Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models

Bothe, Dieter; Druet, Pierre-Etienne

doi:10.1016/j.na.2021.112389

Cited by 13 publications

(84 citation statements)

References 33 publications

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“…Comparing with the paper [5] on compressible class-one models based on a similar reformulation, we see that the incompressible limit corresponds structurally to the case that one of the relative chemical potentials is subject to an elliptic-instead of a parabolic-equation, and the total mass density is confined to a bounded interval.…”

Section: A Review Of Prior Investigations and Our Methodsmentioning

confidence: 74%

“…In this paper, we study the well-posedness analysis in classes of strong solutions of class-one models 1 of mass transport in isothermal, incompressible multicomponent fluids. This investigation is a direct continuation of results obtained recently concerning the compressible case in [5], and the weak solvability of the incompressible model in [14]. Performing the incompressible limit (the low-Mach number limit) in models for fluid mixtures and for multicomponent fluids is desirable both from the practical and the theoretical viewpoint.…”

Section: Multicomponent Diffusion In An Incompressible Fluidmentioning

confidence: 70%

“…Our main method to approach the PDE problem is a switch of variables in the transport problem as already applied in [5]. Instead of the original variables (ρ 1 , .…”

Section: A Review Of Prior Investigations and Our Methodsmentioning

confidence: 99%

“…For an overview of possible methods to study the transformed PDE system, we refer to our study [5]. We shall follow here the same principal road map, but profound transformations are necessary to deal with the constraint on , since it implies that the nonlinear functions occurring in the transformed system are singular for dist( , { min , max }) → 0.…”

Section: A Review Of Prior Investigations and Our Methodsmentioning

confidence: 99%

“…Concerning the presence of reaction terms in (2), we have to mention in respect with the compressible systems considered in [5] a subtle difference of the incompressible models. For the compressible case, the reactions densities r i in (2) are allowed to be general functions r i = r i (ρ 1 , .…”

Section: Multicomponent Diffusion In An Incompressible Fluidmentioning

confidence: 99%

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Well-posedness analysis of multicomponent incompressible flow models

Bothe

Druet

2021

J. Evol. Equ.

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In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities stays constant. In this type of models, the velocity field in the Navier–Stokes equations is not solenoidal and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.

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Section: A Review Of Prior Investigations and Our Methodsmentioning

confidence: 74%

Section: Multicomponent Diffusion In An Incompressible Fluidmentioning

confidence: 70%

“…Our main method to approach the PDE problem is a switch of variables in the transport problem as already applied in [5]. Instead of the original variables (ρ 1 , .…”

Section: A Review Of Prior Investigations and Our Methodsmentioning

confidence: 99%

Section: A Review Of Prior Investigations and Our Methodsmentioning

confidence: 99%

Section: Multicomponent Diffusion In An Incompressible Fluidmentioning

confidence: 99%

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Well-posedness analysis of multicomponent incompressible flow models

Bothe

Druet

2021

J. Evol. Equ.

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Multicomponent incompressible fluids—An asymptotic study

2022

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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$^\textrm {nd}$ 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.

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GENERIC framework for reactive fluid flows

2022

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We describe reactive fluid flows in terms of the formalism General Equation forNon-Equilibrium Reversible-Irreversible Coupling also known as GENERIC. We present the thermodynamical and mechanical foundations for the treatment of fluid flows using GENERIC and present a framework for noninvertible transformations, that is in particular applied to transformations of variational approaches between Lagrangian and Eulerian coordinates. We bring the abstract framework to life by providing many physically relevant examples for reactive compressible fluid flows.

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Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models

Cited by 13 publications

References 33 publications

Well-posedness analysis of multicomponent incompressible flow models

Well-posedness analysis of multicomponent incompressible flow models

Multicomponent incompressible fluids—An asymptotic study

GENERIC framework for reactive fluid flows

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