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

TL;DR

This paper introduces the moment method using Generalized Collision Invariants to rigorously derive the Ericksen-Leslie model from the Doi-Navier-Stokes model of liquid crystals, accounting for non-uniform densities.

Contribution

It develops the GCI concept and applies it to derive the liquid crystal model limit in arbitrary dimensions with non-uniform densities, extending previous methods.

Findings

GCI relates to collision operator structures

Derivation valid in any spatial dimension

Includes effects of non-uniform polymer density

Abstract

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.