Fingering instabilities in driven thin nematic films

Motivated by experimental work (Cazabet et al., unpublished), we consider the possibility of fingering instabilities in thin films of nematic liquid crystals. We use lubrication theory on the flow equations for nematic liquid crystals to derive a simple model describing the evolution of the film height. As far as we are aware, this is the first time such a systematically derived, time-dependent thin film model for nematics has been presented. Simple "leading-order" solutions (depending on only one spatial coordinate) are found for two different flow driving mechanisms: (i) gravity perpendicular to the film and (ii) gravity parallel to the film (capillarity is also included in both cases). The effect of imposing two-dimensional perturbations to these solutions is studied. We find that for case (i) instability is possible, depending on whether or not there is complete wetting (i.e., whether or not the equilibrium contact angle of the droplet with the substrate is zero). For case (ii) we always have instability, as we would expect from the analogous result for Newtonian fluids [Phys. Fluids 8, 460 (1996); Europhys. Lett. 10, 25 (1989)].