Feedback control of flow alignment in sheared liquid crystals

TL;DR

This paper demonstrates how time-delayed feedback control can stabilize flow alignment in sheared liquid crystals, transforming oscillatory director dynamics into a steady state under specific control parameters.

Contribution

It introduces a novel application of TDFC to stabilize flow alignment in liquid crystals, providing exact conditions for control parameter selection.

Findings

TDFC successfully stabilizes flow alignment.

Small control strength suffices for isotropic initial states.

Control parameters are determined by solving an eigenvalue equation.

Abstract

Based on a continuum theory, we investigate the manipulation of the non-equilibrium behavior of a sheared liquid crystal via closed-loop feedback control. Our goal is to stabilize a specific dynamical state, that is, the stationary "flow-alignment", under conditions where the uncontrolled system displays oscillatory director dynamics with in-plane symmetry. To this end we employ time-delayed feedback control (TDFC), where the equation of motion for the ith component, q_i(t), of the order parameter tensor is supplemented by a control term involving the difference q_i(t)-q_i(t-\tau). In this diagonal scheme, \tau is the delay time. We demonstrate that the TDFC method successfully stabilizes flow alignment for suitable values of the control strength, K, and \tau; these values are determined by solving an exact eigenvalue equation. Moreover, our results show that only small values of K are needed when the system is sheared from an isotropic equilibrium state, contrary to the case where the equilibrium state is nematic.