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Case when surface tension and difference of density are absent

In this case,where $ \rho_{1}=\rho_{2}=\rho$, there is continuity of pressure by crossing the interface: 
$\displaystyle P_{1}(x,\xi,t)=P_{2}(x,\xi,t) $
With Bernoulli equation, we can write for each fluid: 
$\displaystyle P_{i}+\rho \frac{\partial \Phi}{\partial t} + \rho gy+\frac{1}{2} \rhov^{2}_{i}=Cte $
With these two equations, we obtain for terms of first order: 
$\displaystyle \left( \frac{\partial \Phi_{1}}{\partial t} \right)_{y=0} + \frac......_{y=0} - \frac{U}{2}\left(\frac{\partial \Phi_{2}}{\partial x} \right)_{y=0} $
We look for solutions with the following form: 
$\displaystyle \xi=A e^{ikx+\sigma t} $
$\displaystyle \Phi_{1}=B_{1} e^{ikx-ky+\sigma t} $
$\displaystyle \Phi_{2}=B_{2} e^{ikx+ky+\sigma t} $
with $ \sigma$ which is the rate of development of perturbation and k the wave length.

By injecting these solutions in last equations we obtain a linear homogeneous system with the unknown variables A,$ B_{1}$ and $ B_{2}$. The condition of compatibility is obtain by saying the determinant of the matrix of the system equal to zero so :

$\displaystyle \sigma = \pm k \frac{U}{2} = \pm \frac{2 \pi}{\lambda} \frac{U}{2} $
which is the relation of dispersion for wave. Like $ Re(\sigma)$ can be positive,we can say that there's always an instable state for this king of flow, and amplification of perturbation is more important when wave lengths are little.


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Next:Case when surface tensionUp:Theory of the Kelvin-HelmholtzPrevious:General studyContents
Stephanie Terrade

Julien Delbove
2000-11-06