Viscous flow in smectic A liquid crystals

Smectics A are fluid, layered, materials which obey Darcy's law for flow normal to the layers, and an equation of the Navier-Stokes type in the other two directions. The slow motions of various solid bodies inside a smectic are discussed, assuming that the layers are not distorted around the solid at rest. The flow pattern depends on a dimensionless parameter κL = S where L is the size of the object and κ−1 is a characteristic length of the smectic fluid. Only the limit S ≫ 1 is considered. Depending on the shape of the object and on the direction of motion, either a velocity field confined to the smectic planes, but not very different from the Stokes flow in viscous fluids, or a “permeation layer” of thickness δ ∼ κ−1/2L1/2 is found. In the latter case the drag is strongly enhanced. The idea of a permeation layer is also applied to a discussion of the undulation mode of a smectic sample between fixed plates, which has recently been studied by inelastic scattering of light.