Arnold tongues in liquid crystals under a periodic shear flow

Lucia Russo¹, Pier Luca Maffettone² and Silvestro Crescitelli²

¹ University of Salerno, Department of Food and Chemical Engineering,

via Ponte don Melillo I-84084, Fisciano (SA), Italy

² University Federico II di Napoli, Department of Chemical Engineering,

Piazzale Tecchio 80, 80125, Napoli, Italy

It is well known that polymers in liquid crystalline nematic phase exhibit a complex rheological behavior. Under steady shear flow and in homogeneous conditions, theoretically and experimental studies have proved the existence of a variety of periodic regimes as well as aperiodic behaviour. In particular, periodic regimes have been predicted both with mesoscopic models for the second moments of the orientation distribution and with continuous kinetic models based on a Smoluchowski equation.

In this work we study the influence of a parallel sinusoidal field superimposed on top of steady shear. The mesoscopic model adopted to describe the liquid crystalline nematic phase of finite aspect-ratio rodlike molecules is based on a quadratic closure capable of reproducing the complex dynamics predicted by the kinetic theory. The analysis is performed through bifurcation analysis applying a parameter continuation technique to the Poincare map of the system. When the unforced system is in a periodic regime, the interaction of the forcing frequency with the natural one gives rise the frequency locking phenomena. We delimited parameter regions (Arnold tongues) in the plane of forcing parameters (amplitude and frequency) where periodic frequency locked regimes can been found. Finally we discuss how periodic shear flow experiments can be performed to give more insights on the dynamics of the liquid crystals under steady state shear flow.