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Some New Applications of Conformal Mapping

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).

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Some New Applications of Conformal Mapping

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  1. 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)

  2. Motivation: An advection-diffusion problem Why is there a similarity solution of this form?

  3. Conformal mapping Maybe we don’t need Laplace’s equation…

  4. Conformal mapping of non-harmonic functions? Textbook mantra: Conformal mapping preserves harmonic functions, since they are the real (or imaginary) parts of analytic functions. Alternate perspective: Laplace’s equation is conformally invariant. This property could be more general…

  5. A class of non-Laplacian, nonlinear, conformally invariant systems M. Z. Bazant, Proc. Roy. Soc. A 460, 1433 (2004). Examples in physics (transport theory): • Advection-diffusion in potential flows • Ion transport in bulk (quasi-neutral) electrolytes • Forced gravity currents in porous media

  6. Proof: All “two-gradient” operators transform under conformal mapping, w=f(z), in the same way, multiplied by the Jacobian factor:

  7. A simple consequence: Boussinesq’s transformation M. J. Boussinesq, J. Math. 1, 285 (1905). PDE for steady (linear) advection-diffusion 2d potential flow Transformation to streamline coordinates Arbitrary shape, finite absorber Strip in streamline coordinates Other conformal maps…

  8. New applications in electrochemistry M. Z. Bazant, Proc. Roy. Soc. A 460, 1433 (2004). Nernst-Planck equations for (steady) ion transport in a neutral electrolyte: A class of exact solutions Misaligned coaxial electrodes Parallel plate electrodes Fringe currents

  9. New applications in multiphase flows I. Eames, M. A. Gilbertson & M. Landeryou, J. Fluid Mech. 523, 265 (2005). Spreading of viscous gravity currents below ambient (potential) flows Conformal mappings Experiments in Hele-Shaw cells Point source, uniform flow Obstacles Straining flows

  10. New applications in viscous fluid mechanics M. Z. Bazant & H. K. Moffatt, J. Fluid Mech. 541, 55 (2005). Burgers vortex sheet The similarity solution, revisited. J. M. Burgers, Adv. Appl. Mech. 1, 171 (1948). Out-of-plane velocity (shearing 1/2 planes) Transverse in-plane velocity potential Pressure Seek new solutions for vorticity “pinned” by transverse flow.

  11. Exact solutions to the Navier-Stokes equations having steady vortex structures We seek solutions to the steady 3d Navier-Stokes equations for 2d vortex structures stabilized by planar potential flow Then, the non-harmonic out-of-plane velocity satisfies an advection-diffusion problem (where ) with the pressure given by .

  12. New solutions: I. Mapped vortex sheets Non-uniformly strained “wavy vortex sheets” For each f(z), these “similarity solutions” have the same isovorticity (v=const) lines for all Reynolds numbers. A six-pointed “vortex star”

  13. Towards a Class of Non-Similarity Solutions… “The simplest nontrivial problem in advection-diffusion” An absorbing cylinder in a uniform potential flow. Maksimov (1977), Kornev et al. (1988, 1994) Choi, Margetis, Squires & Bazant (2005) Very accurate uniformly valid matched asymptotic approx. in streamline coordinates Numerical solution by spectral method after conformal mapping inside the disk

  14. Transition from “clouds to wakes” Choi, Margetis, Squires & Bazant, J. Fluid Mech. 536, 155 (2005). Pe = 0.01 Diffusive flux versus angle Pe = 1 Pe = 100 Critical Peclet number = 60

  15. New Navier-Stokes solutions: II. Vortex avenues 1. Analytic continuation of potential flow inside the disk 2. Continuation of non-harmonic concentration by circular reflection • 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”)

  16. Mapped vortex avenues A “vortex wheel” A “vortex butterfly” These new exact solutions show how arbitrary 2d vorticity patterns can be “pinned” by transverse flows, although instability is likely at high Re.

  17. 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)

  18. Applications in pattern formation M. Z. Bazant, J. Choi, B. Davidovitch, Phys. Rev. Lett. 91, 045503 (2003). • Quasi-steady, conformally invariant transport processes • Continuous interfacial dynamics • Stochastic interfacial dynamics

  19. Continuous Laplacian GrowthViscous fingering & solidification (without surface tension) • Conformal-map dynamics Polubarinova-Kochina, Galin (1945) • “Finger” solutions Saffman & Taylor (1958) • Finite-time cusp singularities Shraiman & Bensimon (1984) Viscous fingering with surface tension (M. Siegel)

  20. Stochastic Laplacian Growth:Diffusion-Limited Aggregation (DLA) T. Witten & L. M. Sander, Phys. Rev. Lett. (1981). Off-lattice cluster of 1,000,000 “sticky” random walkers (Sander)

  21. 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)

  22. Laplacian field driving DLA Random-walk simulation Mandelbrot, Evertsz 1990 Conformal-mapping simulation

  23. T. Halsey, Physics Today (2000). Iterated conformal maps for DLA M. Hastings & L. Levitov, Physica D (1998). Stepanov & Levitov, Phys. Rev. E (2001)

  24. Effects of fluid flow, electric fields, and surface curvature? Infer ancient geological conditions? George Rossman, Caltech http://minerals.gps.caltech.edu Mineral Dendrites

  25. Advection-Diffusion-Limited Aggregation (ADLA) M. Z. Bazant, J. Choi, B. Davidovitch, Phys. Rev. Lett. 91, 045503 (2003). w plane z plane

  26. Advection-diffusion-limited aggregation (ADLA) M. Z. Bazant, J. Choi, B. Davidovitch, Phys. Rev. Lett. 91, 045503 (2003). w plane Pe = 0.1 Pe = 1 Pe = 10 z plane Same fractal dimension as DLA, but time-(Peclet-)dependent anisotropy.

  27. ADLA Morphology and Dynamics Same fractal dimension as DLA in spite of changing anisotropy and growth rate

  28. Dynamical Fixed Point of ADLA as How does this compare to the long-time limit of continuous growth?

  29. Continuous growth by advection-diffusion Davidovitch, Choi & Bazant, Phys. Rev. Lett. 95, 075504 (2005). 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 the high-Pe (long time) limit: Exact self-similar limiting shape How does this compare to the average shape of stochastic ADLA clusters?

  30. The average shape of transport-limited aggregates Davidovitch, Choi & Bazant, Phys. Rev. Lett. 95, 075504 (2005). 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.

  31. Transport-limited growth on curved surfaces V. 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)

  32. Jaehyuk Choi, PhD Thesis (2005). DLA on curved surfaces “Circle Limit III” M.C. Escher Sphere (k = 1) Pseudosphere (k = -1) • The fractal dimension is independent of curvature, but.. • Multifractal exponents of the harmonic measure do depend on curvature.

  33. 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 Last term: Todd Squires Some new applications of conformal mapping: f http://math.mit.edu/~bazant

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