Models of Two-Phase Fluid Dynamics à la Allen-Cahn, Cahn-Hilliard, and ... Korteweg!

One purpose of this paper on the Navier-Stokes-Allen-Cahn (NSAC), the Navier-Stokes-Cahn-Hilliard (NSCH), and the Navier-Stokes-Korteweg (NSK) equations consists in surveying solution theories that one of the authors, M. K., has developed for these three evolutionary systems of partial differential equations. All three theories start from a Helmholtz free energy description of the compressible two-phase fluids whose dynamics they describe in various ways. While a diphasic fluid composed from two constituents of individually constant density is still compressible as long as these two densities are different from each other, the abovementioned solution theories for NSAC and NSCH do not apply in this “quasi-incompressible” case, as the Helmholtz-energy framework degenerates. The second purpose of the paper is to present an observation made by both authors together that shows how to fill these gaps. As ‘by-products’ one obtains (a) in the case that the phases can transform into each other, a justification of NSK, and (b) in the case that they cannot, a new Korteweg type system with non-local ‘viscosity’.