Consistent closure schemes for statistical models of anisotropic fluids
Identifieur interne : 008E15 ( Main/Exploration ); précédent : 008E14; suivant : 008E16Consistent closure schemes for statistical models of anisotropic fluids
Auteurs : Martin Kröger [Suisse] ; Amine Ammar [France] ; Francisco Chinesta [France]Source :
- Journal of non-newtonian fluid mechanics [ 0377-0257 ] ; 2008.
Descripteurs français
- Pascal (Inist)
English descriptors
- KwdEn :
Abstract
We propose a rational approach to approximating the various alignment tensors. It preserves the correct symmetry and leads to consistent results. For the case of uniaxial nematic fluids, the decoupling approximation for a tensor of rank n involves (n - 2)/2 scalar functions Sn(S2) in terms of a scalar argument S2, with Sn (0) = 0 and Sn (1) = 1. Nothing else can be concluded about the mathematical relationship between moments of the distribution function, and in particular, all consistent decoupling approximations for fourth-order moment in terms of second-order moments can be characterized by a single S4(S2) function. We propose using the simple model dependent convex shaped equilibrium relationship between S4 and S2 to characterize new (and simple) decoupling approximations K-I and K-II for the biaxial (including uniaxial) phase. In order to test the new against earlier proposed approximations rigorously, and to discuss consistency issues, we solve the Hess-Doi Fokker-Planck equation for nematic and nematic-discotic liquid crystals efficiently for a wide range of (2300 distinct) possible conditions including mixed shear and elongational flows, diverse field strengths, and molecular shapes. As a result, we confirm the closures K-I and K-II with correct tensorial symmetry; they are valid under arbitrary conditions to high precision, exact in the isotropic and totally aligned phases, improve upon earlier parameter-free closures in particular in the temperature regime T∈ [0.6, ∞] x TNI with the nematic-isotropic transition temperature TNI (or alternatively, for mean-field strengths U ∈ [0, 8]). K-II performs as good as the so-called Bingham closure, which usually requires 30 empirical coefficients, while K-I and K-II are essentially parameter-free, and their quality can be expected to be insensitive to the particular model.
Affiliations:
- France, Suisse
- Auvergne-Rhône-Alpes, Canton de Zurich, Rhône-Alpes, Île-de-France
- Grenoble, Paris, Zurich
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10</s1><s2>8093 Zürich</s2><s3>CHE</s3><sZ>1 aut.</sZ></inist:fA14><country>Suisse</country><placeName><settlement type="city">Zurich</settlement><region nuts="3" type="region">Canton de Zurich</region></placeName></affiliation></author><author><name sortKey="Ammar, Amine" sort="Ammar, Amine" uniqKey="Ammar A" first="Amine" last="Ammar">Amine Ammar</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Laboratoire de Rhéologie, UMR 5520 CNRS-UJF-INPG, 1301 Rue de la Piscine, BP 53 Domaine Universitaire</s1><s2>38041 Grenoble</s2><s3>FRA</s3><sZ>2 aut.</sZ></inist:fA14><country>France</country><wicri:noRegion>38041 Grenoble</wicri:noRegion><placeName><settlement type="city">Grenoble</settlement><region type="region" nuts="2">Auvergne-Rhône-Alpes</region><region type="old region" nuts="2">Rhône-Alpes</region></placeName></affiliation></author><author><name sortKey="Chinesta, Francisco" sort="Chinesta, Francisco" uniqKey="Chinesta F" first="Francisco" last="Chinesta">Francisco Chinesta</name><affiliation wicri:level="1"><inist:fA14 i1="03"><s1>LMSP UMR 8106 CNRS-ENSAM-ESEM, 151 Boulevard de l'Hôpital</s1><s2>75013 Paris</s2><s3>FRA</s3><sZ>3 aut.</sZ></inist:fA14><country>France</country><wicri:noRegion>75013 Paris</wicri:noRegion><placeName><settlement type="city">Paris</settlement><region type="région" nuts="2">Île-de-France</region></placeName></affiliation></author></analytic><series><title level="j" type="main">Journal of non-newtonian fluid mechanics</title><title level="j" type="abbreviated">J. non-newton. fluid mech.