On a Navier–Stokes–Fourier-like system capturing transitions between viscous and inviscid fluid regimes and between no-slip and perfect-slip boundary conditions

Maringová, Erika; Žabenský, Josef

doi:10.1016/j.nonrwa.2017.10.008

Cited by 24 publications

(20 citation statements)

References 25 publications

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“…The numerical solution of the corresponding governing equations is investigated in Janečka et al [29], while a rigorous numerical analysis for various models that fall into this class is discussed in Diening et al [30], Stebel [31], Hirn et al [32], Süli and Tscherpel [33], Farrell et al [34], Farrell and Gazca-Orozco [35]. A rigorous mathematical theory for some of the aforementioned models is developed in Bulíček et al [21,36] and Maringová and Žabenský [37]; see also Blechta et al [38] for a recent review.…”

Section: Fluidsmentioning

confidence: 99%

Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

Průša

Rajagopal

2021

Fluids

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Viscoelastic fluids are non-Newtonian fluids that exhibit both “viscous” and “elastic” characteristics in virtue of the mechanisms used to store energy and produce entropy. Usually, the energy storage properties of such fluids are modeled using the same concepts as in the classical theory of nonlinear solids. Recently, new models for elastic solids have been successfully developed by appealing to implicit constitutive relations, and these new models offer a different perspective to the old topic of the elastic response of materials. In particular, a sub-class of implicit constitutive relations, namely relations wherein the left Cauchy–Green tensor is expressed as a function of stress, is of interest. We show how to use this new perspective in the development of mathematical models for viscoelastic fluids, and we provide a discussion of the thermodynamic underpinnings of such models. We focus on the use of Gibbs free energy instead of Helmholtz free energy, and using the standard Giesekus/Oldroyd-B models, we show how the alternative approach works in the case of well-known models. The proposed approach is straightforward to generalize to more complex settings wherein the classical approach might be impractical or even inapplicable.

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Section: Fluidsmentioning

confidence: 99%

Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

Průša

Rajagopal

2021

Fluids

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“…It was well documented that in certain situations the Navier slip boundary conditions are more appropriate than no slip boundary conditions, we refer e.g. to [12,20,23,25] or [36] and references therein. In addition, it was shown that the Navier slip boundary condition can be understood as an asymptotic limit of no slip boundary conditions in case we consider rough and highly oscillating boundary, see e.g.…”

Section: First Of All Letmentioning

confidence: 99%

“…Nevertheless, since we shall always deal with formulation without the pressure (see the De nition), we can also treat the Dirichlet boundary condition, as well as very general implicitly speci ed boundary conditions see e.g. [12,13,36] or [8]. The Neumann boundary condition for B is considered here only for simplicity and without any speci c physical meaning.…”

Section: First Of All Letmentioning

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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“…As to the question of existence of solutions, for any time interval and any reasonable set of data, a satisfactory theory, within the context of weak solutions, has been established during the last fifty years for a general class of fluids given by (2.2) with a continuous monotone function G (see [30,8,25,27,28,31,4,2,34,35,19,7,43,13,3,5,6,40]).…”

Section: Incompressible Non-newtonian Fluids: a Brief Overviewmentioning

confidence: 99%

PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion

Bulíček¹,

Málek²,

Průša³

et al. 2018

Mathematical Analysis in Fluid Mechanics

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We establish the long-time existence of large-data weak solutions to a system of nonlinear partial differential equations. The system of interest governs the motion of non-Newtonian fluids described by a simplified viscoelastic rate-type model with a stress-diffusion term. The simplified model shares many qualitative features with more complex viscoelastic rate-type models that are frequently used in the modeling of fluids with complicated microstructure. As such, the simplified model provides important preliminary insight into the mathematical properties of these more complex and practically relevant models of non-Newtonian fluids. The simplified model that is analyzed from the mathematical perspective is shown to be thermodynamically consistent, and we extensively comment on the interplay between the thermodynamical background of the model and the mathematical analysis of the corresponding initial-boundary-value problem.2000 Mathematics Subject Classification. Primary 35Q35, 76A05, 76A10. The authors acknowledge the support of the ERC-CZ project LL1202, financed by MŠMT.

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On a Navier–Stokes–Fourier-like system capturing transitions between viscous and inviscid fluid regimes and between no-slip and perfect-slip boundary conditions

Cited by 24 publications

References 25 publications

Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

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

PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion

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