Unifying binary fluid diffuse-interface models in the sharp-interface limit

Flows involving free boundaries occur widely in both nature and technological applications, existing at liquid-gas interfaces (e.g. between liquid water and water vapour) or between different immiscible fluids (e.g. oil and water, and termed a binary fluid). To understand the asymptotic behaviour near a contact line, a liquid-gas diffuse-interface model has been investigated recently [1]. In contrast, here we investigate the behaviour between two ostensibly immiscible fluids, a binary fluid, using related models where the interface has a thin but finite thickness. Quantities such as the mass fraction of the two fluid components are modelled as varying smoothly but rapidly in the interfacial region. There has been a wide variety of models used for this situation, based on Cahn--Hilliard or Allen--Cahn theories coupled to hydrodynamic equations, and we consider the effect of these differences using matched asymptotic methods in the important sharp-interface limit, where the interface thickness goes to zero. Our aim is to understand which models represent better the classical hydrodynamic model and associated free-surface boundary conditions. \\[4pt] [1] Sibley, Nold, Savva, Kalliadasis. Eur. Phys. J. E 36, 26 (2013)