Existence and stability of kayaking orbits for nematic liquid crystals in simple shear flow

TL;DR

This paper proves the existence and stability of kayaking orbits in nematic liquid crystals under shear flow, using geometric dynamical systems methods to analyze symmetry-breaking bifurcations.

Contribution

It provides a rigorous proof of kayaking orbit existence and stability, addressing a long-standing open problem in nematic liquid crystal theory.

Findings

Existence of kayaking periodic orbits established

Stability of these orbits demonstrated

Bifurcation analysis from symmetric states performed

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 for which the principal axis of orientation of the molecular field (the director) rotates around the vorticity axis in response to steady shear flow. 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 kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.