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Some New Applications of Conformal Mapping

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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)

Why is there a similarity solution of this form?

Maybe we don’t need Laplace’s equation…

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…

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

All “two-gradient” operators transform under conformal mapping,

w=f(z), in the same way, multiplied by the Jacobian factor:

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…

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

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

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.

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 .

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”

“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

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

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”)

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.

- Generalization of Burgers vortex sheet
- Nontrivial dependence on Reynolds number
- Exact solutions everywhere, free of singularities
(useful for testing numerics or rigorous analysis)

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

- 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)

- 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

T. Halsey, Physics Today (2000).

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

M. Z. Bazant, J. Choi, B. Davidovitch, Phys. Rev. Lett. 91, 045503 (2003).

w plane

z plane

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.

Same fractal dimension as DLA

in spite of changing anisotropy and growth rate

How does this compare to the long-time limit of continuous growth?

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?

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.

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).

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.

- 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