Effect of Vertical Periodic Oscillation with Two Commensurate Frequencies on the Rayleigh-Taylor Instability

The aim of this work is to examine the effect of a periodic vertical oscillation with two frequencies on the instability of a thin horizontal fluid layer in a Rayleigh Taylor configuration. The linear stability analysis leads to the periodic Mathieu equation, which describes the evolution of the interface amplitude. To solve the linear problem numerically, a combination of Floquet’s theory and the Runge-Kutta method is used. Numerical results show, as indicated in previous works for a single frequency, that the Rayleigh Taylor instability is not affected by the oscillation, while the resonances are. Moreover, the incorporation of two frequencies produces a richer dynamic, in terms of parametric resonances, than in the case of a single-frequency oscillation. The most unstable parametric resonances depend on the frequency ratio and can occur over a wide range of wavenumbers (large or small). Thus, the wavelength of the waves can be selected according to this ratio. It should also be noted that this ratio can have a stabilizing or destabilizing effect. The effect of fluid layer thickness on the threshold of parametric resonances is also discussed in this study.