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Pattern Formation in Convection: The Legacy of Henry Benard and Lord Rayleigh Guenter Ahlers Department of Physics University of California Santa Barbara CA USA

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The Legacy of Henry Benard and Lord Rayleigh Henri Benard Lord Rayleigh Rayleigh-Benard Convection

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Count Rumford (Benjamin Thompson) Recognized the phenomenon of convection in his development of an improved fireplace

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William Prout ( ) Identified a new heat transport mechanism (as opposed to radiation or conduction) and called it convection, from “Convectio”: to carry observed a similar hexagonal pattern in soapy water cooled from the open top surface J. Thompson, Proc. Glasg. Phil. Soc. 13, 464 (1882) E.H. Weber, Ann. Phys. Chemie 94, 447 (1855) reported that a flow pattern consisting of an array of polygonal cells developed in a droplet of an alcohol-water solution containing a tracer, sitting in ambient air

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Henri Benard (1900) 1.) he had a free upper surface 2.) he found an hexagonal pattern 3.) used various fluids of different viscosity 4.) studied surface deformation due to convection 5.) measured the characteristic length scales of the patterns 6.) determined the direction of flow within the convection cells Carried out the first systematic and quantitative study of convection in a shallow layer heated from below, and studied the associated formation of convection PATTERNS systematically and quantitatively

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Ph.D. March College de France

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William Nusselt ( ) W. Nusselt, Forsch.-Arbeit auf dem Gebiete des Ing. Wesens, Heft 63,64 (1907); Gesundh.-Ing. 42,43, 477 (1915). Heat transport by air between rigid plates. Motivated by technological issue of insulation. N = Nusselt Number eff = effective thermal conductivity in the presence of convection = thermal conductivity of quiescent fluid

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Lord Rayleigh (John William Strutt, third Baron Rayleigh) Fellow, Trinity College, FRS : Professor of Experimental Physics and Head of the Cavendish Laboratory (successor of Maxwell) 1887 to 1905: Professor of Natural Philosophy, Royal Institution of Great Britain (successor of Tyndall) Nobel Prize 1904 “for his investigation of the density of the most important gases and for his discovery of argon in connection with these studies” Chancellor of Cambridge University 1908

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Lord Rayleigh

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Q L TT Rayleigh carried out a stability analysis of the quiescent fluid layer. He used slip boundary conditions at top and bottom in order to permit an analytic solution T/ T c - 1 He did not recognize that Benard’s convection actually was driven primarily by surface-tension gradients, as pointed out much later by J.R.A. Pearson of Cambridge “On convection cells induced by surface tension”, JFM 4, (1958).

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Rayleigh’s results:

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Sir Harold Jeffreys, Fellow, St.Johns College, Cambridge 1914 to 1989 Mathematics, Geophysics Plumian Professor of Astronomy Knighted 1951 No-slip (rigid) boundary conditions

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nonlinear effects: nature of the bifurcation pattern just above onset Next milestones: Malkus and Veronis (1958): slip (free) BCs super-critical rolls (or stripes)

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nonlinear effects: nature of the bifurcation pattern just above onset Next milestones: Malkus and Veronis (1958): slip (free) BCs super-critical rolls (or stripes) Schluter, Lortz, and Busse (1965): ridgid BCs super-critical rolls (or stripes)

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Open circles: increasing Solid circles: decreasing T Supercritical! Non-BoussinesqBoussinesq Binary mixtures (diffusion driven) Swift and Hohenberg showed that the bifurcation becomes subcritical in the presence of additive (thermal) noise !!!

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Swift and Hohenberg showed that the bifurcation becomes subcritical in the presence of additive (thermal) noise !!! J. Swift and P.C. Hohenberg, “Hydrodynamic fluctuations at the convective instability”, Phys. Rev. A 15, 319 (1977). For most systems, this effect was expected to be observable only for < 10 -6

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Pierre C. Hohenberg 1934 NYU Senior Vice Provost for Research J. Oh and G.A., Phys. Rev. Lett. 91, (2003). Convection in SF 6 near its critical point = /

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Fritz Busse 1936

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Stable States No extremum principle Any state inside the Busse Balloon is attainable if the phase of the pattern is pinned, e.g. by sidewalls, in an experiment. Wave-number selection processes occur when the phase can slip at some point In the pattern In the 1970’s and thereafter the Busse Balloon became the “playground” of experimentalists and theorists with an interest in nonlinear physics and pattern formation

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Four uniquely nonlinear issues: 1.) Wavenumber selection 2.) Spatio-temporal chaos (STC) 3.) Coarsening 4.) Localized structures (Pulses)

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Wave-number selection by curved rolls: Curved rolls normally induce mean flow. For the rotationally symmetric target pattern mean flow can not occur. Thus the mean flow must be balanced precisely by a pressure gradient. This condition leads to a unique wave number unrelated to any extremum principle. L. Koschmieder and S. Pallas, Int. J. Heat Mass Trans. 17, 991 (1974); M. Cross, Phys. Rev. A 27, 490 (1983); P. Manneville and J.M. Piquemal, Phys. Rev. A 28, 1774 (1983); M.C. Cross and A.C. Newell, Physica D 10, 299 (1984); J. Buell and I. Catton, Phys. Fluids 29, 1 (1986) A.C. Newell, T. Passot, and M. Souli, Phys. Rev. Lett. 64, 2378 (1990); J. Fluid Mech. 220, 187 (1990). J.R. Royer, P. O'Neill, N. Becker, and G. A., Phys. Rev. E 70, (2004).

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Cross-Newell equation for the phase of the pattern [M.C. Cross and A.C. Newell, Physica D 10, 299 (1984)]. Photo credit: Ekaterina G. Tribelskaya

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Wave-number selection by a domain wall. The wall can be diplaced, thereby changing the wave number of the rolls parallel to it. This yields a wave number that differs from the one selected by the target patterns. M.C. Cross and G. Tessauro, Phil. Mag. A 56, 703 (1987).

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Spatio-temporal chaos Experiment: S. Morris, E. Bodenschatz, D.S. Cannell, and G.A., Phys. Rev. Lett. 71, 2026 (1993). Bi-stability (straight rolls and Spiral-Defect-Chaos) Dynamics driven by competition between different wavenumber-selection mechanisms and mean flow [Chiam, M. R. Paul, M. C. Cross, and H. S. Greenside, Phys. Rev. E67, (2003)].

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Q d TT

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Prandtl = 0.9 CO 2 Omega = 17 Movies by N. Becker Spatio-temporal chaos at onset above a supercritical bifurcation

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Coarsening in a non-potential system Much numerical work (Vinals, Cross, Paul, and others) starting from random small-amplitude initial conditions. Amplitude evolves fast to a highly disordered pattern. Pattern evolves from extreme disorder, perhaps to stripes or possibly to a ‘glassy” state retaining some disorder. What are the time scales and the dominant mechanisms of this evolution? Experiments are difficult because random initial conditions are difficult to maintain. Instead: Quench from spiral-defect chaos down to a small where straight rolls are known to be the only statistically stationary state.

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N. Becker and G.A., unpublished. Roll curvature in the bulk heals quickly; disclinations disappear. Dislocations become scarce soon after. Domain walls seem to dominate the late pattern evolution. Some dislocations are lingering at late times. 0.5 to 0.05

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Localized structures (Pulses, dissipative solitons) convect. with rotation (Ecke, Knobloch et al.)

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Localized structures (Pulses, dissipative solitons) Binary mixture convection in an annulus Experiments: E. Moses, J. Fineberg, and V. Steinberg, Phys. Rev. 35, 2757 (1987); R. Heinrichs, G.A., and D.S. Cannell, Phys. Rev. A 35, 2761 (1987); Numerous subsequent papers by Kolodner and others. Theory: O. Thual and S. Fauve, Europhys. Lett. 49, 749 (1988); and numerous papers thereafter.

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Localized structures (Pulses, dissipative solitons) Binary mixture convection Experiment: K. Lerman, D.S. Cannell, and G. A., Phys. Rev. E 53, R2041 (1996). Theory: I. Mercader, M. Net, and E. Knobloch, Phys. Rev. E 51, 339 (1995). Binary mixture convection Experiment: K. Lerman, E. Bodenschatz, D.S. Cannell, and G. A, Phys. Rev. Lett. 70, 3572 (1993).

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Count Rumford Henri BenardLord RayleighWilliam NusseltSir Harold Pierre Hohenberg Mike Cross Alan Newell Fritz Busse Our Playground!

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