THE RAYLEIGH-BENARD INSTABILITY

        Typically, Rayleigh-Benard instabilities occur when you consider a viscous fluid (conducting heat) placed between two plates at different temperatures.
If the bottom plate is cooler than the upper one (T2<T1), the system is stable as buoyancy forces and gravitational forces are acting in the same direction.
When the bottom plate is hotter than the upper plate, there can be an apparition of Rayleigh-Benard instabilities, partly depending on the temperature difference between the two plates and the distance between the two plates. Indeed, with a sufficient temperature difference, the particles at the bottom gets hot enough to rise up (buoyancy forces are strong enough to push up the particles), whereas the coldest ones tend to fall with the effects of the gravity, then heat at the contact of the hot layer and so rise up again leading to the formation of non stable convective rolls, as can be seen in the following visualization.

     







To study this kind of phenomenon, it is interesting to consider the Rayleigh number which is an indicator of the vigor of the convection. It is defined as follows:

In fact, even if the hottest plate is under the coldest one, the system can be stable. In fact, instabilities occur from a certain temperatues difference, ie a certain Rayleigh number. In fact, a particule is under the influence of two opposite forces: buoyancy forces (destabiliting forces) and draft forves (stabilisating forces). When the buoyancy forces get high enough compared to the draft forces, the system is not stable anymore. This theorically occurs after a critical Rayleigh number of 1708.