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### Some New Applications of Conformal Mapping

### Stochastic Laplacian Growth:Diffusion-Limited Aggregation (DLA)

### Laplacian field driving DLA

### DLA on curved surfaces

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

Conformal mappings

Experiments in

Hele-Shaw cells

Point source,

uniform flow

Obstacles

Straining flows

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.

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

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”

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 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.

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 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)

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)

Random-walk simulation

Mandelbrot, Evertsz 1990

Conformal-mapping simulation

T. Halsey, Physics Today (2000).Iterated conformal maps for DLA

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

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

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

Infer ancient geological conditions?

George Rossman, Caltech

http://minerals.gps.caltech.edu

Mineral DendritesAdvection-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)

Jaehyuk Choi, PhD Thesis (2005).

“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.

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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