Benard convection

The same / lias also boon used as a model for spatiotemporal interniittency in Rayleigh-Benard convection ([cili88], [davl89]). 

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. 

W. J. Goux, L. A. Verkruyse, S. J. Salter 1990, (The impact of Rayleigh-Benard convection on NMR pulsed-field-gra-dient diffusion measurements), J. Mag. Reson. 88, 609. 

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. 

This principle is very general, relating neither to the linearity nor to the symmetry of the transport laws. On the other hand, it is difficult to attribute a physical meaning to dxP- The authors later attempted to derive a local potential from this property, and they applied this concept to the study of the chemical and hydrodynamical stability (e.g., the Benard convection). The results of this approach were published in Glansdorff and Prigogine s book Thermodynamic Theory of Structure, Stability and Fluctuations (LS.IO, 10a), published in 1971. 

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). 

The density at the surface increases during evaporation thus giving rise to the onset of Benard convection. 

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. 

V. Croquette, P. Le Gal, A. Pocheau, and R. Guglielmetti, Large-scale characterization in a Rayleigh-Benard convective pattern, Europhys. Lett., 1, 393-399 (1986). 

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. 

In this framework an interesting example is the Lagrangian motion in velocity field given by a simple model for Rayleigh-Benard convection [31], which is given by the stream function ... 

Plapp B. R, Egolf D. A., Bodenschatz E. and Pesch W., Dynamics and Selection of Giant Spirals in Rayleigh-Benard Convection, Phys. Rev. Lett. 81, 5334 (1998). 

Weidman, P.D., Linde, H., and Velarde, M.G., Evidence for solitary wave behavior in Marangoni-Benard convection, Phys. Fluids A, 4, 921-926, 1992. 

The first problem considered is the classic problem of Rayleigh-Benard convection -namely the instability that is due to buoyancy forces in a quiescent fluid layer that is heated... 

Problem 12-13. Raleigh-Benard Convection - Nonisothermal Boundaries. The assumption of isothermal boundaries in the buoyancy-driven convection instability problem is generally an oversimplification. A more realistic picture is that the upper and lower surfaces are in contact with reservoirs that are maintained at a constant temperature so that the thermal boundary conditions are better approximated as... 

Problem 12-14. Rayleigh-Benard Convection - One Free and One Rigid Boundary. We... 

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

T.H. Solomon and J.P. Gollub. Passive transport in steady Rayleigh-Benard convection. Phys. Fluids, 31 1372-1379, 1988. 

H. T. Rossby, A Study of Benard Convection With and Without Rotation, J. Fluid Mech. (36/2) 309-335,1969. 

The most well-known example of pattern formation is Rayleigh-Benard convection which appears when a fluid layer is uniformly heated from below... 

If condition (29) is violated, more complicated patterns may appear. As an example, let us consider the Gertsberg-Sivashinsky equation that was derived in the problem of Rayleigh-Benard convection in a layer between weakly conducting boundaries [41],... 

We come to the conclusion that only the roll patterns inside the stability interval 0 < K < 1/ /3 are stable. The stability interval is also called the Busse balloon, for it was first discovered by Busse et al. in the context of the Rayleigh-Benard convection patterns [40]. See the diagram in Fig. 12. 

Spiral-defect chaos in Rayleigh-Benard convection. The most remarkable phenomenon that needs an extension of the Swift-Hohenberg model for its explanation, is the development of spiral-defect chaos in Rayleigh-Benard convection [56], [6], which involves rotating spirals, target patterns, dislocations etc. The origin of this complicated behavior is the creation of a two-dimensional mean flow... 

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