1 / 46

Fractal growth of viscous fingers

Developments in Experimental Pattern Formation Isaac Newton Institute, Cambridge 12 August 2005. Fractal growth of viscous fingers. Harry Swinney University of Texas Center for Nonlinear Dynamics and Department of Physics. Matt Thrasher Leif Ristroph (now Cornell U)

westra
Download Presentation

Fractal growth of viscous fingers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Developments in Experimental Pattern Formation Isaac Newton Institute, Cambridge 12 August 2005 Fractal growth of viscous fingers 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. Velocity of the interface • Pressure field 2P=0 Viscous fingering in Hele-Shaw cell m1< m2 b << w Flow w LAPLACIAN GROWTH PROBLEM Saffman-Taylor (1958): finger width → ½ channel width

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

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

  6. air air air air 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. E72 (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 > 1.85 and is not in same universality class as DLA

  10. N(e)e–D0 Fractal dimension of viscous fingering pattern N(e) number of boxes of size e needed to cover the entire object

  11. N(e) e-D0 D0 = 1.70±0.02 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 D8, 435 (1983) Grassberger, Phys. Lett. A97, 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 dimensions Generalized Dimension Dq 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. r generalized dimensions Dq  f(a) spectrum Halsey, Jensen, Kadanoff, Procaccia, Shraiman, Phys. Rev. A33 (1986) • Pi(r) ~ ra, a – singularity strength • with values amin < a < 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 viscous fingering clusters of increasing size DLA 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 Θ P(Q) → P(Q) is independent of forcing but depends on r/b

  20. Asymptoticscreening angle PDF P(Q) 484 322 644 160 r/b 806 Invariant distribution at large r/b

  21. Exponential convergence to invariant distribution Dp=1.75 atm G(r) 1.25 atm conver- gence length x=200 0.5 atm 0.25 atm r/b

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

  23. Unscreened angle PDF P(Q DLA on-lattice algorithm Kaufman, Dimino, Chaikin, Physica A157 (1989) viscous fingering experiment v. f.DLA <>: 146o127o σ: 36o51o Skewness: 0.060.3 Kurtosis: 2.33.8

  24. CoarseningDLA with diffusion & viscous fingering patterns DLA plus diffusion Lipshtat, Meerson, & Sarasov (2002) t=0 54 516 4900 EXPT 115 s t=0 s 1040 s 10040 s

  25. Coarsening:length L1below 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 pattern:described by two lengths L1 and L2

  28. Non-self-similar coarsening:lengths L1 and L2power 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. E50 (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.

  32. A fit with 43 sets of complex constants ak and bk

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

  34. 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 t=0

  39. time dependence curvature k (cm-1) t=0 tip velocity V (cm/s) 5 10 15 -5 0 time (s)

  40. Channel base width: W0 = W(ℓ=0, t=0) 4 W 5 10 15 0 fjord lengthℓ (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. E69 (2004) channel wall

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

  44. Opening angle probability distributionRESULT: <  > = 8.0  1.0 deg p(y) • Invariant • with fjord • width • length • direction • forcing • geometry rectangular cell <y> = 7.90.8 deg circular cell <y >= 8.21.1 deg y (degrees)

  45. Electrodeposition Dielectric breakdown Brady & Ball, Nature (1983) Niemeyer et al. PRL (1984) Fractal growth phenomena: same universality class ? Bacterial growth Diffusion Limited Aggregation Viscous fingers DLA Matsushita (2003) U Texas (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

More Related