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
Artikel, Zweitveröffentlichung, Verlagsversion
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Kurzbeschreibung (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 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.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2024 |
| Autor(en): | Bothe, Dieter ; Dreyer, Wolfgang ; Druet, Pierre‐Etienne |
| Art des Eintrags: | Zweitveröffentlichung |
| Titel: | Multicomponent incompressible fluids — An asymptotic study |
| Sprache: | Englisch |
| Publikationsjahr: | 9 Februar 2024 |
| Ort: | Darmstadt |
| Publikationsdatum der Erstveröffentlichung: | 2023 |
| Ort der Erstveröffentlichung: | Weinheim |
| Verlag: | Wiley-VCH |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik |
| Jahrgang/Volume einer Zeitschrift: | 103 |
| (Heft-)Nummer: | 7 |
| Kollation: | 48 Seiten |
| DOI: | 10.26083/tuprints-00024685 |
| URL / URN: | https://tuprints.ulb.tu-darmstadt.de/24685 |
| Zugehörige Links: | |
| Herkunft: | Zweitveröffentlichung DeepGreen |
| Kurzbeschreibung (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 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. |
| ID-Nummer: | Artikel-ID: e202100174 |
| Status: | Verlagsversion |
| URN: | urn:nbn:de:tuda-tuprints-246858 |
| Zusätzliche Informationen: | Special Issue: Energy‐Based Mathematical Methods for Reactive Multiphase Flows |
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Analysis 04 Fachbereich Mathematik > Analysis > Mathematische Modellierung und Analysis |
| Hinterlegungsdatum: | 09 Feb 2024 13:50 |
| Letzte Änderung: | 13 Feb 2024 15:03 |
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