**Theory
of the Kelvin-Helmholtz instability**

__Modelization of the flow__

We can modelize the Kelvin-Helmholtz instability by the flow of two fluids with different densities and axial velocities in a rectangular channel. We replace this flow by the superposition of two flows: a global translation with the velocity U and a symetrical flow with a velocity of .

This modelization corresponds to a spatial study of stability.

__Boundary conditions at the interface__

On the interface, the two normal velocities must be equal to the velocity of the interface: (1)

There is also a condition with the pressures: (2)

__Condition of stability__

After some transformations and the writing of the compatibility relation between the equations, we obtained the following stability conditions:

This condition can be written in another way:

(3)

__Supplementary conditions__

The Reynolds number of the flow must be superior than 1: Re>>1

The first instability which develops (the studied case here) is bidimensionnal.

Another sufficient condition for instability is to put the heaviest fluid in the top layer. As it is not very physical and to avoid simulations problems, we do not use this condition in our calculations.

__Minima__

The condition (3) permits to determine the minimum condition for the appearance of the instability: the minimum wave propagation velocity is and the corresponding wave number is .

__Stability curve__

With the condition of stability and the chosen fluids for our simulations (see Parameters), we can draw the theoretical curve of stability versus the wave number (and thus versus the wave length). We obtain the following curve:

This curve corresponds to the awaited results for a spatial study.

__Stability conditions for a test case__

For the chosen fluids, we calculate the stability conditions for a chosen wave length of 0.4 m.

With the formula (3), we obtain that U must be higher than 0.226 m/s to observe the instability.

__Other way to obtain the instability curve__

Another way to determine the instability curve is to place a beater at the inlet of the channel with the two fluids. The frequency and the amplitude of the beater can be fixed to desired values. Then, for a fixed frequency the point of unstability can be found by means of variations of the velocity of the upper fluid. For many frequencies, it is then possible to obtain the stability curve, in function of the frequency.

This study cannot be compared with the study above: it is indeed the other part of the stability theory, the frequencial study, for convective flows. In this case, the limit of stability is different from the limit given by the spatial theory, except for little mean velocities.

In our work, we will use this type of analysis, with a basis velocity for the lower fluid of 1 m/s.