Phase-Field and Korteweg-Type Models for the Time-Dependent Flow of Compressible Two-Phase Fluids

Abstract

Various versions of the Navier–Stokes–Allen–Cahn (NSAC), the Navier–Stokes–Cahn–Hilliard (NSCH), and the Navier–Stokes–Korteweg (NSK) equations have been used in the literature to model the dynamics of two-phase fluids. One main purpose of this paper consists in (re-)deriving NSAC, NSCH and NSK from first principles, in the spirit of rational mechanics, for fluids of very general constitutive laws. For NSAC, this deduction confirms and extends a proposal of Blesgen. Regarding NSCH, it continues work of Lowengrub and Truskinovsky and provides the apparently first justified formulation in the non-isothermal case. For NSK, it yields a most natural correction to the formulation by Dunn and Serrin. The paper uniformly recovers as examples various classes of fluids, distinguished according to whether none, one, or both of the phases are compressible, and according to the nature of their co-existence. The latter is captured not only by the mixing energy, but also by a ‘mixing rule’—a constitutive law that characterizes the type of the mixing. A second main purpose of the paper is to communicate the apparently new observation that in the case of two immiscible incompressible phases of different temperature-independent specific volumes, NSAC reduces literally to NSK. This finding may be considered as an independent justification of NSK. An analogous fact is shown for NSCH, which under the same assumption reduces to a new non-local version of NSK.