Global–in–time existence for liquid mixtures subject to a generalised incompressibility constraint

Druet, Pierre-Etienne

doi:10.1016/j.jmaa.2021.125059

Cited by 8 publications

(25 citation statements)

References 23 publications

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“…of which we know the asymptotic behaviour. As already shown in the papers [DDGG20,Dru21], in the case of ideal mixtures (92), we can achieve partially explicit formula.…”

Section: Proof Of the Maximum Principle For A Special Ideal Mixturementioning

confidence: 64%

“…, ρ N ), is to introduce the thermodynamic pressure as a Lagrange multiplier directly for the algebraic constraint (160) instead of the differential constraint div v = 0. We refer to [BDD] for a recent model derivation -similar observations had been done in [JHH96] and [LT98] with the notion of quasi-incompressible fluid -and to [FLM16], [Dru21], [BD21b] for first analytical results. Here too, div v vol = 0 results as a trivial consequence of the mass conservation equations ∂ t ρ i + div  i = 0, the equation of state (160) (= the incompressibility constraint), and the assumption that the constituents are incompressible -that is, ρi (T, p) = ρ R i constant.…”

Section: Proof Of the Maximum Principle For A Special Ideal Mixturementioning

confidence: 99%

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Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics

Druet

2021

Preprint

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We consider a Navier-Stokes-Fick-Onsager-Fourier system of PDEs describing mass, energy and momentum balance in a Newtonian fluid with composite molecular structure. For the resulting parabolic-hyperbolic system, we introduce the notion of optimal regularity of mixed type, and we prove the short-time existence of strong solutions for a typical initial boundary-value-problem. By means of a partial maximum principle, we moreover show that such a solution cannot degenerate in finite time due to blow-up or vanishing of the temperature or the partial mass densities. This second result is however only valid under certain growth conditions on the phenomenological coefficients. In order to obtain some illustration of the theory, we set up a special constitutive model for volume-additive mixtures.

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“…of which we know the asymptotic behaviour. As already shown in the papers [DDGG20,Dru21], in the case of ideal mixtures (92), we can achieve partially explicit formula.…”

Section: Proof Of the Maximum Principle For A Special Ideal Mixturementioning

confidence: 64%

Section: Proof Of the Maximum Principle For A Special Ideal Mixturementioning

confidence: 99%

Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics

Druet

2021

Preprint

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“…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 [BDb], and the weak solvability of the incompressible model in [Dru19]. 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: 67%

Well-posedness analysis of multicomponent incompressible flow models

Bothe¹,

Druet²

2020

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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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“…) is continuously differentiable in the whole of R N . From the point of view of mathematical analysis of models for multicomponent mass transport in fluids, this property is very important, as shown in [DDGG20], [Dru21], [BDa],…”

Section: Additive Decomposition Of the Free Energymentioning

confidence: 99%

“…That term leads to a considerable complication of the mathematical analysis. Some analysis of weak solutions of the same kind of model was also performed in [Dru21]. So far we already introduced two constitutive functions, viz.…”

Section: Introductionmentioning

confidence: 99%

Multicomponent incompressible fluids -- An asymptotic study

Bothe¹,

Dreyer²,

Druet³

2021

Preprint

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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 2 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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Global–in–time existence for liquid mixtures subject to a generalised incompressibility constraint

Cited by 8 publications

References 23 publications

Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics

Maximal mixed parabolic-hyperbolic regularity for the full equations of multicomponent fluid dynamics

Well-posedness analysis of multicomponent incompressible flow models

Multicomponent incompressible fluids -- An asymptotic study

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