Nonlinear continuum theory of smectic-\(A\) liquid crystals
type: Item
instance of: scholarly article
title: Nonlinear continuum theory of smectic-\(A\) liquid crystals
zbMATH Open document ID: 0891.76009
published in: Archive for Rational Mechanics and Analysis
publication date: 1998-07-19T00:00:00Z
review text: This paper presents a general nonlinear continuum theory for smectic-\(A\) liquid crystals. In particular, the author seeks to incorporate into the model a unique permeation process which is responsible for much of the unusual behaviour observed in these layered structures. Adopting a form of the free energy \(f\) proposed by \textit{M. Kleman} and \textit{O. Parodi} [J. Physique 36, 671-681 (1975)], namely \(2f= B(|\nabla\phi|- 1)^2+ k(\nabla\cdot {\mathbf n})^2\), where the isosurfaces of a layer variable \(\phi\) parametrize the smectic layers, \({\mathbf n}\) is the unit director and \(B\) and \(k\) are material constants, the author proceeds to present the hydrostatic part of the theory. A computation of the variation of \(f\) with respect to the layer position without deforming the material and the deformation of the material without altering the position of the layers leads to expressions for the `permeation force' and the static stress, respectively. To derive the hydrodynamic equations, the author starts with a set of basic variables consisting of density, velocity \({\mathbf u}\), total energy density, and layer variable \(\phi\). Then the familiar balance equations and entropy production inequality are written down together with an equation for \(\phi\). Restricting attention to a linear constitutive theory, the author proceeds to derive the final form of the equations for compressible materials by utilizing the parity of various quantities under time reversal. It is observed that in the isothermal case, the system of equations is among the simplest for nonlinear anisotropic fluids. An appropriate set of equations in the incompressible limit is then derived by two distinct methods. One employs the constraints on the fluid velocity \((\text{div }{\mathbf u}= 0)\) and the smectic layers \((|\nabla\phi|= 1)\) but otherwise follows the lines indicated for the compressible case; the other uses a systematic asymptotic analysis of the equations for a compressibl…
Mathematics Subject Classification ID: 76A15
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 1045412
zbMATH Keywords: permeation process
zbMATH Keywords: free energy
zbMATH Keywords: layer variable
zbMATH Keywords: static stress
zbMATH Keywords: hydrodynamic equations
zbMATH Keywords: balance equations
zbMATH Keywords: entropy production inequality
zbMATH Keywords: incompressible limit
zbMATH Keywords: asymptotic analysis
zbMATH Keywords: compressible material
zbMATH Keywords: compression modulus
zbMATH Keywords: linear stability analysis
author: Weinan E
reviewed by: P. J. Barratt
MaRDI profile type: Publication
full work available at URL: s002050050026
OpenAlex ID: W2085607931
DOI: 10.1007/S002050050026
description: scientific article; zbMATH DE number 1045412