Multicomponent incompressible fluids -- An asymptotic study

TL;DR

This paper explores the asymptotic behavior of multicomponent incompressible fluids, revealing limitations of the incompressible assumption and its thermodynamic implications through rigorous mathematical analysis.

Contribution

It provides a new classification of admissible data for thermodynamically consistent models and analyzes the incompressible limit's effects on free energy and volume dependence.

Findings

Incompressibility cannot hold for large deviations unless volume is linear in composition.

Incompressible limit implies molar volume becomes temperature-independent, which is often unrealistic.

The free energy converges in the sense of epi- or Gamma-convergence as compressibility vanishes.

Abstract

This paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems: (i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition. (ii) As a consequence of the second law of thermodynamics, the incompressible limit implies that the molar volume becomes independent of temperature as well. Most applications, however, reveal the non-appropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi- or Gamma-convergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDE-system relying on the equations of balance for partial masses, momentum and the internal energy.