Simiple Shear Flow of a Non-Aligning Nematic
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SXMIPLE SHEAR FLOW OF A NON-ALOGNHING NEMA1IC W. H. HAN and A. D. REY
Department of Chemical Engineering, McGill University 3480 University Street, Montreal, Quebec, Canada H3A 2A7
ABSTRACT This paper presents a nonlinear numerical analysis of the orientational instabilites, textures, and flow patterns of a characteristic rod-like nematic liquid crystal in steady simple shear flow. The parameter vector V=(A,E) is given by the reactive parameter (A) and the Ericksen number (E). There are two stationary solutions: in-plane(IP) and out-of-shear-plane(OP), according to whether the average orientation lies in or out of the plane of shear. For a given A the in-plane stationary solutions may undergo a continuous transition (second order) to two dissipatively equivalent OP solutions when E=E. or a discontinuous transition (first order) to a highly distorted IP solution when E=Ej. The continuous transition at E=E. is a supercritical (pitchfork) bifurcation and the discontinuous transition at E=Ei is a limit point instability. Nonlinear numerical analysis shows that for the studied liquid crystal E. < Ej. The orientation phase diagram for stable stationary solutions therefore consist of a region of OP solutions separated from a region of IP solutions by the curve V. = (A,E,), describing a set of continuous supercritical bifurcations. These stable OP solutions are characterzied by the absence of sharp splay and bend deformations and by the presence of complex secondary flows arising from the three dimensional orientation. Presented results also include the dependence of the three dimensional texture, primary and secondary velocity fields on E.
1[NTRODUCTION The dynamics of nematic liquid crystals[l have complex features due to the coupling of the average molecular orientation, defined by a unit vector the director, and the velocity field. The two possible orientation modes under a steady shear deformation, aligning and nonaligning are characterized by the reactive parameter A[l]. When A > 1, simple shear causes homogeneous director alignment close to the flow direction at which viscous torques vanish. If 0 < A < 1 the viscous torques no longer vanish, and balancing elastic torques always arise. In this case, Manneville[2] showed IP solutions are characterized by a cascade of limit point instabilities at a set of critical Ejj where i denotes the IP solution and n denotes the nth transition; the solutions are characterized by high elastic curvature strains. Zuniga and Leslie[3], using linear stability theory, have shown that the IP solutions are unstable to OP perturbations for E < Eij. In this paper we show that for the material parameters of a characteristic nematic liquid crystal and parallel boundary conditions, two equivalent OP solutions bifurcate from the IP solution branch before any IP instability takes place. In contrast to the highly distorted textures of the IP solutions, the OP solution exhibit relatively weak elastic distorsions. The less distorted texture of the OP solutions is preferable to the highly distort
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