Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
MiS Preprint
120/2005
On Generalized Solutions of Two-Phase Flows for Viscous Incompressible Fluids
Helmut Abels
Abstract
We discuss the existence of generalized solutions of the flow of two immiscible, incompressible, viscous Newtonian and Non-Newtonian fluids with and without surface tension. In the case without surface tension, the existence of weak solutions can be shown, but little is known about the interface between both fluids. If surface tension is present, the energy estimates gives an a priori bound on the $(d-1)$-dimensional Hausdorff measure of the interface, but the existence of weak solutions is open. This might be due to possible oscillation and concentrations effect of the interface related to instabilities of the interface as f.e. fingering, emulsification or just cancellation of area, when two parts of the interface meet. Nevertheless one can show the existence of so-called measure-valued varifold solutions, which are weak solution if an energy equality is satisfied.
