On Generalized Solutions of Two-Phase Flows for Viscous Incompressible Fluids

revised version: October 2006
Helmut Abels
(Please use for correspondence this email).

Submission date: 19. Dec. 2005
Pages: 43
published in: Interfaces and free boundaries, 9 (2007) 1, p. 31-65 
MSC-Numbers: 35Q30, 35Q35, 76D27, 76D45, 76T99
Keywords and phrases: two-phase flow, free boundary value problems, varifold solutions, measure-valued solutions, surface tension
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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.