Rayleigh-Taylor instability for the two-phase Navier-Stokes equations with surface tension in cylindrical domains

This article is concerned with the dynamic behaviour of two immiscible and incompressible fluids in a cylindrical domain, which are separated by a sharp interface. In case that the heavy fluid is situated on top of the light fluid, one expects that the fluid on top sags down into the lower phase. This effect is known as the Rayleigh-Taylor-Instability. We present a rather complete analysis of the corresponding free boundary problem which involves a contact angle. Our main result implies the existence of a critical surface tension with the following property. In case that the surface tension of the interface separating the two fluids is smaller than the critical surface tension, one has Rayleigh-Taylor-Instability. On the contrary, if the interface has a greater surface tension than the critical value, the instability effect does not occur and one has exponential stability of a flat interface. The last part of this article is concerned with the bifurcation of nontrivial equilibria in multiple eigenvalues. The invariance of the corresponding bifurcation equation with respect to rotations and reflections yields the existence of bifurcating subcritical equilibria. Finally it is proven that the bifurcating equilibria are unstable.