1. Theorical study:
1.1 Presentation of the problem
 

We consider a viscous fluid placed between two plates at different temperatures.
 


 
 

If T1>T2:  Case of stability
The system is stable as buoyancy forces and gravitationnal forces are acting in the same direction.

If T1<T2:
There can be apparition of instability, partly depending on the temperature difference between the two plates.
When the temperature difference is great enough, fluid particles at the bottom get 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 lading to the formulation of non stable convecting rolls.
 
 


 

1.2 Criterion of instability: The Rayleigh Number

When the temperature difference between the two plates is great enough, the stability effects of viscosity and thermal conductivity are overcome by the destabilizing buoyancy, and an overturning instability ensues as thermal convection.
It is interesting to consider the following Rayleigh number which is an indicator of the vigor of the convection:


 
 
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