Mixture-aware closure of the N-phase Navier--Stokes--Cahn--Hilliard mixture model

Diffuse-interface (phase-field) models are widely used to describe multiphase mixtures and their interfacial dynamics. In multiphase settings, however, the constitutive closure should remain meaningful across different representations of the same mixture. Existing N-phase phase-field constructions commonly enforce reduction only when a phase is absent (restriction to a face of the Gibbs simplex), but do not address the natural requirement that physically identical phases can be merged without changing the governing equations. This requires characterizing thermodynamically admissible, mixture-aware constitutive closures that are consistent with merging identical phases at the PDE level. Here, we show that, under a small set of structural axioms, PDE-level reduction consistency uniquely fixes the admissible free-energy structure to an ideal-mixing contribution to an ideal-mixing contribution, a symmetric mean-field interaction term, and a constant-coefficient quadratic gradient penalty. yielding a thermodynamic closure that includes Maxwell--Stefan-type mobilities as a special case. The same requirement constrains the Onsager mobility matrix to a pairwise-exchange form with bilinear degeneracy in the volume fractions, yielding a thermodynamic closure that includes Maxwell--Stefan-type mobilities as a special case. These results provide a consistent closure for N-phase Navier--Stokes--Cahn--Hilliard mixture models and, in the bulk-only setting, for multiphase Maxwell--Stefan diffusion systems. Numerical experiments confirm the predicted mixture-aware reduction properties and illustrate the capabilities of the N-phase Navier--Stokes--Cahn--Hilliard framework in representative multiphase-flow computations.