  We analyze a solenoidal motion in a vertically vibrated freely suspended thin
smectic film. We demonstrate analytically that transverse oscillations of the
film generate two-dimensional vortices in the plane of the film owing to
hydrodynamic nonlinearity. An explicit expression for the vorticity of the
in-plane film motion in terms of the film displacement is obtained. The air
around the film is proven to play a crucial role, since it changes the
dispersion relation of transverse oscillations and transmits viscous stresses
to the film, modifying its bending motion. We propose possible experimental
observations enabling to check our predictions.
