Existence and stability of kayaking orbits for nematic liquid crystals in simple shear flow
Existence and stability of kayaking orbits for nematic liquid crystals in simple shear flow
We use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits in response to steady shear flow. These are orbits for which the principal axis of orientation of the molecular field (the director) rotates out of the plane of shear and around the vorticity axis. With a small parameter attached to the symmetric part of the velocity gradient, the problem can be viewed as a symmetry-breaking bifurcation from an orbit of the rotation group SO (3) that contains both logrolling (equilibrium) and tumbling (periodic rotation of the director within the plane of shear) regimes as well as a continuum of neutrally stable kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.
1229-1287
Chillingworth, David
39d011b7-db33-4d7d-8dc7-c5a4e0a61231
Forest, M. Gregory
85996fea-52e1-4c12-a957-8efcdea212b5
Lauterbach, Reiner
f541c8b3-23d9-4282-b431-7e01e3060937
Wulff, Claudia
08fbf431-3386-4a1a-b5c7-ec1416f18486
November 2021
Chillingworth, David
39d011b7-db33-4d7d-8dc7-c5a4e0a61231
Forest, M. Gregory
85996fea-52e1-4c12-a957-8efcdea212b5
Lauterbach, Reiner
f541c8b3-23d9-4282-b431-7e01e3060937
Wulff, Claudia
08fbf431-3386-4a1a-b5c7-ec1416f18486
Chillingworth, David, Forest, M. Gregory, Lauterbach, Reiner and Wulff, Claudia
(2021)
Existence and stability of kayaking orbits for nematic liquid crystals in simple shear flow.
Archive for Rational Mechanics and Analysis, 242 (2), .
(doi:10.1007/s00205-021-01703-x).
Abstract
We use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits in response to steady shear flow. These are orbits for which the principal axis of orientation of the molecular field (the director) rotates out of the plane of shear and around the vorticity axis. With a small parameter attached to the symmetric part of the velocity gradient, the problem can be viewed as a symmetry-breaking bifurcation from an orbit of the rotation group SO (3) that contains both logrolling (equilibrium) and tumbling (periodic rotation of the director within the plane of shear) regimes as well as a continuum of neutrally stable kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.
Text
Chillingworth2021_Article_ExistenceAndStabilityOfKayakin
- Version of Record
More information
Accepted/In Press date: 6 August 2021
e-pub ahead of print date: 7 September 2021
Published date: November 2021
Additional Information:
Funding Information:
This collaboration arose during a workshop at the Mathematics of Liquid Crystals Programme at the Isaac Newton Institute in Cambridge in 2013 where the problem of existence and stability of the kayaking orbit was raised by GF, remaining open in spite of decades of overwhelming numerical evidence together with convincing experimental evidence. An active discussion followed and co-authors DC, RL and CW continued to work, with intermittent exchanges with GF, toward the resolution presented here. The research was supported by the Isaac Newton Institute, Cambridge and (DC) a Leverhulme Emeritus Research Fellowship; in addition CW was grateful to the Free University Berlin for hospitality. The authors also express thanks to Jaume Llibre for helpful conversations about higher-order averaging, and to Stefano Turzi for valuable input concerning invariants.
Publisher Copyright:
© 2021, The Author(s).
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
Identifiers
Local EPrints ID: 453452
URI: http://eprints.soton.ac.uk/id/eprint/453452
ISSN: 0003-9527
PURE UUID: 769918a8-c4f3-4c7e-b50e-6ac94c9894bc
Catalogue record
Date deposited: 17 Jan 2022 17:50
Last modified: 11 Sep 2024 17:23
Export record
Altmetrics
Contributors
Author:
M. Gregory Forest
Author:
Reiner Lauterbach
Author:
Claudia Wulff
Download statistics
Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.
View more statistics