  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.
