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:

where and

and the boundary conditions are:

• , on the two walls (z=0 and z=d):

• u^*= 0 (rigid) or u^*.n= 0; ;

(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.