This paragraph aims at present a short linear study of the Rayleigh-Bernard instabilities.
The equations of this problem is the Navier-Stockes equations coupled with energy equations (and we use the Boussinesq approximation). The dimensionless equations are:
and the boundary conditions are:
(free), on the inlet and the outlet (rigid or free).
After an (long) analyse, we found the stability:
for the rigid conditions
and R=1707 for the free conditions.