Description of the Book's Front Cover Figure:
Disordered Traveling Waves in a Binary Fluid Convection Experiment


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This picture is a computer-enhanced image of experimental data taken from a binary-fluid Rayleigh-Benard convection experiment carried out at the University of California San Diego by Arthur La Porta and Cliff Surko. You are looking down into a wide shallow horizontal cylindrical cell (so gravity points into the screen) such that the diameter of the cylinder is 52 times its height. The cylinder is filled completely with a mixture of ethanol in water, so the cylinder height is also the fluid depth. The bottom and top surfaces of this cylinder were kept warm and cold respectively, which caused the ethanol-water mixture to start convecting, with warmer fluid rising and cooler fluid descending. After a transient time, a time-dependent pattern spontaneously formed as shown, consisting of stripe-like convection rolls. A novelty of binary fluid convection (as opposed to convection of a pure fluid like water or nitrogen gas) is that, when convection first occurs, there is an intrinsic oscillation and this results in the formation of traveling waves that propagate horizontally (in the plane of this image).

The La Porta-Surko experiment was important because it was one of the first to investigate what kind of dynamics arise from an oscillatory instability at a finite wavelength of a uniform state (a so-called type-Io instability, see Chapter 2 of the book) when an experimental cell has two large extended directions. La Porta and Surko discovered that traveling waves still occur (as they do in a system with one large extended direction) but now multiple traveling waves coexist and these can have different orientations, frequencies, and speeds. Here the color encodes the local frequencies of the traveling waves (roughly the same as the speeds of the wave), with blue and yellow areas oscillating respectively faster and slower than the average frequency. The range of frequencies ω varies from 0.25 to 0.6 radians per vertical diffusion time.

There are numerous details of this experiment which remain poorly understood and so remain challenges to understand. Analysis of the time-dependent pattern suggests that the dynamics is not periodic in time nor space and so empirically is an example of spatiotemporal chaos. As is the case with many spatiotemporal chaotic states, it is not clear why the pattern fails to become more regular over time, say ordered in space or periodic in time. It is currently not understood what determines the number of traveling waves (for example, does the number increase with increasing aspect ratio of the cell or with increasing Rayleigh number?) nor their local frequencies, nor how these details depend on experimental parameters or on the geometry of the cell, say a cylinder versus a square box of comparable aspect ratio.

Further details of the image and experiment: The colored stripes correspond to regions of cooler descending fluid, which act as weak converging lenses when light is directed through a transparent upper surface of the cell (the experiment used a thin sapphire plate) and reflected off the mirror-like bottom surface of the cell (which was a highly polished plate of pure silicon). (See Figure 1.13 in the book which shows a schematic diagram of the shadowgraphy method.) The black stripes correspond to regions of warmer ascending fluid, that act as weak diverging lenses that deflect light reflected from the bottom plate away from the recording CCD camera above the cylinder. The light intensity at any given point represents a rather complicated average over the depth of the fluid of the temperature-dependent and so spatially varying index of refraction of the fluid. From the time-dependent movie of the patterns, La Porta and Surko used the technique of complex demodulation (see the paper below) to extract a local complex-valued amplitude field A(x,y,t)=|A|ei φ whose phase φ(x,y,t) was used to compute and assign a frequency to the local oscillations of each roll. A computer program then assigned colors to each bright stripe, depending on its frequency relative to the mean frequency over the pattern.

We note that numerous parameters need to be specified so that the experiment that produced this figure can be replicated, or for a computer simulation to be carried out of this experiment. La Porta and Surko used the following values:

  1. Aspect ratio Γ = "cylinder radius"/"cylinder height"= 26.
  2. A binary fluid of 8% by weight ethanol in water.
  3. Mean fluid temperature ⟨T⟩ = 26o C.
  4. Rayleigh number   R = 1.28 Rc.
  5. Prandl number (ratio of kinematic viscosity ν to thermal diffusivity κ)   σ = 9.15.
  6. Lewis number (ratio of diffusivity D of alcohol in water to thermal diffusivity κ)   L = 0.01.
  7. Separation ratio ψ = -0.24.
  8. Non-forcing lateral walls made of plastic. A non-forcing lateral wall was achieved by inserting a small thin fin into the fluid from the lateral external wall at half the fluid depth. This made the lateral walls have the same thermal diffusivity as the fluid itself. The upper horizontal surface was made of sapphire, a transparent material with a high thermal conductivity (which leads to an isothermal surface). The lower horizontal surface was made of a plate of pure silicon and polished to reflect light transmitted through the upper surface.
  9. The pattern was created by initiating convection at a larger Rayleigh number R = 1.9 Rc, followed by decreasing this parameter to the final constant value of 1.28 Rc.

Some related images with comments are available on the Surko website http://physics.ucsd.edu/research/surkogroup/fluids. Further details such as how the local frequency was extracted from the recorded spatiotemporal data can be obtained from the the paper Dynamics of two-dimensional traveling-wave convection patterns by A. La Porta and C. M. Surko, Physical Review E 53(6):5916-5934 (1996).


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