ROLL-CELL INSTABILITIES IN SHEARING FLOWS OF NEMATIC POLYMERS

Manneville and Dubois-Violette's linear stability analysis of the roll-cell instability in nematics is extended to account for the effect of finite rates of roll-cell growth. The extended theory is applied to shearing flows of tumbling liquid crystalline polymers, in which the Ericksen number is typically a decade or more above the critical value for formation of the roll cells. It is found that, starting from a monodomain with director oriented perpendicular to both the flow and the flow gradient, at asymptotically high Ericksen numbers a wide band of wave numbers is unstable and the width l(m) of the fastest growing roll cells is proportional to gamma-1/4D1/2, where gamma is the shear rate and D is the gap between the shearing surfaces. This prediction of thin roll cells whose width depends only weakly on shear rate is consistent, in part, with roll-cell observations of Srinivasarao and Berry for poly(1,4-phenylene-2,6-benzobisthiazole), and with characteristic texture spacings measured in sheared poly(gamma-benzyl-glutamate) solutions. These results suggest that the roll-cell instability may provide a mechanism for texture refinement and director turbulence in shearing flow of tumbling nematics.