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Multicomponent incompressible fluids — An asymptotic study

Bothe, Dieter ; Dreyer, Wolfgang ; Druet, Pierre‐Etienne (2024)
Multicomponent incompressible fluids — An asymptotic study.
In: ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2023, 103 (7)
doi: 10.26083/tuprints-00024685
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Item Type: Article
Type of entry: Secondary publication
Title: Multicomponent incompressible fluids — An asymptotic study
Language: English
Date: 9 February 2024
Place of Publication: Darmstadt
Year of primary publication: 2023
Place of primary publication: Weinheim
Publisher: Wiley-VCH
Journal or Publication Title: ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
Volume of the journal: 103
Issue Number: 7
Collation: 48 Seiten
DOI: 10.26083/tuprints-00024685
Corresponding Links:
Origin: Secondary publication DeepGreen

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 2nd 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.

Identification Number: Artikel-ID: e202100174
Status: Publisher's Version
URN: urn:nbn:de:tuda-tuprints-246858
Additional Information:

Special Issue: Energy‐Based Mathematical Methods for Reactive Multiphase Flows

Classification DDC: 500 Science and mathematics > 510 Mathematics
Divisions: 04 Department of Mathematics > Analysis > Mathematical Modeling and Analysis
Date Deposited: 09 Feb 2024 13:50
Last Modified: 18 Apr 2024 06:15
SWORD Depositor: Deep Green
URI: https://tuprints.ulb.tu-darmstadt.de/id/eprint/24685
PPN: 517194392
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