</title><idno type="ISSN">0377-0257</idno><imprint><date when="2008">2008</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">Journal of non-newtonian fluid mechanics</title><title level="j" type="abbreviated">J. non-newton. fluid mech.</title><idno type="ISSN">0377-0257</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Closure model</term><term>Distribution functions</term><term>Ferromagnetic fluid</term><term>Fokker-Planck equation</term><term>Liquid crystals</term><term>Liquid nematic transformation</term><term>Magnetorheological fluid</term><term>Mathematical models</term><term>Nematic crystals</term><term>Orientational order</term><term>Rational approximation</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Ordre orientationnel</term><term>Fonction distribution</term><term>Modèle mathématique</term><term>Equation Fokker Planck</term><term>Transformation liquide nématique</term><term>Fluide ferromagnétique</term><term>Approximation rationnelle</term><term>Modèle fermeture</term><term>Cristal liquide</term><term>Cristal nématique</term><term>Fluide magnétorhéologique</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">We propose a rational approach to approximating the various alignment tensors. It preserves the correct symmetry and leads to consistent results. For the case of uniaxial nematic fluids, the decoupling approximation for a tensor of rank n involves (n - 2)/2 scalar functions S<sub>n</sub>(S<sub>2</sub>) in terms of a scalar argument S<sub>2</sub>, with S<sub>n</sub> (0) = 0 and S<sub>n</sub> (1) = 1. Nothing else can be concluded about the mathematical relationship between moments of the distribution function, and in particular, all consistent decoupling approximations for fourth-order moment in terms of second-order moments can be characterized by a single S<sub>4</sub>(S<sub>2</sub>) function. We propose using the simple model dependent convex shaped equilibrium relationship between S<sub>4</sub> and S<sub>2</sub> to characterize new (and simple) decoupling approximations K-I and K-II for the biaxial (including uniaxial) phase. In order to test the new against earlier proposed approximations rigorously, and to discuss consistency issues, we solve the Hess-Doi Fokker-Planck equation for nematic and nematic-discotic liquid crystals efficiently for a wide range of (2300 distinct) possible conditions including mixed shear and elongational flows, diverse field strengths, and molecular shapes. As a result, we confirm the closures K-I and K-II with correct tensorial symmetry; they are valid under arbitrary conditions to high precision, exact in the isotropic and totally aligned phases, improve upon earlier parameter-free closures in particular in the temperature regime T∈ [0.6, ∞] x T<sub>NI</sub> with the nematic-isotropic transition temperature T<sub>NI</sub> (or alternatively, for mean-field strengths U ∈ [0, 8]). K-II performs as good as the so-called Bingham closure, which usually requires 30 empirical coefficients, while K-I and K-II are essentially parameter-free, and their quality can be expected to be insensitive to the particular model.</div></front></TEI><affiliations><list><country><li>France</li><li>Suisse</li></country><region><li>Auvergne-Rhône-Alpes</li><li>Canton de Zurich</li><li>Rhône-Alpes</li><li>Île-de-France</li></region><settlement><li>Grenoble</li><li>Paris</li><li>Zurich</li></settlement></list><tree><country name="Suisse"><region name="Canton de Zurich"><name sortKey="Kroger, Martin" sort="Kroger, Martin" uniqKey="Kroger M" first="Martin" last="Kröger">Martin Kröger</name></region></country><country name="France"><region name="Auvergne-Rhône-Alpes"><name sortKey="Ammar, Amine" sort="Ammar, Amine" uniqKey="Ammar A" first="Amine" last="Ammar">Amine Ammar</name></region><name sortKey="Chinesta, Francisco" sort="Chinesta, Francisco" uniqKey="Chinesta F" first="Francisco" last="Chinesta">Francisco Chinesta</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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