Consistent closure schemes for statistical models of anisotropic fluids
Identifieur interne : 009753 ( Main/Merge ); précédent : 009752; suivant : 009754Consistent 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.
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<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>
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<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>
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<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>
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