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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 $ U_{1}$ and $ U_{2}$ and densities $ \rho_1$ and $ \rho_2$, one stream above the other. The two fluids are unmiscible.

This flow can be remplaced by the superposition of a global translation of velocity $ \left( U_{1}+U_{2} \right)$ and of a symetric flow from an horizontal plane y=0 and with velocity $ \pm \frac{U}{2}$ with $ U=U_{1}-U_{2}$. (see figure 1.1)

\includegraphics [scale=0.6]{schtheo.eps}
Figure 1.1: Illustration of the studied case 

We have to solve classic 2D Navier-Stokes equations in each fluid. So, for incompressible inviscid fluids we have the Euler model:

$\displaystyle \frac{\partial U_{j}}{\partial x_{j}}=0 $
$\displaystyle \rho \frac{\partial U_{i}}{\partial t}=\rho \left( \frac{\partial...... U_{j}}{\partial x_{i}} \right)=\frac{\partial P}{\partial x_{i}}+\rho F_{i} $
We look for a solution using potentials $ \Phi_{1}$ and $ \Phi_{2}$ associated to velocities $ v_{1}$ and $ v_{2}$
$\displaystyle v_{1}=grad \left [ \frac{Ux}{2} + \Phi_{1}(x,y,t) \right] $
$\displaystyle v_{2}=grad \left [ -\frac{Ux}{2} + \Phi_{2}(x,y,t) \right] $
We consider we have a linear approximation which means magnitude of perturbations $ \xi(x,t)$ 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: 

$\displaystyle [v_{1\bot}]_{y=\xi} = [v_{2\bot}]_{y=\xi} \cong \frac{\partial \xi}{\partial t} $
Projecting velocity of each fluid onto the normal of the interface we have now (i=1 or 2): 
$\displaystyle v_{iy} \cong \frac{\partial \xi}{\partial t} + v_{ix} \frac{\partial \xi}{\partial x} $
By retaining only terms of first order in perturbation $ \xi$$ \Phi_{1}$ and $ \Phi_{2}$, relations above become: 
$\displaystyle v_{1y}=\frac{\partial \Phi_{1}}{\partial y} = \frac{\partial \xi}{\partial t} + \frac{1}{2} \frac{\partial \xi}{\partial x} $
$\displaystyle v_{2y}=\frac{\partial \Phi_{2}}{\partial y} = \frac{\partial \xi}{\partial t} + \frac{1}{2} \frac{\partial \xi}{\partial x} $

nextuppreviouscontents
Next:Case when surface tensionUp:Theory of the Kelvin-HelmholtzPrevious:Theory of the Kelvin-HelmholtzContents
Stephanie Terrade

Julien Delbove
2000-11-06