Shear flow instability in nematic liquids : theory steady simple shear flows

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Abstract
We study shear flow instabilities in nematic liquids of planar geometry (molecules parallel to the plates limiting the sample) and where molecules are aligned (along Ox) perpendicular to the shear flow direction (along Oy). An external magnetic field reinforces the alignement imposed by surface effects. We develop a bidimensional model which takes into account exactly boundary conditions on the director and the velocity fields. The two types of instability which have been observed by Guyon and Pieranski (homogeneous distortion and convective rolls parallel to Oy) are interpreted within the framework of this model. We obtain instability thresholds together with the value of the critical wave vector qcx at threshold. Under weak fields the homogeneous distortion (qcx = 0) is achieved, whereas in high fields a roll instability develops (qcx ~ 0). The crossover from one regime to the other takes place at H ~ 1 kG. Moreover, we show that in either case, as far as the director is concer- ned, the aspect of the distortion along the z-direction is dominated by only one wave vector qz. Contrary to the case of the homogeneous distortion where qz varies strongly with H, for the roll instability qz remains nearly constant and close to 03C0/2a (2 a is the thickness of the sample). This allows us to develop an approximation which leads to results in good agreement with those obtained exactly.