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Sharp Interface Limits for Diffuse Interface Models

By Yuning Liu, NYU Shanghai

We consider the mathematical relation between diffuse interface and sharp interface models for the flow of two viscous, incompressible Newtonian fluids like oil and water. In diffuse interface models a partial mixing of the macroscopically immiscible fluids on a small length scale ε > 0 is taken into account. These models are capable to describe 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  is of interest.  We discuss several rigorous mathematical results on convergence of diffuse interface to sharp interface models in dependence of the scaling.