Experimental illustrations of pattern-forming phenomena: Examples from Rayleigh-Benard convection, Taylor-vortex flow, and electro convection

Guenter Ahlers

Department of PhysicsUniversity of CaliforniaSanta Barbara CA USA

Q

d T

T/Tc - 1

Prandtl numberkinematic viscosity

thermal diffusivity

z

x

k = (q, p) T = Tcond + T sin( z) exp i(q x + p y ) exp( t )

Neutral curve

k = (q, p)

Fluctuations

Patterns

Equilibrium

<T

>

T sin( z ) exp[ i ( q x + p y ) ]

Temperature

FerromagnetParamagnet

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Fluctuations well below the onset of convection

R / Rc = 0.94

Snapshot in real space

Structure factor =square of the modulus of the Fourier transformof the snapshot

Movie by Jaechul Oh

p

p

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Shadowgraph image of the pattern. The sampleis viewed from the top.In essence, the methodshows the temperature field.

Experiment: J. Oh and G.A., cond-mat/0209104.Linear Theory: J. Ortiz de Zarate and J. Sengers, Phys. Rev. E 66, 036305 (2002).

ST ~ k2

ST ~ k-4

k k

= -0.57

-0.68

-0.78

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

-0.14

-0.70

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

C = C0 exp( -k) t )

k)

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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).

k

T

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

Taylor vortex flowFirst 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 asub-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 surfacethe pinning of the phaseis weak and a vortexpair can be gained orlost. The EckhausInstability 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).

Shadowgraph image ofthe pattern. The sampleis viewed from the top.In essence, the methodshows the temperature field.

Back to Rayleigh-Benard !

Wavenumber Selection byDomain 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).

†= 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).

†= 0

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Movie by N. Becker

†= 0

Movie by Nathan Becker

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

Q

d T

T/Tc - 1

= 2 f d2/

Prandtl number

kinematic viscosity

thermal diffusivity

c†= 16

Movies by Nathan Becker

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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).

Electroconvection in a nematic liquid crystal

Director

PlanarAlignment

V = V0 cos(t )

Convection for V0 > Vc = (V0 / Vc) 2 - 1

Anisotropic !

Oblique rolls

zig

zag

Director

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

Rayleigh-Benard convectionFluctuations and linear growth rates below onsetRotational invarianceNeutral curveStraight rolls above onsetStability range above onset, Busse Balloon

Taylor-vortec flowEckhaus instabilityNarrower band due to reduced phase pinning at a free surfaceWavenumber selection by a ramp in epsilon

More Rayleigh-BenardWavenumber selection by a domain wallWavenumber determined by skewed-varicose instabilityOnset of spiral-defect chaos

Rayleigh-Benard with rotationKuepers-Lortz or domain chaos

Electro-convection in a nematicLoss of rotational invariance

Summary: