Benard cells

At low Rayleigh numbers, Wragg (W6) found a smaller Ra dependence, resembling more the dependence in laminar free convection. In this range of Ra numbers, a cellular flow pattern is believed to exist, analogous to that of thermal and surface tension-driven cellular convection (Benard cells F3). In the range where the convection is turbulent, the Ra1/3 dependence has been confirmed over seven powers of Ra by Ravoo (R9), who used a centrifuge to vary the body force at constant bulk composition. 

A pot of water, heated gradually on a stove, is seen to develop highly structured convection patterns ( Benard cells ), resembling a checkerboard of ascending and descending columns. Clearly, such order out of chaos represents a process with A S < 0, proving that the second law is invalid. 

Remark. Instability and bistability are defined as properties of the macroscopic equation. The effect of the fluctuations is merely to make the system decide to go to one or the other macroscopically stable point. Similarly the Taylor instability and the Benard cells are consequences of the macroscopic hydrodynamic equations. ) Fluctuations merely make the choice between different, equally possible macrostates, and, in these examples, determine the location of the vortices or of the cells in space. (In practice they are often overruled by extraneous influences, such as the presence of a boundary.) Statements that fluctuations shift or destroy the bistability are obscure, because on the mesoscopic level there is no sharp separation between stable and unstable systems. Some authors call a mesostate (i.e., a probability distribution P) bistable when P has two maxima, however flat. This does not correspond to any observable fact, however, unless the maxima are well-separated peaks, which can each be related to separate macrostates, as in (1.1). 

Remark. A great deal of attention has been paid in recent years to non-equilibrium stationary processes that are unstable and also extended in space. They can give rise to different phases that exist side by side, so that translation symmetry is broken. The name dissipative structures has been coined for them, and the prime examples are the Benard cells and the Zhabotinski reactions, but they also occur in biology and meteorology. However, these are features of the macroscopic equations. They are only relevant for fluctuation theory inasmuch as the fluctuation becomes very large at the point where the instability sets in. The critical fluctuations in XIII.5 are an example. There are many more varieties, in particular in the case of more variables. 

When the Rayleigh number exceeds the critical value, fluid motion develops. Initially, this consists of a series of parallel two-dimensional vortices as indicated in Fig. 8.35a. However at higher Rayleigh numbers a three-dimensional cellular flow of the type indicated in Fig. 8.35b develops. These three-dimensional cells have a hexagonal shape as indicated in the figure. This type of flow is termed Benard cells or Benard convection. 

Heat transfer in horizontal enclosed spaces involves two distinct situations. If the upper plate is maintained at a higher temperature than the lower plate, the lower-density fluid is above the higher-density fluid and no convection currents will be experienced. In this case the heat transfer across the space will be by conduction alone and Nus = 1.0, where 8 is still the separation distance between the plates. The second, and more interesting, case is experienced when the lower plate has a higher temperature than the upper plate. For values of Grs below about 1700, pure conduction is still observed and Nu = 1.0. As convection begins, a pattern of hexagonal cells is formed as shown in Fig. 7-12. These patterns are called Benard cells [33]. Turbulence begins at about Gr6 = 50,000 and destroys the cellular pattern. 

Flfl. 7- 12 Benard-cell pattern in enclosed fluid layer heated from below, from Ref. 33... 

Systems that exchange entropy with their surroundings may undergo spontaneous transformation to dissipative structures and self-organization. The forces that exist in irreversible processes create these organized states, which range from convection patterns of Benard cells to biological cycles. 

One of the best-known physical ordering phenomena is the Benard cells, which occur during the heating a fluid held between two parallel horizontal plates separated by a small distance. The lower plate is heated, and the temperature is controlled. The upper plate is kept at a constant temperature. When the temperature difference between the two plates reaches a certain critical value, the elevating effect of expansion predominates, and the fluid starts to move in a structured way the fluid is divided into horizontal cylindrical convection cells, in which the fluid rotates in a vertical plane. At the lower hot plate, the hot fluid rises later, it is cooled at the upper plate, and its density increases again this induces a movement downward, as seen in Figure 13.2. The Benard cells are one of the best-known physical examples of spontaneous structurization as a result of sufficient distance from equilibrium, which is the large temperature difference between the plates. The critical temperature difference ( A 7 )c can be determined from the... 

