Sharp interface limits for diffuse interface models for two-phase flows of viscous incompressible fluids

我们寻求一个严格的理解，所谓的扩散界面模型的两个粘性流体，如油和水的流动的尖锐的界面限制。在扩散界面模型中，考虑了宏观上不混溶的流体在小尺度ε > 0上的部分混合和质量颗粒的扩散。这些模型能够描述这种两相流超出发生的拓扑奇点的接口，由于碰撞或液滴形成。为了理论和数值目的，更深入地理解依赖于迁移率mε的标度的极限ε → 0是有意义的。这里的迁移率是Peclet数的倒数，并控制扩散的强度。特别地，我们希望严格地理解迁移率系数mε的标度随着ε → 0对极限系统和收敛速度的影响。通过这种方式，我们希望澄清文献中的数值观测和形式推导，并获得新的见解。

We seek a rigorous understanding of sharp interface limits of so-called diffuse interface models for the flow of two viscous fluids like oil and water. In diffuse interface models a partial mixing of the macroscopically immiscible fluids on a small length scale ε > 0 and diffusion of the mass particles are taken into account. These models are capable of describing such two-phase flows beyond the occurrence of topological singularities of the interface due to collision or droplet formation. Both for theoretical and numerical purposes a deeper understanding of the limit ε → 0 in dependence of the scaling of the mobility mε is of interest. Here the mobility is the inverse of the Peclet number and controls the strength of the diffusion. In particular, we want to understand the influence of the scaling of the mobility coeffcient mε as ε → 0 on the limit system and on the convergence rates rigorously. This way we want to clarify numerical observations and formal derivations in the literature and obtain new insights.