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Title Rayleigh-Taylor instability for the two-phase Navier-Stokes equations with surface tension

Name Mathias Wilke
Country Germany
Email mathias.wilke@mathematik.uni-halle.de
Submit Time 2014-02-26 04:41:41
Special Session 2: Nonlinear evolution PDEs and interfaces in applied sciences
This talk is concerned with the dynamic behaviour of two immiscible and incompressible fluids in a cylindrical container, e.g. a capillary, which are separated by a sharp interface. In case that the heavier fluid overlies the lighter fluid one expects that the heavier fluid sinks down into the lighter one. This effect is known as Rayleigh-Taylor instability. The main result yields the existence of a critical surface tension with the following property. In case that the surface tension of the interface between 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 the interface. The last part of the talk is concerned with the bifurcation of nontrivial equilibria in multiple eigenvalues. The invariance of the 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.