  Motivated by the variety of applications in which nematic Hele-Shaw flow
occurs, a theoretical model for Hele-Shaw flow of a nematic liquid crystal is
formulated and analysed. We derive the thin-film Ericksen-Leslie equations that
govern nematic Hele-Shaw flow, and consider two important limiting cases in
which we can make significant analytical progress. Firstly, we consider the
leading-order problem in the limiting case in which elasticity effects dominate
viscous effects, and find that the nematic liquid crystal anchoring on the
plates leads to a fixed director field and an anisotropic patterned viscosity
that can be used to guide the flow of the nematic. Secondly, we consider the
leading-order problem in the opposite limiting case in which viscous effects
dominate elasticity effects, and find that the flow is identical to that of an
isotropic fluid and the behaviour of the director is determined by the flow. As
an example of the insight which can be gained by using the present approach, we
then consider the flow of nematic according to a simple model for the squeezing
stage of the One Drop Filling method, an important method for the manufacture
of Liquid Crystal Displays, in these two limiting cases.
