WIAS Preprint No. 660, (2001)

Equilibrium figures of viscous fluids governed by external forces and surface tension


  • Guckel, Ralf

2010 Mathematics Subject Classification

  • 35R35 76D05 76D45 76U05


  • Equilibrium Figures, Viscous Fluids, External Forces, Surface Tension, Rotating Drops, Free Boundary Problem, Navier-Stokes Equations, Perturbation Problem, Implicit Function Theorem




We reconsider the problem of determining equilibrium figures of an isolated drop of an incompressible viscous liquid. The fluid body is subject to an external force density, and, along the free boundary, to surface tension. Here the term "equilibrium figure" means that the whole configuration is assumed to be stationary with respect to a uniformly rotating reference frame. Moreover, the pressure outside the fluid body is assumed to be close to a constant, and the fluid body itself is assumed to be close to the unit ball. The existence of such configurations is proved by applying successive approximations, under certain smallness and symmetry conditions on the external and inertial forces. The smallness assumptions are in some sense stronger, while the symmetry assumptions are weaker compared to previous results.

In case surface tension is no longer present and is (or is not) replaced by self-gravitation, the perturbation problem degenerates. The mathematical difficulties are sketched, along with a proposal of how to overcome these difficulties. Details will be presented in forthcoming papers.

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