Bothe, Dieter ; Dreyer, Wolfgang ; Druet, Pierre‐Etienne (2023)
Multicomponent incompressible fluids — An asymptotic study.
In: ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 103 (7)
doi: 10.1002/zamm.202100174
Article, Bibliographie
This is the latest version of this item.
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.
Item Type:  Article 

Erschienen:  2023 
Creators:  Bothe, Dieter ; Dreyer, Wolfgang ; Druet, Pierre‐Etienne 
Type of entry:  Bibliographie 
Title:  Multicomponent incompressible fluids — An asymptotic study 
Language:  English 
Date:  2023 
Place of Publication:  Weinheim 
Publisher:  WileyVCH 
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:  10.1002/zamm.202100174 
Corresponding Links:  
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. 
Identification Number:  ArtikelID: e202100174 
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 04 Department of Mathematics > Analysis 04 Department of Mathematics > Analysis > Mathematical Modeling and Analysis 
Date Deposited:  13 Feb 2024 15:03 
Last Modified:  07 Mar 2024 14:46 
PPN:  516076671 
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Multicomponent incompressible fluids — An asymptotic study. (deposited 09 Feb 2024 13:50)
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