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On a diffuse interface approach to PDEs on surfaces and networks

Banff International Research Station for Mathematical Innovation and Discovery

Diffuse interface models based on the phase field methodology have been developed for various free boundary problems. They involve representing the interfaces by thin layers. Some applications feature phenomena on the interfaces described by PDEs for interface resident fields. We will address the questions of how to model such phenomena in the diffuse interface setting and how to numerically approximate the solutions, which may require special consideration due to degeneracies. The approach can be generalised to networks and bubble clusters. One key challenge then is to correctly recover the conditions in the triple junctions formed by three interfaces. The research is motivated by surface active agents (surfactants) in multi-phase flow which can be effectively modelled by Cahn-Hilliard-Navier-Stokes systems. Some preliminary results for a novel numerical scheme will be presented, as well as some simulations to support the theoretical findings.

Mathematics; Partial differential equations; Numerical analysis

video/mp4

Author affiliation: University of Warwick

2019-03-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68729

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/