We consider a viscous fluid, conduct heat, flow between two parallels plaques, distant of a length d and at different temperatures (see figure). , we try to determine the thermo-dynamic coupling : knowing the heat flux exchanged we can determine the dynamic flow regime.
II. Theoretical outline
Boussinesq approximation : To simplify N&S equations we make the following hypothesis :
r and T are constants except in the buoyancy term :
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:
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.