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)

Motivation: An advection-diffusion problem

Why is there a similarity solution of this form?

Conformal mapping

Maybe we don’t need Laplace’s equation…

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…

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

Proof:

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

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…

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

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 Point source, uniform flow -1 -2 1 0 2 Conformal mappings 3 Straining flows Y 2 1 X 0 -1 -3 0 5 10 15 20 -2 -2 0 2 Experiments in Hele-Shaw cells 4 6 8 X

New applications in viscous fluid mechanics

M. Z. Bazant & H. K. Moffatt, J. Fluid Mech.

541

, 55 (2005).

The similarity solution, revisited.

Burgers vortex sheet 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.

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 .

New solutions: I. Mapped vortex sheets A six-pointed “

vortex star

” Non-uniformly strained “

wavy vortex sheets

” For each f(z), these “similarity solutions” have the same isovorticity (v=const) lines for all Reynolds numbers.

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

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

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”)

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 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

Continuous Laplacian Growth

Viscous 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)

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)

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 simulation Mandelbrot, Evertsz 1990 Conformal-mapping simulation

Iterated conformal maps for DLA

M. Hastings & L. Levitov, Physica D (1998).

T. Halsey, Physics Today (2000).

Stepanov & Levitov, Phys. Rev. E (2001)

Mineral Dendrites

• Effects of fluid flow, electric fields, and surface curvature?

• Infer ancient geological conditions?

George Rossman, Caltech http://minerals.gps.caltech.edu

Advection-Diffusion-Limited Aggregation (ADLA)

M. Z. Bazant, J. Choi, B. Davidovitch, Phys. Rev. Lett.

91

, 045503 (2003).

w

plane

z

plane

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.

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

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?

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.

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)

DLA on curved surfaces

Jaehyuk Choi , PhD Thesis (2005).

Sphere (k = 1) Pseudosphere (k = -1) “Circle Limit III” M.C. Escher • • 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.

Some new applications of conformal mapping: • Steady 2d transport processes • Electrochemical transport • • Gravity currents in ambient flows in porous media Last term: Todd Squires • Quasi-steady 2d transport-limited growth • Continuous growth: fiber coating from flows, electrodeposition • Stochastic growth: ADLA, DLA on curved surfaces

http://math.mit.edu/~bazant