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Case when surface tension and difference of density are not absent

Phenomenons of difference of density and surface tension are known as to stabilize the flow.

Now, at the interface we don't have equality of pressure because of surface tension: 

$\displaystyle (P_{1})_{y=\xi}=(P_{2})_{y=\xi} + \gamma \frac{\partial^{2} \xi}{\partial^{2} x}=(P_{2})_{y=\xi}-\gamma k^{2} \xi $
$ \gamma$ is the value of surface tension between the two fluids.

In Bernoulli equation hydrostatic term $ \rho_{i}g\xi$ (i=1 or 2) must be introduced.

Like in the last section we have a condition of stability by saying the determinant of the matrix of the system is equal to zero: 

$\displaystyle 4\rho_{1}\rho_{2} \left( \frac{U}{2(\rho_{1}+\rho_{2})^{2}} \righ......frac{\rho_{2}-\rho_{1}}{\rho_{2}+\rho_{1}}+\frac{\gamma k}{\rho_{2}+\rho_{1}} $
with $ c_{0min}=\sqrt{\frac{4\gamma g (\rho_{2}-\rho_{1})}{(\rho_{1}+\rho_{2})^{2}}}$ and $ k_{c}=\sqrt{\frac{g(\rho_{2}-rho{1})}{\gamma}}$.

Stephanie Terrade

Julien Delbove