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General study
Consider the basic flow of incompressible inviscid fluids (1) and (2) in
two horizontal parallel infinite streams of different velocities
and
and densities
and ,
one stream above the other. The two fluids are unmiscible.
This flow can be remplaced by the superposition of a global translation
of velocity
and of a symetric flow from an horizontal plane y=0 and with velocity
with .
(see figure 1.1)

Figure 1.1: Illustration of the studied
case
We have to solve classic 2D NavierStokes equations in each fluid. So,
for incompressible inviscid fluids we have the Euler model:
We look for a solution using potentials
and
associated to velocities
and :
We consider we have a linear approximation which means magnitude of perturbations
is little considering their wave length.
The velocities of fluids perpendicular to the interface must be equal
for every fluid and equal to the velocity of the interface:
Projecting velocity of each fluid onto the normal of the interface we have
now (i=1 or 2):
By retaining only terms of first order in perturbation ,
and ,
relations above become:
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when surface tensionUp:Theory
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Stephanie Terrade
Julien Delbove
20001106