Some%20unresolved%20issues%20in%20pattern-forming

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Some unresolved issues in pattern-forming
non-equilibrium systems
Guenter Ahlers Department of
Physics University of California Santa
Barbara CA
Kapil Bajaj, Nathan Becker, Eberhard
Bodenschatz Robert Ecke, Yuchou Hu, Jun Liu,
Worawat Meevasana Stephen Morris,
Jaechul Oh, Xin-Liang Qiu
Sheng-Qi Zhou, and others
Supported by the National Science Foundation
And the US Department of Energy
2
Some unresolved issues in pattern-forming
non-equilibrium systems
or Some questions to which I
really would like to know
the answer
Guenter Ahlers Department of
Physics University of California Santa
Barbara CA
Kapil Bajaj, Nathan Becker, Eberhard
Bodenschatz Robert Ecke, Yuchou Hu, Jun Liu,
Worawat Meevasana Stephen Morris,
Jaechul Oh, Xin-Liang Qiu
Sheng-Qi Zhou, and others
Supported by the National Science Foundation
And the US Department of Energy
3
Rayleigh-Benard convection
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DT 0.1320 K e 0.003
0.1311 K -0.003
0.1248 K -0.051
1.28x1.28x0.0343 mm3
J. Swift, P. C. Hohenberg, Phys. Rev. A 15, 319
(1977). P. C. Hohenberg, J. Swift, Phys. Rev. A
46, 4 773 (1992)
J. Oh and G.A., Phys. Rev. Lett. 91, 094501
(2003).
5
a
b
a
c
d
c
b
d
ltTgt 46.2oC
6
DT 145.5 mK
7
ec 6x10-3
Fexp 7.1x10-4 Fth 5.1x10-4
d 34.3mm
8
q
621.999.bmp
e 0.009
021212a/621/.999
9
q
Director Angle
621.999.bmp
e 0.009
J. Toner and D.R. Nelson, Phys. Rev. B 23, 316
(1981).
021212a/621/.999
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Do we know why the fluctuations are
symmetric About onset? Can we understand the
Phonons quantitatively (e.g. Derive their
corelation functions)? The dislocation density?
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Electroconvection in a nematic liquid crystal
z
Planar Alignment
y
x
Director
V V0 cos( wt )
Convection for V0 gt Vc
e (V0 / Vc) 2 - 1
Anisotropic !
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Oblique -roll fluctuations
Normal-roll fluctuations
Director
f 4000 Hz e -1.3x103
f 25 Hz e -1.3x103
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Time series of S(k0, e, t) at e -1.0x10-2 for
(a) f 25 Hz, k0 (3.936, 1.968) and (b) f
4000 Hz, k0 (4.830, 0)
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f 25 Hz, k0 (3.936, 1.968)
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p direction (all f)
q direction (f 25 Hz)
q direction (f 4 kHz)
f 4000 Hz
f 25 Hz
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Normal rolls
f 4000 Hz
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Normal rolls
Oblique rolls
f 4000 Hz
f 25 Hz
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Alternatively fits of S(k)
Oblique
Normal
Oblique
Normal
X.-L. Qiu and G.A., unpublished.
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Open symbols from S(k) Solid symbols from
growth rates
4 kHz
25 Hz
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Summary

• At large frequency (i.e. for normal rolls), the
  results are consistent with the prediction of
  linear theory.
• At low frequency (i.e. for oblique rolls), there
  were deviations from linear theory. S0, s0, and
  x tended toward a finite limit as e vanished.

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What is the problem?

• At low frequency (i.e. for oblique rolls), there
  were deviations from linear theory. S0, s0, and
  x tended toward a finite limit as e vanished.
• What is the critical behavior expected from the
  two coupled stochastic GL equations

(E. Bodenschatz, W. Zimmermann, and L. Kramer, J.
Phys. (Paris) 49, 1875 (1988)
?
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What is the problem?

