From kinetic to fluid models of liquid crystals by the moment method

Abstract

<p style='text-indent:20px;'>This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.</p>

A Molecular-Based Q-Tensor Hydrodynamic Theory of Smectic Liquid Crystals

“…By the Hilbert expansion, Li and Wang [27] rigorously justified the connection between the inertial Qian-Sheng theory and the full Ericksen-Leslie theory. Based on the moment method, Degond, Frouvelle and Liu [28] presented the convergence of the Doi-Onsager theory to the Ericksen-Leslie theory. These works were mainly concerned with the hydrodynamic theory of nematic phases.…”