__Theorical study__

__3)
Mecanisms and criteria of instability__

We consider a
layer of a heat conducting, viscous fluid, contained between two plane
plaques, distant of a length d and

at different temperatures (see
figure). Initially the fluid is immobile or in motion

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If TUp>TDown , then there is no motion and the temperature distribution is linear.

If TUp<TDown,
then 2D stationary contra-rotative rolls appear if the difference of temperature
is over a critical value

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There are three different forces which act on particules :

Archimède force which is an instability source :

Drag force which is a stability force

Thermical diffusion which plays the same role as drag force

The balance
between this three forces introduces the stability condition on the Rayleigh
number

This competition of forces is parameterized by the Rayleigh number, which is the temperature difference, but appropriately

normalized to take into account the geometry of the convection cell and the physical properties of the fluid.

If the Rayleigh
number is greater than 1708, then convection occurs. If it is below this
value called the critical Rayleigh

number Rac, there is no convective
flow.

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4-1)The exact equations

The exact
equations of motion of a heat conducting viscous fluid under the action
of gravity are

in addition to the state equqtions.

4.1) The Boussinesq approximation

The basis
of this approximation is that there are flows in which the temperature
varies little, and therefore the density varies

little, yet in which the buoyancy
drives the motion. Then the variation of density is neglected everywhere
except in the buoyancy.

On the basis of this approximation
for small temperature difference between the bottom and the top of the
layer of fluid .

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