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Case without surface tension

In this case the condition of instability is :

$\displaystyle 4\rho_1 \rho_2 \left( \frac{U}{2(\rho_1+ \rho_2)}\right)^2>\frac{g}{k}\frac{\rho_2- \rho_1}
{\rho_2 + \rho_1}

We have the objective to observe $ 20$ spacial periods of the phenomenon, so the wave number is $ k=\frac{2 \pi}{\lambda} $ with $ \lambda =\frac{1}{10}$. So $ k \approx 125 $Hz.

The stability curve (see figure 3.2), indicates if a perturbation of wave number $ k $ superimposed to the basic flow characterized by the speed $ U $ will be amplify or attenuate. If the couple of values is above the curve, the perturbation on this flow will be amplified, whereas if the couple of values is below the curve, the perturbation will be attenuated.

For $ k \approx 125 Hz$, the speed of the flows must be heigher than $ 0.0783 m/s$. So for our simulation, we took $ U $ of the order of $ 0.1 m/s$

Figure 3.2: Stability curve
\includegraphics [scale=0.6]{stabcur.eps}

Stephanie Terrade
Julien Delbove