Compressible N-phase fluid mixture models
M. F. P. ten Eikelder, E. H. van Brummelen, D. Schillinger
TL;DR
This work develops a thermodynamically consistent theory for compressible, isothermal $N$-phase mixtures rooted in continuum mixture theory. It introduces a reduced system consisting of $N$ mass balance laws and a single momentum balance, closed by Coleman–Noll derived constitutive relations that ensure energy dissipation. The framework unifies phase-field and compressible two-phase approaches by showing how appropriate closures recover NSK and NSCH/AC-type models, while preserving thermodynamic consistency across regimes and limits. It analyzes the first-order hyperbolic structure, equilibrium states, and binary mixtures, and discusses incompressible limits and connections to existing models, providing a foundation for spinodal dynamics and future numerical developments.
Abstract
Fluid mixture models are essential for describing a wide range of physical phenomena, including wave dynamics and spinodal decomposition. However, there is a lack of consensus in the modeling of compressible mixtures, with limited connections between different classes of models. On the one hand, existing compressible two-phase flow models accurately describe wave dynamics, but do not incorporate phase separation mechanisms. On the other hand, phase-field technology in fluid dynamics consists of models incorporating spinodal decomposition, however, a general phase-field theory for compressible mixtures remains largely undeveloped. In this paper, we take an initial step toward bridging the gap between compressible two-phase flow models and phase-field models by developing a theory for compressible, isothermal N-phase mixtures. Our theory establishes a system of reduced complexity by formulating N mass balance laws alongside a single momentum balance law, thereby naturally extending the Navier-Stokes Korteweg model to N-phases and providing the Navier-Stokes Cahn-Hilliard/Allen-Cahn model for compressible mixtures. Key aspects of the framework include its grounding in continuum mixture theory and its preservation of thermodynamic consistency despite its reduced complexity.