Benard Cells

The second example is the formation of Benard cells. Consider a thin layer of liquid between two large parallel plates as shown in the figure. If the system is in equilibrium, with the liquid and the two plates at the same temperature and the liquid motionless, then the properties of the system are homogeneous. Suppose now the bottom plate is heated slowly. The heat will pass from the bottom plate to the liquid and will be transferred through the liquid to its upper layer by the process of thermal conduction. In thermal conduction there is no bulk motion of the liquid but rather a greater thermal motion of the molecules that causes the transfer of heat from the warmer layers to adjacent cooler layers. However as the temperature of the bottom layer is increased, a stage is reached (critical temperature) where the liquid overcomes its viscosity (the internal friction which opposes movement) and begins to undergo bulk motion. This results in a transport of heat by convection currents. It is interesting to note that the currents are not random but rather they lead to the formation of patterns. Often one first sees small convection cells (Benard cells) as shown in the figure. Thus this is another example of dynamic pattern formation. Amusingly in this example the driving force is temperature: that is, temperature differences, or the transport of heat by a medium, can cause non-equilibrium structure formation on macroscopic scales (the size of Bernard cells is about 1mm, which is large compared to the scale of intermolecular forces, $10^{-7}$ mm). When viewed from above the cells formed striped patterns, thus breaking the symmetry of the uniform state. As the temperature is raised, rolls can also appear in a perpendicular direction so that viewed from above one has square patterns.