On generalized solutions of two-phase flows for viscous incompressible fluids

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 a domain $\Omega \subseteq {\mathbb{R}}^d$, $d=2,3$. In the case without surface tension, the existence of weak solutions is 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 concentration effects of the interface related to instabilities of the interface as for example fingering, emulsification or just cancellation of area, when two parts of the interface meet. Nevertheless we will show the existence of so-called measure-valued varifold solutions, where the interface is modeled by an oriented general varifold $V(t)$ which is a non-negative measure on $\Omega \times {\mathbb{S}}^{d-1}$, where ${\mathbb{S}}^{d-1}$ is the unit sphere in ${\mathbb{R}}^d$.

Moreover, it is shown that measure-valued varifold solutions are weak solution if an energy equality is satisfied.