Some New Applications of Conformal Mapping Martin Z. Bazant Department of Mathematics, MIT Collaborators: Jaehyuk Choi (PhD’05), Benny Davidovitch (Harvard), Keith Moffatt (DAMTP, Cambridge), Dionisios Margetis (MIT), Darren Crowdy (Imperial), Todd Squires (UCSB)
Some New Applications of Conformal Mapping. Martin Z. Bazant Department of Mathematics, MIT Collaborators: Jaehyuk Choi (PhD’05), Benny Davidovitch (Harvard), Keith Moffatt (DAMTP, Cambridge), Dionisios Margetis (MIT), Darren Crowdy (Imperial), Todd Squires (UCSB). - PowerPoint PPT Presentation
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Numerical solution by spectralmethod after conformal mapping inside the disk
Very accurate uniformly valid matched asymptotic approx.in streamline coordinates
Diffusive flux versus angle
Pe = 1
Pe = 0.01
Pe = 100
Critical Peclet number = 60
Transition from “clouds to wakes”Choi, Margetis, Squires & Bazant, J. Fluid Mech. 536, 155 (2005).
New Navier-Stokes solutions: II. Vortex avenues
• An exact steady solution for a cross-flow jet• Vorticity is pinned between flow dipoles at zero and infinity (uniform flow)• Nontrivial dependence on Reynolds number (“clouds to “wakes”)
1. Analytic continuation of potential flow inside the disk
2. Continuation of non-harmonic concentration by circular reflection
Mapped vortex avenues
A “vortex butterfly” A “vortex wheel”
These new exact solutions show how arbitrary 2d vorticity patterns can be “pinned” by transverse flows, although instability is likely at high Re.
Vortex fishbones
• Generalization of Burgers vortex sheet• Nontrivial dependence on Reynolds number • Exact solutions everywhere, free of singularities (useful for testing numerics or rigorous analysis)
Applications in pattern formationM. Z. Bazant, J. Choi, B. Davidovitch, Phys. Rev. Lett. 91, 045503 (2003).
• Quasi-steady, conformally invariant transport processes
Off-lattice cluster of 1,000,000 “sticky” random walkers (Sander)
T. Witten & L. M. Sander, Phys. Rev. Lett. (1981).
Some DLA-like clusters in nature
• Electrodeposits
(CuSO4 deposit, J. R. Melrose)
• Thin-film surface deposits (GeSe2/C/Cu film, T. Vicsek)
• Snowflakes (Nittman, Stanley)
Laplacian field driving DLA
Random-walk simulationMandelbrot, Evertsz 1990
Conformal-mapping simulation
Iterated conformal maps for DLA
T. Halsey, Physics Today (2000). Stepanov & Levitov, Phys. Rev. E (2001)
M. Hastings & L. Levitov, Physica D (1998).
Mineral Dendrites• Effects of fluid flow, electric fields,
and surface curvature?• Infer ancient geological conditions?
George Rossman, Caltechhttp://minerals.gps.caltech.edu
Advection-Diffusion-Limited Aggregation (ADLA)
w plane z plane
M. Z. Bazant, J. Choi, B. Davidovitch, Phys. Rev. Lett. 91, 045503 (2003).
Advection-diffusion-limited aggregation (ADLA)
w plane
z plane
Pe = 0.1 Pe = 1 Pe = 10
M. Z. Bazant, J. Choi, B. Davidovitch, Phys. Rev. Lett. 91, 045503 (2003).
Same fractal dimension as DLA, but time-(Peclet-)dependent anisotropy.
ADLA Morphology and Dynamics
Same fractal dimension as DLA in spite of changing anisotropy and growth rate
Dynamical Fixed Point of ADLA as
How does this compare to the long-time limit of continuous growth?
Continuous growth by advection-diffusion
Generalized Polubarinova-Galin equation (1945) for the time-dependent conformal map from the exterior of the unit disk to the exterior of the growth.
Flux profile on the disk in thehigh-Pe (long time) limit:
How does this compareto the average shape ofstochastic ADLA clusters?
Davidovitch, Choi & Bazant, Phys. Rev. Lett. 95, 075504 (2005).
Exact self-similar limiting shape
The average shape of transport-limited aggregates
An integral equation for average conformal map:
• We show that the continuous dynamics is the “mean-field approximation” of the stochastic dynamics, but the average shape is not the same for ADLA.• Suggests that Arneodo’s conjecture (that the average DLA in a channel is a Saffman-Taylor viscous finger) is false.
Davidovitch, Choi & Bazant, Phys. Rev. Lett. 95, 075504 (2005).
Transport-limited growth on curved surfacesV. Entov & P. Etingov (1991): viscous fingering (Laplacian growth) on a sphere. J. Choi, M. Z. Bazant & D. Crowdy, in preparation: DLA on curved surtaces
Our “two-gradient” equations are invariant under any conformal mapping (e.g. including stereographic projections to curved surfaces)
Motivation: Mineral dendrites(G. Rossman, Caltech)
“Circle Limit III” M.C. Escher
DLA on curved surfaces
Sphere (k = 1) Pseudosphere (k = -1)
Jaehyuk Choi, PhD Thesis (2005).
• The fractal dimension is independent of curvature, but..• Multifractal exponents of the harmonic measure do depend on curvature.
Conclusion “Two-gradient” equations are conformally invariant.
• Steady 2d transport processes• Electrochemical transport• Gravity currents in ambient flows in porous media• Navier-Stokes vortex structures
• Quasi-steady 2d transport-limited growth• Continuous growth: fiber coating from flows, electrodeposition• Stochastic growth: ADLA, DLA on curved surfaces