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 structures.

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.

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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.

A small perturbation will not be able to change the rotation of the cells, but a larger one could very well do it: the cells exhibit hysteresis, i.e. they have a memory of their history.

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 is enough to produce a macroscopic effect: this is the Butterfly effect.

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

If we further increase the temperature of the bottom plane, the structure becomes more complex in space and time: the turbulent flow becomes 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.

Web-Links

• Juanita Lofthouse - a fascinating series of papers which present a qualitative argument suggesting Benard-Marangoni convection in the viscous fluids of biological cells forms patterned templates for the assembly of cytoskeletal proteins, playing a crucial role in Morphogenesis[ &

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• A. Getling, O. Brausch: Cellular flow patterns
• J. Rogers, M. Schatz, O. Brausch, W. Pesch: Oscillated Rayleigh-Bénard Convection
• K. Daniels, E. Bodenschatz, B. Plapp, W.Pesch, O. Brausch, R.Wiener: Localization and Bursting in Inclined Layer Convection
• K. Daniels, B. Plapp, W.Pesch, O. Brausch, E. Bodenschatz: Undulation Chaos in inclined Layer Convection