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 TT  T/  T c - 1  Prandtl number  kinematic viscosity  thermal diffusivity z x

k = (q, p) T = T cond +  T sin(  z) exp i(q x + p y ) exp(  t )

Neutral curve

k = (q, p) 

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

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

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.

Experiment: J. Oh and G.A., cond-mat/ Linear Theory: J. Ortiz de Zarate and J. Sengers, Phys. Rev. E 66, (2002).  S T ~ k 2  S T ~ k -4 k k  =

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

J. Oh, J. Ortiz de Zarate, J. Sengers, and G.A., Phys. Rev. E 69, (2004)  C(k,  ) = / C = C 0 exp( -  k) t )  k)

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T F. Busse and R.M. Clever, J. Fluid Mech. 91, 319 (1979); and references therein.

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

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).

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 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

J.R. Royer, P. O'Neill, N. Becker, and G.A., Phys. Rev. E 70, (2004).

Experiment: J. Royer, P. O’Neill, N. Becker, and G.A., Phys. Rev. E 70, (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

Movie by N. Becker

 = 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).

Q d TT  T/  T c - 1  = 2  f d 2 /  Prandtl number  kinematic viscosity  thermal diffusivity

 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).

Electroconvection in a nematic liquid crystal Director Planar Alignment V = V 0 cos(  t ) Convection for V 0 > V c  = (V 0 / V c ) Anisotropic !

Oblique rolls zig zag Director

X.-L. Qiu + G.A., Phys. Rev. Lett. 94, (2005)

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: