Bénard cells

Bénard cells are convection cells that appear spontaneously in a liquid layer when heat is applied from below. They can be obtained using a simple experiment first conducted by Henri Bénard, a French physicist, in 1900. The experiment illustrates the theory of dissipative systems.

The experimental set-up uses a layer of liquid, e.g. water, between 2 parallel planes. The height of the layer is small compared to the horizontal dimension.

Equilibrium and thermal conduction

At first, the temperature of the bottom plane is the same as the top plane. The liquid will then tend towards an equilibrium, where its temperature is the same as the one outside. Once there, the liquid is perfectly uniform: an observer in it would see the same environment in any spot, and in any direction. This equilibrium is also asymptotically stable: after a local, temporary perturbation of the outside temperature, it will go back to its uniform state, in line with the second law of thermodynamics.

Then, the temperature of the bottom plane is increased slightly: a permanent flow of energy will occur through the liquid. The system will begin to have a structure of thermal conductivity: the temperature, and the density and pressure with it, will vary linearly between the bottom and top plane. This system is modelled very well in Statistical mechanics.

Far from equilibrium: convection and turbulence

If we progressively increase the temperature of the bottom plane, there will be a temperature at which something dramatic happens in the liquid: convection cells will appear. The microscopic random movement spontaneously became ordered on a macroscopic level, with a characteristic correlation length. The rotation of the cells is stable and will alternate from clock-wise to counter-clockwise as we move along horizontally: there is a spontaneous symmetry breaking.

Bénard cells are metastable. This means that a small perturbation will not be able to change the rotation of the cells, but a larger one could affect the rotation; they exhibit a form of hysteresis.

Moreover, the deterministic law at the microscopic level produces a non-deterministic arrangement of the cells: if you reproduce the experiment many times, a particular position in the experiment will be in a clockwise cell in some cases, and a counter-clockwise cell in others. Microscopic perturbations of the initial conditions are enough to produce a macroscopic effect: this is an example of the Butterfly effect from Chaos theory.

The temperature at which convection appears is thus a bifurcation point, hence the system can be analyzed via bifurcation diagrams. The bifurcation temperature depends on the viscosity and thermal conductivity of the liquid, and on the physical dimensions of the experiment.

If the temperature of the bottom plane was to be further increased, the structure would become more complex in space and time; the turbulent flow would become chaotic.

Rayleigh-Bénard and Bénard-Marangoni convection

In the case of two plates between which a thin liquid layer is placed, only buoyancy is responsible for the appearance of convection cells. This type of convection is called Rayleigh-Bénard convection. The initial movement is the upwelling of warmer liquid from the heated bottom layer.

In case of a free liquid surface in contact with air also surface tension effects will play a role, besides buoyancy. It is known that liquids flow from places of lower surface tension to places of higher surface tension. This is called the Marangoni effect. When applying heat from below, the temperature at the top layer will show temperature fluctuations. With increasing temperature, surface tension decreases. Thus a lateral flow of liquid at the surface will take place, from warmer areas to cooler areas. In order to preserve a horizontal (or nearly horizontal) liquid surface, liquid from the cooler places on the surface have to go down into the liquid. Thus the driving force of the convection cells is the downwelling of liquid.

A very detailed mathematical treatment of this from the point of view of the Navier-Stokes equations and the equations of Thermodynamics can be found in the first third of Subrahmanyan Chandrasekhar's [http://www.amazon.ca/Hydrodynamic-Hydromagnetic-Stability-S-Chandrasekhar/dp/048664071X Hydrodynamic and Hydromagnetic Stability] , originally published in the early 1960s by Oxford, and republished by Dover in 1981.

Shape

Convective Bénard cells tend to approximate regular right hexagonal prisms, particularly in the absence of turbulence [http://www.etl.noaa.gov/about/eo/science/convection/RBCells.html] [http://www.edata-center.com/proceedings/1bb331655c289a0a,088ce8ea747789cd,59d115f133a4fd07.html] [http://cat.inist.fr/?aModele=afficheN&cpsidt=17287579] , although certain experimental conditions can result in the formation of regular right square prisms [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=13973] or spirals [http://www.psc.edu/science/Gunton/gunton.html] .

References

External links

* [http://www.magnetosphere.ru/~avg/publications/PRE_046313_v67_2003.pdf A. Getling, O. Brausch: Cellular flow patterns]
* [http://www.physics.gatech.edu/schatz/pubs/PRL04281.pdf J. Rogers, M. Schatz, O. Brausch, W. Pesch: Oscillated Rayleigh-Bénard Convection]
* [http://131.111.145.118/webseminars/pg+ws/2005/pfd/0818/daniels/all.pdf K. Daniels, E. Bodenschatz, B. Plapp, W.Pesch, O. Brausch, R.Wiener: Localization and Bursting in Inclined Layer Convection]
* [http://arxiv.org/abs/nlin/0702006v1 K. Daniels, B. Plapp, W.Pesch, O. Brausch, E. Bodenschatz: Undulation Chaos in inclined Layer Convection]