Stability of the interface between two thin liquid layers under tangential high frequency vibrations
In this paper, a system of two equally thin layers of immiscible incompressible isothermal ideal liquids under high frequency horizontal harmonic vibrations is considered theoretically. The vessel containing the liquids is assumed to be closed, of rectangular form with weakly-deformable side borders and infinitely long in horizontal direction. Previous studies showed that, for significantly thin layers, the main instability in the system, oscillatory Kelvin-Helmholtz instability, should be the long-wave instability. Therefore, the problem was solved analytically using “shallow water” approximation. For all equations, a formal expansion with respect to two small parameters was used: one associated with a small ratio of the vertical to horizontal scale and another with small perturbations of the flat interface. Evolutionary equations were derived for the interface in the main order of expansion for vibration intensity less than a threshold value for oscillatory Kelvin-Helmholtz instability (subcritical area). The solutions found for these evolutionary equations correspond to traveling waves with soliton or cnoidal interfacial surface. The soliton profile is a limiting case of the cnoidal profile. The maximum speed of these waves was determined. It was demonstrated that the obtained solutions exist only in the subcritical area of parameters and from the critical level subcritical bifurcation emerges. For special case of traveling waves in a quasi-stationary mode (i.e. with static interface) also referred to as a “frozen wave”, a numerical analysis of linear stability was performed using a Fourier series expansion in a horizontal coordinate. The instability of quasi-stationary modes to small disturbances was demonstrated.