Undulation instability under shear in smectic A liquid crystals

2014 We study the undulation instability in a smectic A in the presence of shear flow parallel to the layers. We look for the dilation at the threshold where an undulation appears with a wave vector inclined at an angle 03B8 to the direction of flow. We find an integral relation showing that the dilation is necessarily positive. We develop a perturbation method to calculate the dilation explicitly as a function of the angle 03B8 and the shear rate. We show that an undulation with wave vector perpendicular to the shear direction is not affected by the flow and has a minimum threshold dilation equal to the static value. We conclude that a rectangular texture develops at higher dilations in agreement with the rectangular focal domain pattern observed experimentally. J. Physique 43 (1982) 1193-1197 AOÛT 1982, Classification Physics Abstracts 47.15 61.30 Introduction. We deal with a smectic A liquid crystal subjected to a dilative strain perpendicular to the layers. Above a certain threshold value of the dilation, the layers develop a static undulation with some wave vector q of arbitrary orientation within a plane parallel to the layers [1, 2, 3, 4]. We want to ask and find an answer to the question : what happens to the undulation and how is the threshold modified when the sample is dilated while being simultaneously sheared parallel to the layers. 1. Formulation of the problem Consider a smectic sample of thickness 2 d sheared betwen two plane parallel plate moving at equal and opposite velocities ± Vo. Homeotropic anchoring keeps the smectic layers parallel to the plates (Fig. 1). The flow is described by the equation of Navier-Stokes [5] : where V is the velocity field, P the pressure, and G = (0, 0, G) is a volume force density due to the Fig. 1. Sample geometry. smectic layer elasticity. Equation (1) is valid under the assumptions that all viscosities can be replaced by a single coefficient ?1, that the mass density p is constant and thus the incompressibility condition can be written as : Let u be the displacement of the layers in the z-direction ; then the elastic free energy density F can be written as [6] : Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043080119300