Orientation instability of shear flow of a nematic liquid crystal

The problem of instability of shear flow of a nematic liquid crystal layer is considered. The case when the director and the velocity vector are parallel is studied. It is shown that the instability of this flow is possible in the case of the weak anchoring boundary condition and if the splay-bend constant in the Frank energy is taken into account. For this type of instability, periodical structures are possible to appear. Their wave vector is in plane of flow and is perpendicular to the velocity. An estimate for the medium constants is obtained from the existence condition for this instability. The wave number of periodical structures is evaluated.