Undulation instability under shear: A model to explain the different orientations of a lamellar phase under shear?

We study the effects of a dilation on a sheared smectic A phase. Through a linear analysis, we show that undulation may grow in the direction of the flow and of the vorticity as found in previous works. At higher shear rates, we evidence that the undulation along the flow disappears whereas it persists in the vorticity direction. We determine the stable or unstable zone as a function of the shear rate and of the lamellar spacing. This allows us to draw a theoretical shear diagram of the instability. Finally we compare our results with the orientations found experimentally in a lyotropic lamellar phase under shear. Our diagram describes qualitatively and quantitatively the transition observed at high sheer rate.