• At low frequency (i.e. for oblique rolls), there
  were deviations from linear theory. S0, s0, and
  x tended toward a finite limit as e vanished.
• What is the critical behavior expected from the
  two coupled stochastic GL equations

(E. Bodenschatz, W. Zimmermann, and L. Kramer, J.
Phys. (Paris) 49, 1875 (1988)
?
But some skeptics may still believe that the
finite limits of S0 , s0, and x are due to
some kind of undetermined experimental
problem. So let us do one more thing
MODULATE e
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X.-L. Qiu and G.A., unpublished.
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Vary d
squares zag circles zig
W/2p 0.1 Hz Black d 0.000 Red d
0.014 Blue d 0.021
And similarly for the correlation lengths
X.-L. Qiu and G.A., unpublished.
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Vary W
squares zag circles zig
d 0.021 Black W/2p 1.0 Hz Red W/2p 0.2
Hz Blue W/2p 0.1 Hz
And similarly for the correlation lengths
X.-L. Qiu and G.A., unpublished.
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How about more complicated cases where A
and B are amplitudes of travelling waves? Oblique
and normal rolls, Lifshitz points? Etc?
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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
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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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J.R. Royer, P. O'Neill, N. Becker, and G.A.,
Phys. Rev. E 70 , 036313 (2004).
L.S. Tuckerman and D. Barkley, Phys. Rev. Lett.
61, 408 (1988).
J. Buell and I. Catton, Phys. Fluids 29, 1
(1986) A.C. Newell, T. Passot, and M. Souli,
Phys. Rev. Lett. \bf 64, 2378 (1990)
J.
Fluid Mech. \bf 220, 187 (1990).
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S.W. Morris, E. Bodenschatz, D.S. Cannell, and
G.A., Phys. Rev. Lett. 71, 2026 (1993).
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Can we understand the wavenumber selection by
domiain walls? In large samples?
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0.14 /- 0.03
Gamma 30 sigma 0.32
J. Liu and G.A., Phys. Rev. Lett. 77, 3126 (1996)
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S.W. Morris, E. Bodenschatz, D.S. Cannell, and
G.A., Phys. Rev. Lett. 71, 2026 (1993). Y.-C.
Hu, R. Ecke, and G.A., Phys. Rev. E 51, 3263
(1995).
Movie by N. Becker
SDC
s 0.8
Crosses Aspect ratio lt 60 Solid circles Aspect
ratio gt 60
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s 0.8
Data from several sources, Bodenschatz, Ecke,
Hu, SB group
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Circles G 30 triangles G 70
J. Liu and G.A., Phys. Rev. Lett. 77, 3126
(1996) and unpub.
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Can we understand the onset of spiral-defect chaos
as a function of aspect ratio and Prandtl
number?
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Gamma 28, sigma 1.0
J. Liu and G.A., unpublished.
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R. Krishnamurty and L.N. Howard, Proc. Nat. Acad.
Sci. 78, 1981 (1981) Large Scale Circulation
(Wind of Turbulence)
R 6.8x108
s 596

• 1
• cylindrical
• slightly tilted
• in real time

Movie from the group of K.-Q Xia, Chinese Univ.,
Hong Kong
Why up one side and down the other (rather than
in the middle)?
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Do we have anything to contribute to
the understanding of the formation of relatively
simple patterns at very large Rayleigh numbers?
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Prandtl 0.9 CO2 Omega 17
Movies by N. Becker
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Prandtl 0.51 He-SF6 Omega 260 X300
K. M. S. Bajaj, J. Liu, B. Naberhuis, and G. A,
Phys. Rev. Lett.81, 806 (1998).
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Prandtl 0.17 H2-Xe
K.M.S. Bajaj, G. A., and W. Pesch, Phys. Rev. E
65 , 056309 (2002).
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X10 Omega 42
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Summary I. Fluctuations 1. Rayleigh-Benard a
. Fluctuations above onset b. "Phonons" and
dislocations 2. Electro-Convection Finite
correlation lengths etc. at onset II.
Deterministic patterns, RB 1. Wavenumber
selection by domain walls 2. Wavenumber
selection in large samples 3. Onset of
spiral-defect chaos a.) as a function od
aspect ratio b.) as a function of Prandtl
number 4. "Simple" patterns at very large
Rayleigh a.) R 20,000 and large aspect
ratio b.) R 108 III. Deterinistic
patterns, RB with rotation 1. Squares at
onset 2. First-order transition without
hysteresis