
emergent fjord
Conclude W (1/2)?c
emergent finger
37
Wavelength of instability of an interface
Chuoke, van Meurs, van der Pol, Petrol. Trans.
AIME 216 (1959) (fluid) Mullins Sekerka, J.
Appl. Phys. 35 (1964) (solidification front)
surface tension
interface velocity
viscosity
38
Tip splits and forms a fjord

• tip
• curvature
• 0
• t0

39
time dependence
curvature
k (cm-1)
t0
tip velocity
V (cm/s)
5
10
15
-5
0
time (s)
40
Channel base width W0 W(l0, t0)
4
W
5
10
15
0
fjord length l (cm)
41
Compare theory and experiment
fjord width (cm)
theory
42
Measure fjord opening angle
channel wall
sequence of snapshots of interface, Dt 50 sec
stagnation point
FJORD
Theory predicts parallel walls of fjord
Mineev, Phys. Rev. Lett. 80 (1998) Pereira
Elezgaray, Phys. Rev. E 69 (2004)
channel wall
43
Opening angle of a fjord rectangular cell
y (deg)
7.5o
Ristroph, Thrasher, Mineev, Swinney 2005
fjord length l (cm)
44
Opening angle probability distributionRESULT lt
? gt 8.0 ? 1.0 deg
p(y)

• Invariant
• with fjord
• width
• length
• direction
• forcing
• geometry

rectangular cell lty gt 7.9?0.8 deg
circular cell lty gt 8.2?1.1 deg
y (degrees)
45
Fractal growth phenomena same universality
class ?
Bacterial growth
Diffusion Limited Aggregation
Viscous fingers
DLA
U Texas (2003)
Matsushita (2003)
Witten Sander (1981)
and metal corrosion, brittle fracture,
46
Conclusions

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

• Viscous fingers and DLA same universality class
• pattern monofractal with Dq 1.70 for all q
  
• harmonic measure same multi-fractal f(a) curve

• Fjord selection rules for viscous fingers
• for all lengths, widths, directions, and
  forcings
• in both circular and rectangular geometries
• width W (1/2)lc
• opening angle 8 ? 1 deg