Experimental illustrations of pattern-forming phenomena: Examples from Rayleigh-Benard convection,...
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xperimental illustrations of pattern-forming phenom Examples from Rayleigh-Benard convection, Taylor-vortex flow, and electro convecti Guenter Ahlers Department of Physics University of California Santa Barbara CA USA Q d T T/T c - 1 Prandtl number kinematic viscosity thermal diffusivity z x
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Transcript of Experimental illustrations of pattern-forming phenomena: Examples from Rayleigh-Benard convection,...
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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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).
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:
