We consider a viscous fluid, conduct heat, flow between two parallels plaques, distant of a length d and at different temperatures (see figure). Initially the fluid is immobile or in motion, we try to determine the thermo-dynamic coupling : knowing the heat flux exchanged we can determine the dynamic flow regime.

Boussinesq approximation : To simplify N&S equations we make the following hypothesis :

r and T are constants except in the buoyancy term :

Rho = Rho_{0(1-alpha(T-T0))}

_{With this hypothesis the solution of N&S equations
can be the following one : if the fluid is immobile initially it keep this
situation, and if it was in motion it stand on moving. In both cases there
is stratification of the temperature from T1 to T2 (there a constant temperature
gradient). This is a state of equilibrium. The problem is that we don't
know if it's stable or not.}

_{We will study the stability by introducing a small
perturbation to the equilibrium state, and we will check if it will be
amplified or not. }

_{The theoretical study of stability is done by the
Fourier analyse. Effectively, and because of the fact that each perturbation
can be represented by analytical function, witch can be decomposed in sum
of sinusoidal terms. This makes us considering mode proper mode notion,
witch will represent elementary perturbations. A proper mode have the following
shape: }

_{V0*sin(k*pi*x/L)*sin(2*pi*y/d)}

_{Here k is the number of rolls introduced by the perturbation.
pi*k/L is the wave number. A graph drawn in the ( k , Ra) plan , represents
the frontier between the stable zone and the unstable one. It enable us
to know the modes gam witch are susceptible to be amplified at a determined
value of Ra.}

_{A mathematical approach of the stability problem
is handled in the hydrodynamics instabilities theory. A theoretical values
of RaC and kC are computed. }

_{RaC = 27}p ^{4}/4
et k_{C }= .