1. Harry Swinney University of Texas Center 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 Formation Isaac Newton Institute, Cambridge 12 August 2005 Fractal growth of viscous fingers

2. Viscous fingering in Hele-Shaw cell Velocity of the interface Pressure field  2 P=0    <  2 b << w Flow w LAPLACIAN GROWTH PROBLEM Saffman-Taylor (1958): finger width → ½ channel width

3.  theoretical assumptions must be re-examined air oil 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

4. Capillary number =  V /  Ca -2/3 Moore, Juel, Burgess, McCormick, Swinney, Phys. Rev. E 65 (2002) tip splitting 10 -1 10 - 2 10 - 4 10 - 3 10 - 2 Scaling of finger width fluctuations 10 -3 For different gaps b, cell widths w, viscosities 

5. Radial geometry: inject air into center of circular oil layer 60 mm gap filled with oil b=0.127 mm ±0.0002 mm P external Silicone oil  VISCOSITY  = 0.345 Pa-s SURFACE TENSION  = 21.0 mN/m oil P in air CCD Camera 1300 x 1000 1 pixel  2b λ MS  3b 288 mm

6. Instability scale depends on pumping rate Forcing air pump out oil slowly pump out oil faster oil AIR

7. Growth of radial viscous fingering pattern strong forcing real time

8. Viscous fingering pattern young old Praud & Swinney Phys. Rev. E 72 (2005)

9. seed particle 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 D 0 > 1.85 and is not in same universality class as DLA

10. N(  ) number of boxes of size  needed to cover the entire object N(  )   –D 0 Fractal dimension of viscous fingering pattern

11. N(  )  -D 0 D 0 = 1.70±0.02 Number of boxes N(  )  Fractal dimension D 0 of viscous fingering pattern

12. Fractal dimension of viscous fingering compared to Diffusion Limited Aggregation ExperimentsD0D0 (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 ± 0.02 1.79 ± 0.07 1.76 1.79 ± 0.04 1200 190 DLA Witten & 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 conformal map theory

13. Generalized dimensions D q Henstchel & Procaccia Physica D 8, 435 (1983) Grassberger, Phys. Lett. A 97, 227 (1983) Is the radial viscous fingering pattern a multifractal or a monofractal ? (i.e., are all D q the same?) fractal dim. q = 0

14. Generalized dimensions Generalized Dimension D q DqDq q Conclude: viscous fingering pattern is a monofractal with D q = 1.70 independent of q (self-similar) DLA is also monofractal: D q = 1.713

15. Harmonic measure harmonic measure -- probability measure for a random walker to hit the cluster. D q 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 D q  f(  ) spectrum r P i (r) ~ r ,  – singularity strength with values  min <  <  max f(  ) – probability of value  f(  ) spectrum of singularities Generalized fractal dimensions D q Legendre transform i Halsey, Jensen, Kadanoff, Procaccia, Shraiman, Phys. Rev. A 33 (1986)

17. harmonic measure f(  ) : viscous fingers & DLA 2 1 0 0 51015 20  1.71 f  DLA viscous fingering clusters of increasing size Tentative conclusion: DLA and viscous fingers are in the same universality class Mathiesen, Procaccia, Thrasher, Swinney --- preliminary results

18. Growth dynamics: unscreened angle   largest angle that does not include pre-existing pattern active region pre-existing pattern

19. Distribution of the unscreened angle Θ → P(  ) is independent of forcing but depends on r/b P(  )

20. Asymptotic screening angle PDF Invariant distribution at large r/b 160 322 484 644 806 r/b P 

21. Exponential convergence to invariant distribution conver- gence length  =200 (r)(r) r/b 0.5 atm  p=1.75 atm 1.25 atm 0.25 atm

22. Asymptotic distribution P(  ): = 145 o  36 o BUT no indication of a critical angle or 5-fold symmetry Gaussian

23. Unscreened angle PDF v. f. DLA : 146 o 127 o σ: 36 o 51 o Skewness: 0.06 0.3 Kurtosis: 2.3 3.8 DLA on-lattice algorithm Kaufman, Dimino, Chaikin, Physica A 157 (1989) viscous fingering experiment P( 

24. Coarsening DLA with diffusion & viscous fingering patterns DLA plus diffusion EXPT t=0545164900 t=0 s 115 s 1040 s10040 s Lipshtat, Meerson, & Sarasov (2002)

25. Coarsening: length L 1 below which viscous fingering pattern is smooth Density- density correlation

26. L 2 : an intermediate length scale -- diluted because small scales thicken while large scales are frozen L 2 defined by minimum in  C  C(r)

27. Non-self-similar coarsening of pattern: described by two lengths L 1 and L 2

28. Non-self-similar coarsening: lengths L 1 and L 2 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, Swinney Phys. Rev. Lett. 91 (2003)

29. Fjords between viscous fingers sector geometry Lajeunesse & Couder J. Fluid Mech. 419 (2000) “A fjord center line follows approximately a curve normal to the successive profiles of stable fingers.” FJORD

30. Can ramified finger be fit to theory for inviscid fingering?

31. 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 ) where  k and  k are complex constants of motion. Mineev & Dawson, Phys. Rev. E 50 (1994)

32. A fit with 43 sets of complex constants  k and  k

33. Evolve solution forward in time preliminary Moore, Thrasher, Mineev, Swinney

34. 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

35. Fjord dependence on forcing Ca = 0.040

36. Predict fjord width W original interface emergent fjord emergent finger Conclude W = (1/2) c V

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 t=0

39. time dependence -5 0 51015 time (s) t=0  ( cm -1 ) V ( cm/s ) curvature tip velocity

40. Channel base width: W 0 = W(ℓ=0, t=0) 0 510 15 fjord length ℓ (cm) W 4

41. fjord width (cm) Compare theory and experiment theory

42. Theory predicts parallel walls of fjord: channel wall Measure fjord opening angle Mineev, Phys. Rev. Lett. 80 (1998) Pereira & Elezgaray, Phys. Rev. E 69 (2004) FJORD stagnation point sequence of snapshots of interface,  t = 50 sec

43. 7.5 o fjord length ℓ (cm) Opening angle of a fjord rectangular cell Ristroph, Thrasher, Mineev, Swinney 2005  (deg)

44. rectangular cell = 7.9  0.8 deg circular cell = 8.2  1.1 deg p(  )  (degrees) Opening angle probability distribution RESULT: = 8.0  1.0 deg Invariant with fjord width length direction forcing geometry

45. 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 Aggregation Viscous fingers Electrodeposition Brady & Ball, Nature (1983) and metal corrosion, brittle fracture, …

46. Viscous fingers and DLA: same universality class pattern: monofractal with D q = 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