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Experimental illustrations of pattern-forming phenomena:

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Experimental illustrations of pattern-forming phenomena:

Examples from Rayleigh-Benard convection,

Taylor-vortex flow, and electro convection

- Guenter Ahlers
- Department of Physics
- University of California
- Santa Barbara CA USA

z

d

DT

Q

x

n = kinematic viscosity

Prandtl number

e = DT/DTc - 1

k = thermal diffusivity

s = n / k

k = (q, p)

T = Tcond + dT sin(p z) exp i(q x + p y ) exp( s t )

Neutral curve

e = 0

k = (q, p)

Fluctuations

Patterns

Equilibrium

Paramagnet

Ferromagnet

<dT>

Temperature

Q = dT sin( p z ) exp[ i ( q x + p y ) ]

Fluctuations well below the onset of convection

Structure factor =

square of the modulus

of the Fourier transform

of the snapshot

Shadowgraph image of the pattern. The sample

is viewed from the top.In essence, the method

shows the temperature field.

p

p

Snapshot in real space

R / Rc = 0.94

Movie by Jaechul Oh

dST ~ k2

e = -0.57

-0.68

-0.78

dST ~ k-4

k

k

Experiment: J. Oh and G.A., cond-mat/0209104.

Linear Theory: J. Ortiz de Zarate and J. Sengers, Phys. Rev. E 66, 036305 (2002).

C(k, t) = < ST (k, t) ST (k, t+ t) > / < ST2 (k, t) >

C = C0 exp( -s(k) t )

-0.14

s(k)

e = -0.70

J. Oh, J. Ortiz de Zarate, J. Sengers, and G.A., Phys. Rev. E 69, 021106 (2004).

Just above onset, straight rolls are stable.

Theory: A. Schluter, D. Lortz, and F. Busse, J. Fluid Mech. 23, 129 (1965).

This experiment: K.M.S. Bajaj, N. Mukolobwiez, N. Currier, and G.A., Phys. Rev. Lett. 83, 5282 (1999).

DT

k

F. Busse and R.M. Clever, J. Fluid Mech. 91, 319 (1979); and references therein.

Taylor vortex flow

First experiments and linear stability analysis by G.I. Taylor in Cambridge

time

Inner cylinder speed

The rigid top and bottom pin the phase of the

vortices. They also lead to the formation of a

sub-critical Ekman vortex.

M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 (1986).

G. A., D.S. Cannell, M.A. Dominguez-Lerma, and R. Heinrichs,

Physica, 23D, 202 (1986).

A.M. Rucklidge and A.R. Champneys, Physica A 191, 282 (2004).

In the interior, a vortex pair is lost or gained

when the system leaves the stable band of states.

Theory:

W. Eckhaus, Studies in nonlinear stability theory, Springer, NY, 1965.

Experiment:

M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 (1986).

G. A., D.S. Cannell, M.A. Dominguez-Lerma, and R. Heinrichs,

Physica, 23D, 202 (1986).

( k - kc ) / kc

M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 1986.

At the free upper surface

the pinning of the phase

is weak and a vortex

pair can be gained or

lost. The Eckhaus

Instability is never

reached.

Experiment:

M. Linek and G.A.,

Phys. Rev. E 58, 3168 (1998).

Theory:

M.C. Cross, P.G. Daniels,

P.C. Hohenberg, and E.D. Siggia,

J. Fluid Mech. 127, 155 (1983).

Free upper surface

Rigid

boundaries

Theory:

H. Riecke and H.G. Paap, Phys. Rev. A 33, 547 (1986).

M.C. Cross, Phys. Rev. A 29, 391 (1984).

P.M. Eagles, Phys. Rev. A 31, 1955 (1985).

Experiment:

M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 (1986).

Back to Rayleigh-Benard !

Shadowgraph image of

the pattern. The sample

is viewed from the top.

In essence, the method

shows the temperature

field.

Wavenumber

Selection by

Domain wall

J.R. Royer, P. O'Neill, N. Becker, and G.A., Phys. Rev. E 70 , 036313 (2004).

Experiment:

J. Royer, P. O’Neill, N. Becker, and G.A., Phys. Rev. E 70, 036313 (2004).

Theory:

J. Buell and I. Catton, Phys. Fluids 29, 1 (1986)

A.C. Newell, T. Passot, and M. Souli, J. Fluid Mech. 220, 187 (1990).

W†= 0

V. Croquette, Contemp. Phys. 30, 153 (1989).

Y. Hu, R. Ecke, and G. A., Phys. Rev. E 48, 4399 (1993);

Phys. Rev. E 51, 3263 (1995).

W†= 0

Movie by N. Becker

Movie by Nathan Becker

W†= 0

Spiral-defect chaos:

S.W. Morris, E. Bodenschatz, D.S. Cannell, and G.A., Phys. Rev. Lett. 71, 2026 (1993).

W = 2p f d2/ n

d

DT

Q

n = kinematic viscosity

Prandtl number

e = DT/DTc - 1

k = thermal diffusivity

s = n / k

Wc < W†= 16

G. Kuppers and D. Lortz, J. Fluid Mech. 35, 609 (1969).

R.M. Clever and F. Busse, J. Fluid Mech. 94, 609 (1979).

Y.-C. Hu, R. Ecke, and G.A., Phys. Rev. Lett. 74 , 5040 (1995);

Y. Hu, R. E. Ecke, and G.A., Phys. Rev. E 55, 6928 (1997)

Y. Hu, W. Pesch, G.A., and R.E. Ecke, Phys. Rev. E 58, 5821 (1998).

Movies by Nathan Becker

Electroconvection in a nematic liquid crystal

Planar

Alignment

Director

V = V0 cos( wt )

Convection for V0 > Vc

e = (V0 / Vc) 2 - 1

Anisotropic !

Oblique rolls

zig

zag

Director

X.-L. Qiu + G.A., Phys. Rev. Lett. 94, 087802 (2005)

Rayleigh-Benard convection

Fluctuations and linear growth rates below onset

Rotational invariance

Neutral curve

Straight rolls above onset

Stability range above onset, Busse Balloon

Taylor-vortec flow

Eckhaus instability

Narrower band due to reduced phase pinning at a free surface

Wavenumber selection by a ramp in epsilon

More Rayleigh-Benard

Wavenumber selection by a domain wall

Wavenumber determined by skewed-varicose instability

Onset of spiral-defect chaos

Rayleigh-Benard with rotation

Kuepers-Lortz or domain chaos

Electro-convection in a nematic

Loss of rotational invariance

Summary: