The Transition to Turbulence in Nematic Liquid Crystals

Abstract

Abstract The article is divided in two parts. In part I we review the present state of knowledge on the problem of the transition to turbulence in isotropic liquids and we discuss analogies/differences with the case of nematics. In the well known Rayleigh-Bénard case, experiments on ordinary liquids show that one must distinguish between confined and extended geometries (small or large “aspect ratios”). This difference is quite general and arises from the more or less large number of variables required to describe the structure. In confined geometry a Ruelle-Takens type of sequence seems relevant, with subharmonic bifurcation or intermittency as special cases. In extended geometry, slowly evolving imperfect structures (which may arise from an instability of the phase of rolls) result in a broad noise which can be viewed as a precursor of turbulence. In nematic liquid crystals (NLC) much less is known on the transition to turbulence. Nematic flows are verystrongly non-linear and are in large part controlled by the director field distortions. We review briefly the transition to turbulence in electrohydrodynamic (EHD) instabilities. It is shown to roughly follow the same type of scheme as ordinary Rayleigh-Btnard convection but mechanisms involved have not yet been unambiguously isolated. The particular role of the director field dynamics is more clearly seen in the case of flow instabilities. We recall the principal situations of interest, namely i) when the viscous torque coefficient a3 is positive and the director lies in the plane of velocity and velocity gradient so that “tumbling” can occur and ii) when the nematic in homeotropicgeometry issubmitted to anelliptical shear: the transition to turbulence then occurs through a gradual loss of spatial coherence analogous to some steps observed in the EHD case. The bibliography though already extended is far from complete. It is mainly experimental but may serve as a useful starting point fora reader who would be interested by a theoretical approach of the transition to turbulence in NLC. In part II we study more thoroughly the “tumbling” phenomenon and conjecture the statistical properties of a turbulent state which would bifurcate quasi directly from the laminar state in a nematic with α1 > 0 and aligned in the direction of the Poiseuille flow. A numerical simulation allows todetermine the “tumbling” threshold withas few simplifyingassumptionsas possibleand starting with Ericksen-Leslie equations. We show that the tumbling instability is first order-like and we give indications on its dynamics. Motion of defects that can nucleate from the “tumbled state” is studied qualitatively. This leads directly to some predictions for the kind of turbulencc obscrvcd by Gahwillcr in his experiment on HBAB.