Case with surface tension

The surface tension changes the condition of instability. $U$ must now verify the condition :

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

As we don't know the real value of the surface tension between the water-liquid and the fuel-oil-liquid, we have taken the value for the air and the water-liquid. It isn't really a problem because we are more interested by the development of the instability than by the exact values of amplification and wave number.

Figure 3.3: Stability curve

The figure 3.3 represents the stability curve for the case with surface tension. We find that the value of the velocity must be heighter than $0.16m/s$.