Existence Results for the Flow of Viscoelastic Fluids with an Integral Constitutive Law

Chupin, Laurent

doi:10.1007/s00021-013-0132-7

Cited by 12 publications

(17 citation statements)

References 31 publications

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“…3. Similar results must be hold introducing a exterior force f in the right hand side of the first equation of (7) as soon as f ∈ L r loc (0, +∞; L q ) (see for instance [12] for similar result).…”

Section: Remark 21mentioning

confidence: 72%

Global Strong Solutions for Some Differential Viscoelastic Models

Chupin¹

2018

SIAM J. Appl. Math.

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The purpose of this article is to show that there are many differential viscoelastic models for which the global existence of a regular solution is possible. Although the problem of global existence in the classic Oldroyd model is still open, we show that by adding a non-linear contribution (proposed by R.G. Larson in 1984), it is possible to obtain more regular and global solutions, regardless of the size of the data (in the two-dimensional and periodic case). Similarly, more complex appearance models such as those related to "pom-pom" polymers are interesting and mathematically richer: some "natural" bounds on the stress make it possible to obtain global results. On the other hand, in the last part, we show that other models clearly do not seem to fit into this framework, and do not even seem to have a global solution in time. These kinds of results allow to highlight the advantages and disadvantages of such or such viscoelastic fluid models. They can thus help rheologists and numericists to make choices with new arguments.

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Section: Remark 21mentioning

confidence: 72%

Global Strong Solutions for Some Differential Viscoelastic Models

Chupin¹

2018

SIAM J. Appl. Math.

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“…In this framework, the following lemma is the analog of the classical Gronwall lemma for functions depending only t. Its proof is based on a change of variable that easily allows to follow the characteristics. It is proved in [2].…”

Section: 3mentioning

confidence: 90%

“…The time T allows to take into account all the history of the motion. The deformation gradient G fulfills the differential equation (see [2]):…”

Section: Constitutive Equationmentioning

confidence: 99%

“…We follow the ideas of [15], generalized by [2] to the d-dimensional case, in order to treat the bilinear term v¨∇v. We have…”

Section: By Assumption We Know Thatmentioning

confidence: 99%

“…The reason of this choice is purely mathematics and it is certainly possible to extend some of the results to the bounded domain case using Dirichlet condition for the velocity field (we note that such a result -local in time -have been obtained for polymer model with integral law, see [2]). …”

Section: It Verifies Limmentioning

confidence: 99%

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Mathematical Existence Results for the Doi–Edwards Polymer Model

Chupin

2016

Arch Rational Mech Anal

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CHUPINÅbstract. In this paper, we present some mathematical results on the Doi-Edwards model describing the dynamics of flexible polymers in melts and concentrated solutions. This model, developed in the late 1970s, has been used and tested extensively in modeling and simulation of polymer flows. From a mathematical point of view, the Doi-Edwards model consists in a strong coupling between the Navier-Stokes equations and a highly nonlinear constitutive law. The aim of this article is to provide a rigorous proof of the well-posedness of the Doi-Edwards model, namely it has a unique regular solution. We also prove, which is generally much more difficult for flows of viscoelastic type, that the solution is global in time in the two dimensional case, without any restriction on the smallness of the data.

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Transport at Interfaces in Lipid Membranes and Enantiomer Separation

Boyarkin

Bürger

Franke

et al. 2017

Transport Processes at Fluidic Interfaces

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We study the dynamics and formation of differently ordered lateral phases of interfacial lipid layers for two types of lipid systems, a vesicle-supported bilayer and a Langmuir-Blodgett monolayer, both in experiment and by simulation. Similarly, we investigate the dynamics of objects embedded in a simpler interface given by an air-water surface and demonstrate the surface-acousticwave-actuated separation of enantiomers (chiral objects) on the surface of the carrier fluid. It turns out that the dynamics and the separation of the phases do not only depend on parameters such as temperature, mobilities and line tension but also on the mechanics of the lipid layers subjected to exterior forces as, for instance, compression, extensional and shear forces in film-balance experiments. Since the mechanical behavior of lipid layers is viscoelastic, we use a modeling approach based on the incompressible Navier-Stokes equations with a viscoelastic stress term and a capillary term, a convective Jeffrey (Oldroyd) equation of viscoelasticity, and the Cahn-Hilliard equation with a transport term. The numerical simulations are based on C 0interior-penalty discontinuous-Galerkin methods for the Cahn-Hilliard equation. Model-validation results and the verification of the simulation results by experimental data are presented. The feasibility of enantiomer separation by

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Existence Results for the Flow of Viscoelastic Fluids with an Integral Constitutive Law

Abstract: We consider the flows of viscoelastic fluid which obey a constitutive law of integral type. Some theoretical results are proved: local existence, global existence with small data and uniqueness results for the initial boundary value problem.

Cited by 12 publications

References 31 publications

Global Strong Solutions for Some Differential Viscoelastic Models

Global Strong Solutions for Some Differential Viscoelastic Models

Mathematical Existence Results for the Doi–Edwards Polymer Model

Transport at Interfaces in Lipid Membranes and Enantiomer Separation

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