Analysis of the effect of fast electric field reversal on smectic C* liquid crystals

For smectic C* (SmC*) liquid crystals, configured in a bookshelf-type geometry between two horizontal parallel plates, with the bottom plate fixed and the top plate free to move, it is known from experiment that pumping can occur when an electric field is applied, i.e. an upward movement of the top plate through mechanical vibrations when the electric field is suddenly reversed. In this paper we revisit an earlier mathematical model for fast electric field reversal by removing an assumption made there on the velocity field; instead, we arrive at a time-dependent, two-dimensional squeeze-film model, which can ultimately be formulated in terms of a highly nonlinear integro-differential equation. Subsequent analysis leads to an unexpected solvability condition involving the five SmC* viscosity coefficients regarding the existence and uniqueness of solutions. Furthermore, we find that, when solutions do exist, they imply that the plate can move down as well as up, with the final resting position turning out to be dependent on the initial conditions; this is in stark contrast to the results of the earlier model.