Benard convection cells

Benard convection cells [27, 28] a liquid with an inverse temperature gradient (hot below and cool on top) may exhibit thermal convection. Less dense parts of the liquid well upward whereas denser parts show down-welling. The convection cells may arrange in hexagonal order in which the center of each cell wells downwards and the rim wells upwards. The cells stem from the concerted movement of many molecules and cease when the temperature gradient is below a threshold at which the thermal equilibrium canbe reached solely bythermalconductionandnotconvection. 

The formation of Benard convection cells takes place as follows if water is heated from below in a vessel, macroscopic convection currents occur under certain conditions seen from above, these have the structure of uniform, honeycombshaped cells. 

BENARD CONVECTION CELLS. When a layer of liquid is heated from below, the onset of convection is marked by the appearance of a regular array of hexagonal cells, the liquid rising in the center and falling near the wall of each cell. The criterion for the appearance of the cells is that the Rayleigh number should exceed 1700 (for rigid boundaries). 

Example 12.3 Stability under both dissipative and convective effects In some cases, both dissipative as well as convective effects determine the stability of a system. Some examples of such stability are the onset of free convection in a layer of fluid at rest, leading to Benard convection cells, and the transition from laminar to turbulent flow. For stability considerations, two limiting cases exist (i) in the case of ideal fluids, dissipative processes are neglected, and (ii) in purely dissipative systems, no convection effects occur. 

B.I. Shraiman. Diffusive transport in a Rayleigh-Benard convection cell. Phys. Rev. A, 36 261-267, 1987. 

The temperature gradient within the droplet perpendicular to the substrate can induce Rayleigh-Benard convection cells. The corresponding convective cells grow stronger... 

Assume that the distance of a system from global equilibrium is jS, (for example, a temperature or a concentration gradient). After a critical value of iSi is reached, the system displays ordering characterized by a certain frequency or a wavelength. Figure 13.1 shows the bifurcation in the velocity in Benard convection cells. If is increased further, we may reach a value of IS2 at which the system is characterized by two frequencies. This behavior can be repeated for the values of jds and 4, leading to increasingly complicated states. 

Methods from heat transfer. The modeling of Benard convection cells in hydrodynamic stability is relevant to our purposes, since it leads to a modal equation identical to Equation 12-9 (Yih, 1969). It turns out interestingly that Equation 12-9 itself can be solved by separation of variables once more. For example, the choice... 

There is another type of bifurcation called Turing bifurcation, which results in a spatial pattern rather than oscillation. A typical example where a new spatial structure emerges from a spatially unique situation is Benard s convection cells. These have been well examined and are formed with increasing heat conduction.53 Prigogine called this type of structure a dissipative structure.54-56... 

Figure 11.2 (a) Microscope image of Benard from Ref 33). (b) Two microscope snapshots convection cells (indicated by the circle in the of an evaporating polystyrene solution on silicon upper left corner) and tears ofwine (indicated wafer. The time between the two frames is by the white arrows) in an evaporating approximately 100 ms. The polymer droplets... 

Figure 2.9.9(a) shows a schematic representation of a thermal convection cell in Rayleigh-Benard configuration [8]. With a downward temperature gradient one expects convection rolls that are more or less distorted by the tortuosity of the fluid filled pore space. In the absence of any flow obstacles one expects symmetrical convection rolls, such as illustrated by the numerical simulation in Figure 2.9.9(b). 

Fig. 2.9.9 (a) Schematic cross section of a compartments at the top and bottom, respec-convection cell in Rayleigh-Benard configura- tively. (b) Velocity contour plot of typical tion. In the version examined in Refs. [8, 44], a convection rolls expected in the absence of any fluid filled porous model object of section flow obstacles (numerical simulation). 

Fig. 2.9.10 Maps of the temperature and of the experimental data. The right-hand column convection flow velocity in a convection cell in refers to numerical simulations and is marked Rayleigh-Benard configuration (compare with with an index 2. The plots in the first row, (al) Figure 2.9.9). The medium consisted of a and (a2), are temperature maps. All other random-site percolation object of porosity maps refer to flow velocities induced by p = 0.7 filled with ethylene glycol (temperature thermal convection velocity components vx maps) or silicon oil (velocity maps). The left- (bl) and (b2) and vy (cl) and (c2), and the hand column marked with an index 1 represents velocity magnitude (dl) and (d2).

Kauffman also considers that autocatalytic reactions are a necessary precondition for biogenesis processes, as they can wind themselves up via self-amplifying feedback processes until a critical boundary has been reached. The next step would then be the transition from autocatalysis to self-organisation, similar to the transition from unstructured water to convection cells (the Benard instability) (Davies, 2000). 

We can illustrate the salient features of convective dispersal by choosing a simple velocity distribution in a rectangular convection cell (0associated with the onset of Benard instability in the conditions of Boussinesq approximations (e.g., Turcotte and Schubert, 1982). Let us make the calculation for the so-called free-slip conditions, which permit free movement along the boundaries, both vertical and horizontal, such as a convection cell which would be limited by no rigid boundary. From Turcotte and Schubert (1982), we take the velocity field to be... 

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. 

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... 

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. 

Figure 10.6.1 Plan photograph taken by Henri Benard of hexagonal cells in a thin film of molten spermaceti from his original experiments on convection cells induced by surface tension gradients. [Courtesy of Prof. Simon Ostrach. From Benard 1900.1...

Fig. 7. Contour lines of the type obtained by Benard for the free surface of a hexagonal convection cell using the Fabry-P6rot interferometer.

Benard employed two variations of the direct-shadow method, using reflected light to obtain photographs of free-surface relief during convection. In particular because the free surfaces of his convection cells were essentially basins separated by ridges, from the optical standpoint they simulated concave mirrors separated by cylindrical convex mirrors. By adjusting the position of the viewing screen, as shown in Fig. 12, the cell centers could be... 

Convection cells can arise at a lower Marangoni number when the coating thickness is above 2 mm. Convection or Benard cells can be reduced or eliminated by adjusting operating conditions and formulation of coating solution. The possibility of having convection cells is reduced at the following conditions ... 

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