On a class of compressible viscoelastic rate-type fluids with stress-diffusion

Bulíček, Miroslav; Feireisl, Eduard; Málek, Josef

doi:10.1088/1361-6544/ab3614

Cited by 16 publications

(30 citation statements)

References 13 publications

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“…While the presence of the centre-of-mass diffusion term is motivated by physical considerations, see for example El-Kareh and Leal [52], Bhave et al [53], Schieber [54] and Degond and Liu [32], it turns out that its presence can be gainfully exploited in the mathematical analysis of the corresponding governing equations in the purely mechanical setting as well, see especially Barrett and Süli [31,33,55,56,57] or Feireisl et al [34]. (See also Bulíček et al [58], Bulíček et al [59] and Bathory et al [60] in the context of macroscopic models with a stress diffusion term.) One might hope that once thermodynamically consistent models for the full thermo-mechanical coupling are derived, similar rigorous mathematical results might also be obtained for the corresponding full system of governing equations, that is including the governing equation for the temperature.…”

Section: Discussionmentioning

confidence: 99%

A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids

Dostalík

Málek

Průša

et al. 2020

Fluids

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We revisit some classical models for dilute polymeric fluids, and we show that thermodynamically consistent models for non-isothermal flows of these fluids can be derived in a very elementary manner. Our approach is based on the identification of energy storage mechanisms and entropy production mechanisms in the fluid of interest, which, in turn, leads to explicit formulae for the Cauchy stress tensor and for all of the fluxes involved. Having identified these mechanisms and derived the governing equations, we document the potential use of the thermodynamic basis of the model in a rudimentary stability analysis. In particular, we focus on finite amplitude (nonlinear) stability of a stationary spatially homogeneous state in a thermodynamically isolated system.

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Section: Discussionmentioning

confidence: 99%

A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids

Dostalík

Málek

Průša

et al. 2020

Fluids

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“…This assumption, however, turns (1.11) into a much simpler scalar equation. Moreover, note that if B = bI, then the equations (1.10) and (1.11) decouple (which is not the case in [14] and [10] since there the considered constitutive relation for T is more complicated than here). Furthermore, in [29], the authors consider yet another class of Peterlin viscoelastic models with stress di usion and prove existence of a global twoor three-dimensional solution.…”

Section: The De Nition Of a Weak Solution And Its Existencementioning

confidence: 91%

“…There are also global large data existence results in three dimensions for slightly di erent classes of di usive rate-type viscoelastic models, but under some simplifying assumptions. For example, in [14] and [10], the authors consider the case where B = bI. This assumption, however, turns (1.11) into a much simpler scalar equation.…”

Section: The De Nition Of a Weak Solution And Its Existencementioning

confidence: 99%

Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion

Bathory

Bulíček

Málek

2020

Advances in Nonlinear Analysis

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We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity v, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus models for a tensor 𝔹. By a proper choice of the constitutive relations for the Helmholtz free energy (which, however, is non-standard in the current literature, despite the fact that this choice is well motivated from the point of view of physics) and for the energy dissipation, we are able to prove that 𝔹 enjoys the same regularity as v in the classical three-dimensional Navier-Stokes equations. This enables us to handle any kind of objective derivative of 𝔹, thus obtaining existence results for the class of diffusive Johnson-Segalman models as well. Moreover, using a suitable approximation scheme, we are able to show that 𝔹 remains positive definite if the initial datum was a positive definite matrix (in a pointwise sense). We also show how the model we are considering can be derived from basic balance equations and thermodynamical principles in a natural way.

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“…, and π 0 ∈ H 1 (Ω). (19) Then: (i) the system (4a) and (15) with the initial and boundary conditions (6) and (16) has a weak solution…”

Section: Remark 1 (Energy Conservation Alternatively) Using the Calcmentioning

confidence: 99%

From quasi-incompressible to semi-compressible fluids

Roubíček¹

2021

DCDS-S

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A new concept of semi-compressible fluids is introduced for slightly compressible visco-elastic fluids (typically rather liquids than gasses) where mass density variations are negligible in some sense, while being directly controlled by pressure which is very small in comparison with the elastic bulk modulus. The physically consistent fully Eulerian models with specific dispersion of pressure-wave speed are devised. This contrasts to the so-called quasi-incompressible fluids which are described not physically consistently and, in fact, only approximate ideally incompressible ones in the limit. After surveying and modifying models for the quasi-incompressible fluids, we eventually devise some fully convective models complying with energy conservation and capturing phenomena as pressure-wave propagation with wavelength (and possibly also pressure) dependent velocity.

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On a class of compressible viscoelastic rate-type fluids with stress-diffusion

Cited by 16 publications

References 13 publications

A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids

A Simple Construction of a Thermodynamically Consistent Mathematical Model for Non-Isothermal Flows of Dilute Compressible Polymeric Fluids

Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion

From quasi-incompressible to semi-compressible fluids

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