experimental illustrations of pattern-forming phenomena: examples from rayleigh-benard convection,...
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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 PhysicsUniversity of CaliforniaSanta Barbara CA USA
Q
d T
T/Tc - 1
Prandtl numberkinematic viscosity
thermal diffusivity
z
x
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k = (q, p) T = Tcond + T sin( z) exp i(q x + p y ) exp( t )
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Neutral curve
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k = (q, p)
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Fluctuations
Patterns
Equilibrium
<T
>
T sin( z ) exp[ i ( q x + p y ) ]
Temperature
FerromagnetParamagnet
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QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
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
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
Shadowgraph image of the pattern. The sampleis viewed from the top.In essence, the methodshows the temperature field.
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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
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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).
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k
T
F. Busse and R.M. Clever, J. Fluid Mech. 91, 319 (1979); and references therein.
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Taylor vortex flowFirst experiments and linear stability analysis by G.I. Taylor in Cambridge
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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).
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( k - kc ) / kc
M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 1986.
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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).
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Free upper surface
Rigid boundaries
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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).
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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
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J.R. Royer, P. O'Neill, N. Becker, and G.A., Phys. Rev. E 70 , 036313 (2004).
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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).
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†= 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).
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†= 0
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QuickTime™ and aYUV420 codec decompressor
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Movie by N. Becker
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†= 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).
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Q
d T
T/Tc - 1
= 2 f d2/
Prandtl number
kinematic viscosity
thermal diffusivity
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c†= 16
Movies by Nathan Becker
QuickTime™ and aYUV420 codec decompressor
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QuickTime™ and aYUV420 codec decompressor
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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).
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Electroconvection in a nematic liquid crystal
Director
PlanarAlignment
V = V0 cos(t )
Convection for V0 > Vc = (V0 / Vc) 2 - 1
Anisotropic !
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Oblique rolls
zig
zag
Director
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X.-L. Qiu + G.A., Phys. Rev. Lett. 94, 087802 (2005)
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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: