# Workshop Programme

## for period 8 - 12 August 2005

### Developments in Experimental Pattern Formation

8 - 12 August 2005

Timetable

 Monday 8 August 08:30-10:00 Registration 10:00-11:00 Ahlers, G (UCSB) Some unresolved issues in pattern-forming non-equilibrium systems Sem 1 This talk will present experimental results about a number of phenomena in pattern-forming non-equilibrium systems that have not yet been elucidated fully from a theoretical viewpoint. The issues to be addressed include critical phenomena near the bifurcation to electro-convection in nematic liquid crystals, wave-number selection in Rayleigh-Benard convection (RBC), the appearance of square patterns near onset in rotating RBC, and the scaling of correlation lengths near onset in systems with supercritical bifurcations to spatio-temporal chaos. 11:00-11:30 Coffee 11:30-12:30 Ecke, R (Los Alamos) Pattern formation in dense granular flow on an inclined plane Sem 1 Experimental results are presented for the periodic patterns formed in flowing granular media on a rough planar surface that was steeply inclined at 41.3$^\circ$ with respect to horizontal. The surface height profile was measured using laser deflection and the velocity field was determined simultaneously using particle image velocimetry. We demonstrate that the structure of the local flow making up the stripes has height maxima for fast flowing regions, that the amplitude of the pattern evolves over downstream length scales that are 50-100 times the lateral wave length, and that the thickness at which the flow becomes unstable to the formation of lateral stripes is quite close to the thickness at which the flow does not have an average terminal velocity. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Schatz, M (Atlanta, Georgia) Homological characterization of complex spatiotemporal patterns Sem 1 Many physical systems exhibit complex spatio-temporal behaviors that are difficult to characterize. We describe an approach that uses topological tools (specifically, computational homology) to connect experimentally observed structures to underlying dynamics. As a specific example, we will discuss homological characterizations of spiral defect chaos, a weakly turbulent state of Rayleigh-Benard convection. We observe asymmetries between hot and cold flows and show novel measures of boundary influence and indicators of system control parameters. We also find the evolution of the global structure of the flow to be primarily stochastic unlike the locally chaotic signatures reported previously. 15:00-15:30 Tea 15:30-16:30 Cross, M (Caltech) Insights from large scale numerical simulations of Rayleigh-Benard convection Sem 1 Experiments on Rayleigh-Bénard Convection have played an important role in the development of our understanding of pattern formation and spatiotemporal chaos. In this talk I will review the work of the Caltech-Argonne-Duke-Virginia Tech collaboration using large scale numerical simulations of experimentally realistic geometries to complement past and ongoing experimental work, and to gain new insights into these phenomena. Topics covered will include pattern chaos in small systems, Lyapunov exponents, mean flow, and domain chaos. 16:30-17:30 Poster session 1 17:30-18:30 Wine and Beer reception 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Tuesday 9 August 09:00-10:00 Steinberg, V (Weizmann Institute) Onset of wave drag due to capillary-gravity surface waves Sem 1 Phase diagrams of the flow states in Couette-Taylor flow of dilute polymer solutions as a function of three controlled parameters, namely the Reynolds number, the elasticity parameter, and the reduced polymer contribution to viscosity, that is proportional to the polymer concentration, are studied in detail. Flow visualization reveals that the Couette flow becomes unstable simultaneously either to one, two or three flow patterns depending on the location in the control parameter space. We investigate the character of flow motion in different patterns and the mechanism and type of transitions between different patterns using laser Doppler velocimetry. 10:00-11:00 Tuckerman, L (LIMSI-CNRS) Turbulent-laminar bands in plane Couette flow Sem 1 Recent experiments by Prigent and Dauchot have shown that the remarkable spiral turbulence state of Taylor-Couette flow also occurs in plane Couette flow. In both cases, a pattern of alternating turbulent and laminar bands appears at a well-defined Reynolds number.The pattern is tilted with respect to the streamwise (or azimuthal) direction and its wavelength is much larger than the gap.The angle and wavelength depend systematically on Reynolds number. We have numerically simulated these turbulent-laminar patterns for plane Couette flow. In our computational approach, we replace the large lateral dimensions of the experiment by a narrow periodically repeating rectangle which is tilted with respect to the streamwise direction. In this way we determine which angles and lengths support turbulent bands. 11:00-11:30 Coffee 11:30-12:30 Knobloch, E (Berkeley) Snaking and its consequences Sem 1 Recent simulations^1 of binary fluid convection in low Prandtl number liquids reveal the presence of multiple numerically stable spatially localized steady states at supercritical Rayleigh numbers. The length of these states decreases as the Rayleigh number decreases; below a critical Rayleigh number the steady states are replaced by relaxation oscillations in which the steady state is gradually eroded until no rolls are present (the slow phase), whereupon a new steady state regrows from small amplitude (the fast phase) and the process repeats. The Swift-Hohenberg equation (both variational and nonvariational) provides much insight into this behavior. This equation contains several classes of localized steady states whose length grows in a characteristic 'snaking' fashion as they approach spatially periodic states, and the associated dynamics resemble the binary fluid simulations. The origin of the snaking and the stability properties of the associated states will be elucidated, and the results used to shed light on the remarkable complexity of these simple systems. This talk is based on joint work with Oriol Batiste and John Burke. ^1 O. Batiste and E. Knobloch, Phys Fluids 17,064102 (2005). 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Purwins, H-G (Muenster) Universal behaviour of self-organized patterns in gas-discharge Sem 1 The understanding of self-organized patterns in spatially extended nonlinear dissipative systems is one of the great challenges in modern natural sciences. Such systems can be investigated in an exemplary manner using planar dc and ac gas-discharge systems. Examples for observed patterns are fronts, stripes, hexagons, targets, spirals, dissipative solitons, fractals and other complex patterns. These patterns are also found in systems that microscopically differ in a fundamental way from plasma systems. It is believed that pattern formation in the above mentioned plasma systems can be described within the scope of the drift-diffusion approximation. In addition, there is strong evidence that in many cases the corresponding equations can be transformed to a set of reaction-diffusion equations. It turn out that plasma specific equations provide insight into the underlying microscopic properties of the plasma involved, whereas reaction-diffusion models offer a convenient background for the identification of relevant pattern forming mechanisms and for the discussion of the experimentally observed universal behaviour. 15:00-15:30 Tea 15:30-16:30 Newell, A (Arizona) Plant patterns and plant phylloFaxis Sem 1 The tiling of plant surfaces into polygonal shapes (stripes, hexagons) and the arrangements of leaves and stickers (phylloFaxis) has intrigued material scientists since the time of Kepler. Drawing on recent ideas of Paul Green and colleagues and of Donady and Conder, Patrick Shipman and I have suggested (J. Theor Bio. 236, 154-197 (2005)) that most observed patterns can be understood as the energy minimizing buckling states of a compressed sheet (the plant's Funica) on an elastic foundation. 16:30-17:30 Bodenschatz, E (Gottingen/Cornell) Solitary states in forced Rayleigh-Benard convection Sem 1 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Wednesday 10 August 09:00-10:00 Firth, W (Strathclyde) Spontaneous patterns in nonlinear optics Sem 1 Spontaneous patterns in optics usually involve diffraction, rather than diffusion, as the primary spatial coupling mechanism. The simplest and most successful system involves a nonlinear medium with a single feedback mirror. The basic theory and experimental status of that system will be reviewed, along with discussion of other systems such as semiconductor micro-resonators, and the closely related topic of dissipative solitons in such systems. 10:00-11:00 Residori, S (INLN, Nice) Patterns and optical structures in liquid-crystal light-valve experiments Sem 1 I will review the basic mechanisms of optical pattern formation, with particular reference to Liquid-Crystal-Light-Valve (LCLV) experiments, where patterns and quasi-patterns have been observed for different settings in the optical feedback loop. I will show dynamical competition between different patterns (hexagons-stripes) and bistability between different states, either homogeneous or spatially periodic. In the bistable regimes, stable localized structures can be excited, that have features of addressable single elements. When bistability is between patterns, a new class of localized structures appear, that are localized peaks of one pattern nucleating over the other pattern state. 11:00-11:30 Coffee 11:30-12:30 Gollub, J (Haverford) Measuring fluid stretching to understand chemical reaction patterns and polymer mixing Sem 1 Normally when we talk about pattern formation in fluid systems, we are considering the spatial structure of the velocity or temperature or composition field. However, many physical processes, especially mixing and chemical reaction, depend on the local stretching or deformation produced by the fluid. We first show how stretching fields can be computed from high-resolution time-dependent velocity fields. Their patterns can be quite different from the velocity field patterns. Yet, the statistical properties of the stretching probability distributions for various two dimensional flows are similar. Then, we consider chemical mixing, and show that the global progress of an acid base reaction that is limited by advection and diffusion can be expressed in terms of a single function that depends only on the mean Lyapunov exponent, not on the details of the structure of the stretching field. Finally, we show how in a non-Newtonian fluid, the process of fluid stretching and the resulting mixing behavior can be substantially modified. Work supported by NSF-DMR. Related Links http://www.haverford.edu/physics-astro/Gollub/lab.html - Nonlinear Physics Lab, Haverford College 12:30-13:30 Lunch at Wolfson Court 19:30-18:00 Conference Dinner at Emmanuel College (Old Library)