According to the hydrodynamics analysis, the approximate velocity distribution in the Benards cells is given by... 

Figure 3.9 The Benard cells in a flat vessel of liquid (A) a schematic of the cell formation due to the self-organized convection of a heated liquid, and (B) the top view of the cells. The convective vertical downstream is centered in the hexagons.

Dissipative structures arise only in strongly nonequilibrium systems, with the states described by nonhnear equations for internal macro parameters. The emergence of the Benard cells in fluids can be described using non hnear differential equations of hydrodynamics coupled with Lyapunov s analysis of the instability of the respective solutions. It is shown that the solution of hydrodynamic equations related to a resting fluid and normal heat transfer becomes unstable at AT > AT, and a new stable convection mode is established in the fluid. 

Daniels et al. also demonstrated that snrface tension plays a controlling role in the formation of striations. They added nnspecified surface leveling agents into their solutions, which reduced the magnitude of the striations. That the cellular pattern found at the wafer center is so similar in appearance to Benard cells is quite interesting as well, since surface tension was ultimately demonstrated to be the cause of the patterns that he was observing a century ago."- ... 

Block, M.J., Surface tension as the cause of Benard cells and surface deformation in a liquid film, Nature, 178, 650, 1956. 

Capillarity phenomena are everyday occurrences that result from the existence of surface tension or interfacial tensions. In addition to the static phenomena discussed herein, surface tension and capillarity are also responsible for numerous dynamic phenomena that may result from localized gradients in temperatures or in compositions the study of dynamic capillary phenomena (e.g., Marangoni flows, Benard cells) is the subject of much literature coverage and is beyond the scope of this survey. 

A well known example of dissipative structures are the Benard cells, also called convection cells. A flat tank is filled with water. The upper and lower surfaces are kept at different temperatures. When the lower surface is slightly warmer than the upper one, the system is in the linear region and one observes a steady and uniform upward flow of heat. 

We have seen in the general introduction that self ordering can occur in two very different situations at equilibrium, when a system goes through a phase transition (example of ferromagnetism) or far away from equilibrium when dissipative structures appear in the system (example of Benard cells). 

When the solvent from paint films evaporates, there may develop a circulatory motion within cells (Benard cells) in the body of the liquid film. This vortex action was described by Bartel 1 and Van Loo in 1925 and has been extensively reviewed and interpreted by Hansen and Pierce (33. 34). Each cell has an hexagonal shape, and the overall pattern of the dried film consists of raised ridges at the outer edges of closely packed hexagons. In each cell liquid moves upward at the center of the cell, spreads outward at the surface to the edges of the hexagon, and then moves downward to the depths of the film and completes the circular motion. As the liquid moves across the surface it cools and becomes more concentrated as solvent evaporates. As a result of both of these factors the surface tension increases from center to edge of each cell. Liquid... 

The vortex action in Benard cells can cause the defects knovm as flooding and floating because pigments of different sizes and weights will move at different velocities, will separate, and show nonuniformity of colors in films. The pigments also can be concentrated at certain local spots and thus leave pure binder at other spots that are vulnerable to film failure upon exposure. 

Although the formation of surface patterns via the Benard cell may occasionally be useful to form special finishes, usually it is desirable to eliminate the situation. Use of higher boiling solvents will retard evaporation rate and the cooling effect that changes surface tension and propels the vortex action. Increase in paint viscosity will inhibit the action, as will decrease in film thickness. Addition of a surfactant will provide a more uniform value of surface tension and also retard evaporation and thereby help to inhibit the vortex motion in cells. 

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