Mathematical Justification of a Baer$-$Nunziato Model for a Compressible Viscous Fluid with Phase Transition
Christian Rohde, Florian Wendt
TL;DR
This work rigorously justifies a one-velocity Baer–Nunziato (BN) multi-fluid model as the macroscopic limit of a non-local Navier–Stokes–Korteweg (NSK) description for isothermal two-phase flow in one dimension. By interpreting density as a parametrized measure and analyzing highly oscillatory initial data, the authors derive a kinetic equation for the measure together with a momentum equation for the velocity, and show that the measure concentrates as a convex combination of Dirac masses, yielding BN in the limit. A global-in-time well-posedness result for the non-local NSK system is established via BD-entropy and refined a-priori estimates, including control of the auxiliary field and the effective viscous flux. The homogenization proceeds through compactness and compensated compactness to produce a BN limit, thereby connecting detailed-phase-scale effects with the macroscopic BN description, and suggesting avenues for extending to higher dimensions and exploring phase-transition closures.
Abstract
In this work, we justify a Baer$-$Nunziato system including appropriate closure terms as the macroscopic description of a compressible viscous fluid that can occur in a liquid or a vapor phase in the isothermal framework. As a mathematical model for the two-phase fluid on the detailed scale we chose a non-local version of the Navier$-$Stokes$-$Korteweg equations in the one-dimensional and periodic setting. Our justification relies on anticipating the macroscopic description of the two-phase fluid as the limit system for a sequence of solutions with highly oscillating initial densities. Interpreting the density as a parametrized measure, we extract a limit system consisting of a kinetic equation for the parametrized measure and a momentum equation for the velocity. Under the assumption that the initial density distributions converge in the limit to a convex combination of Dirac-measures, we show by a uniqueness result that the parametrized measure also has to be a convex combination of Dirac-measures and, that the limit system reduces to the Baer$-$Nunziato system. This work extends existing results concerning the justification of Baer$-$Nunziato models as the macroscopic description of multi-fluid models in the sense, that we allow for phase transition effects on the detailed scale. This work also includes a new global-in-time well-posedness result for the Cauchy problem of the non-local Navier$-$Stokes$-$Korteweg equations.