Layer dynamics of a freely standing smectic-A film

We study the hydrodynamics of a freely-standing smectic-A film in the isothermal, incompressible limit theoretically by analyzing the linearized hydrodynamic equations of motion with proper boundary conditions. The dynamic properties for the system can be obtained from the response functions for the free surfaces. Permeation is included and its importance near the free surfaces is discussed. The hydrodynamic mode structure for the dynamics of the system is compared with that of bulk systems. We show that to describe the dynamic correlation functions for the system, in general, it is necessary to consider the smectic layer displacement $u$ and the velocity normal to the layers, $v_z$, together. Finally, our analysis also provides a basis for the theoretical study of the off-equilibrium dynamics of freely-standing smectic-A films.