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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 Physics University of California Santa Barbara CA USA Q d TT T/ T c - 1 Prandtl number kinematic viscosity thermal diffusivity z x

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k = (q, p) T = T cond + 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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Deterministic equation of motion (e.g. Ginzburg-Landau- Wilson equation for eq. syst. or NS eq. for conv. ) Stochastic “external” force representing the noise acting on the deterministic system due to the Brownian motion of the atoms/molecules Equilibrium phase transitions: Potential system, extremum principle: Non-equilibrium systems: ???? usually small for macroscopic systems

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Fluctuations Patterns Equilibrium T sin( z ) exp[ i ( q x + p y ) ] Temperature Ferromagnet Paramagnet

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Fluctuations well below the onset of convection R / R c = 0.94 Snapshot in real space Structure factor = square of the modulus of the Fourier transform of the snapshot Movie by Jaechul Oh p p Shadowgraph image of the pattern. The sample is viewed from the top.In essence, the method shows 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). S T ~ k 2 S T ~ k -4 k k = -0.57 -0.68 -0.78

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Squares: = 200 ms Circles: = 500 ms = camera exposure time

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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, ) = / C = C 0 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 TT F. Busse and R.M. Clever, J. Fluid Mech. 91, 319 (1979); and references therein.

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Taylor vortex flow First 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 a sub-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 - k c ) / k c M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 1986.

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At the free upper surface the pinning of the phase is weak and a vortex pair can be gained or lost. The Eckhaus Instability 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 of the pattern. The sample is viewed from the top. In essence, the method shows the temperature field. Back to Rayleigh-Benard ! 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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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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Movie by N. Becker

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= 0 Movie by Nathan Becker 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 TT T/ T c - 1 = 2 f d 2 / Prandtl number kinematic viscosity thermal diffusivity

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c = 16 Movies by Nathan Becker 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 Planar Alignment V = V 0 cos( t ) Convection for V 0 > V c = (V 0 / V c ) 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 convection Fluctuations and linear growth rates below onset Rotational invariance Neutral curve Straight rolls above onset Stability range above onset, Busse Balloon Taylor-vortec flow Eckhaus instability Narrower band due to reduced phase pinning at a free surface Wavenumber selection by a ramp in epsilon More Rayleigh-Benard Wavenumber selection by a domain wall Wavenumber determined by skewed-varicose instability Onset of spiral-defect chaos Rayleigh-Benard with rotation Kuepers-Lortz or domain chaos Electro-convection in a nematic Loss of rotational invariance Summary:

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