1 / 38

# Experimental illustrations of pattern-forming phenomena: - PowerPoint PPT Presentation

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. D T. Q. x.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Experimental illustrations of pattern-forming phenomena:' - alyson

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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 )

k = (q, p)

Patterns

Equilibrium

Paramagnet

Ferromagnet

<dT>

Temperature

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

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).

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 flow

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 - k flowc ) / kc

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

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).

Rigid

boundaries

Theory: flow

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).

the pattern. The sample

is viewed from the top.

In essence, the method

shows the temperature

field.

Wavenumber

Selection by

Domain wall

Experiment: flow

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† flow= 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† flow= 0

W†= 0

Spiral-defect chaos:

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

W flow= 2p f d2/ n

d

DT

Q

n = kinematic viscosity

Prandtl number

e = DT/DTc - 1

k = thermal diffusivity

s = n / k

W flowc < 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

Planar

Alignment

Director

V = V0 cos( wt )

Convection for V0 > Vc

e = (V0 / Vc) 2 - 1

Anisotropic !

Oblique rolls flow

zig

zag

Director

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: