Oscillation and critical phenomenon in Rayleigh-Taylor and Richtmyer-Meshkov instabilities with surface tension

The motion of a planar interface in incompressible Rayleigh-Taylor (RT) and Richtmyer-Meshkov instabilities with surface tension is investigated analytically and numerically. Comparison between the analytical and numerical results is made for standing wave solutions in the RT instability, and it is shown that the surface tension stabilizes the gravitational instability. Motion when the linear dispersion relation is equal to zero is discussed and we show that linearly stable but nonlinearly unstable motion appears under this critical condition. When the frequency of the linear dispersion relation satisfies a certain condition, a mode-mode interaction, i.e. a kind of resonance, appears in the interfacial motion. We also discuss this resonant motion in detail.