Some New Applications of Conformal Mapping

Martin Z. BazantDepartment 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

Examples in physics (transport theory):

• Advection-diffusion in potential flows• Ion transport in bulk (quasi-neutral) electrolytes• Forced gravity currents in porous media

M. Z. Bazant, Proc. Roy. Soc. A 460, 1433 (2004).

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

Arbitrary shape, finite absorber Strip in streamline coordinates Other conformal maps…

M. J. Boussinesq, J. Math. 1, 285 (1905).

2d potential flowPDE for steady (linear) advection-diffusion

Transformation to streamline coordinates

New applications in electrochemistry

Nernst-Planck equations for (steady)ion transport in a neutral electrolyte:

Misaligned coaxial electrodes

Fringe currents

Parallel plateelectrodes

M. Z. Bazant, Proc. Roy. Soc. A 460, 1433 (2004).

A class of exact solutions

New applications in multiphase flowsI. Eames, M. A. Gilbertson & M. Landeryou, J. Fluid Mech. 523, 265 (2005).

Spreading of viscous gravity currentsbelow ambient (potential) flows

0 5 10 15 20-3

-2

-1

0

1

2

3Y

X

-2 0 2 4 6 8-2

-1

0

1

2Y

X

Point source,uniform flow

Experiments inHele-Shaw cellsStraining flows Obstacles

Conformal mappings

New applications in viscous fluid mechanics

Seek new solutions for vorticity “pinned” by transverse flow.

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).

Transverse in-plane velocity potential

Out-of-plane velocity (shearing 1/2 planes)

Pressure

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 anadvection-diffusion problem (where )

with the pressure given by .

Exact solutions to the Navier-Stokes equations having steady vortex structures

New solutions: I. Mapped vortex sheets

Non-uniformly strained “wavy vortex sheets”

A six-pointed “vortex star”

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)

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

• 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 fingeringwith surface tension(M. Siegel)

Stochastic Laplacian Growth:Diffusion-Limited Aggregation (DLA)

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

f

Last term: Todd Squires

Some new applications of conformal mapping:

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