Theorical study

1) Physical problem

1) Physical problem

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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2) Condition of motion

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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3) Mecanisms and criteria of instability

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) The equations of motion

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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