experimental illustrations of pattern-forming phenomena:
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/1.jpg)
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
![Page 2: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/2.jpg)
![Page 3: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/3.jpg)
k = (q, p) T = Tcond + T sin( z) exp i(q x + p y ) exp( t )
![Page 4: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/4.jpg)
Neutral curve
![Page 5: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/5.jpg)
k = (q, p)
![Page 6: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/6.jpg)
Fluctuations
Patterns
Equilibrium
<T
>
T sin( z ) exp[ i ( q x + p y ) ]
Temperature
FerromagnetParamagnet
![Page 7: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/7.jpg)
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.
![Page 8: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/8.jpg)
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
![Page 9: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/9.jpg)
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)
![Page 10: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/10.jpg)
QuickTime™ and aYUV420 codec decompressorare needed to see this picture.
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).
![Page 11: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/11.jpg)
k
T
F. Busse and R.M. Clever, J. Fluid Mech. 91, 319 (1979); and references therein.
![Page 12: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/12.jpg)
![Page 13: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/13.jpg)
Taylor vortex flowFirst experiments and linear stability analysis by G.I. Taylor in Cambridge
![Page 14: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/14.jpg)
![Page 15: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/15.jpg)
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).
![Page 16: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/16.jpg)
( k - kc ) / kc
M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 1986.
![Page 17: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/17.jpg)
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).
![Page 18: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/18.jpg)
Free upper surface
Rigid boundaries
![Page 19: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/19.jpg)
![Page 20: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/20.jpg)
![Page 21: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/21.jpg)
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).
![Page 22: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/22.jpg)
![Page 23: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/23.jpg)
![Page 24: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/24.jpg)
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
![Page 25: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/25.jpg)
J.R. Royer, P. O'Neill, N. Becker, and G.A., Phys. Rev. E 70 , 036313 (2004).
![Page 26: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/26.jpg)
![Page 27: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/27.jpg)
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).
![Page 28: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/28.jpg)
†= 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).
![Page 29: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/29.jpg)
†= 0
![Page 30: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/30.jpg)
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
Movie by N. Becker
![Page 31: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/31.jpg)
†= 0
Movie by Nathan Becker
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
Spiral-defect chaos:S.W. Morris, E. Bodenschatz, D.S. Cannell, and G.A., Phys. Rev. Lett. 71, 2026 (1993).
![Page 32: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/32.jpg)
Q
d T
T/Tc - 1
= 2 f d2/
Prandtl number
kinematic viscosity
thermal diffusivity
![Page 33: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/33.jpg)
c†= 16
Movies by Nathan Becker
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
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).
![Page 34: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/34.jpg)
Electroconvection in a nematic liquid crystal
Director
PlanarAlignment
V = V0 cos(t )
Convection for V0 > Vc = (V0 / Vc) 2 - 1
Anisotropic !
![Page 35: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/35.jpg)
Oblique rolls
zig
zag
Director
![Page 36: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/36.jpg)
![Page 37: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/37.jpg)
QuickTime™ and aYUV420 codec decompressorare needed to see this picture.
X.-L. Qiu + G.A., Phys. Rev. Lett. 94, 087802 (2005)
![Page 38: Experimental illustrations of pattern-forming phenomena:](https://reader036.vdocuments.us/reader036/viewer/2022062322/56814ab3550346895db7c608/html5/thumbnails/38.jpg)
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: