Undulation instability under shear in smectic A liquid crystals

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Abstract
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 θ 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 θ 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