Rayleigh–Bénard convection

Bénard–Rayleigh convection is a type of natural convection, occurring in a plane horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Bénard–Rayleigh convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of self-organizing nonlinear systems. Bénard–Rayleigh convection is a type of natural convection, occurring in a plane horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Bénard–Rayleigh convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of self-organizing nonlinear systems. Buoyancy, and hence gravity, are responsible for the appearance of convection cells. The initial movement is the upwelling of lesser density fluid from the heated bottom layer. This upwelling spontaneously organizes into a regular pattern of cells. The features of Bénard convection can be obtained by a simple experiment first conducted by Henri Bénard, a French physicist, in 1900. The experimental set-up uses a layer of liquid, e.g. water, between two parallel planes. The height of the layer is small compared to the horizontal dimension. 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 its surroundings. (Once there, the liquid is perfectly uniform: to an observer it would appear the same from any position. 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 yielding a flow of thermal energy conducted 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. A uniform linear gradient of temperature will be established. (This system may be modelled by statistical mechanics). Once conduction is established, the microscopic random movement spontaneously becomes ordered on a macroscopic level, forming Benard convection cells, with a characteristic correlation length. The rotation of the cells is stable and will alternate from clock-wise to counter-clockwise horizontally; this is an example of 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 the experiment is repeated, 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 non-deterministic macroscopic effect. That is, in principle, there is no way to calculate the macroscopic effect of a microscopic perturbation. This inability to predict long-range conditions and sensitivity to initial-conditions are characteristics of chaotic or complex systems (i.e., the butterfly effect). 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.