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Motivated by the industrial process of blade coating, the two-dimensional flow of a thin film of Newtonian fluid on a horizontal substrate moving parallel to itself with constant speed under a fixed blade of finite length in which the flows upstream and downstream of the blade are coupled via the flow under the blade is analysed. A combination of asymptotic and numerical methods is used to investigate the number and nature of the steady solutions that exist. Specially, it is found that in the presence of gravity there is always at least one, and (depending on the parameter values) possibly as many as three, steady solutions, and that when multiple solutions occur they are identical under and downstream of the blade, but differ upstream of it. The stability of these solutions is investigated, and their asymptotic behaviour in the limits of large and small flux and weak and strong gravity effects, respectively, determined

Motivated by the industrially important processes of blade coating and cavity filling of liquid crystalline materials, we consider steady, two-dimensional shear-driven (Couette) and pressure-driven (plane Poiseuille) flow of a thin film of a nematic liquid crystal in the slowly varying channel formed between a fixed blade of prescribed shape and a planar substrate. Specifically, blade coating motivates the study of shear-driven flow due to the motion of the substrate parallel to itself with constant velocity, while cavity filling motivates the study of pressure-driven flow due to an imposed pressure drop. We use a combination of analytical and numerical techniques to analyse the Ericksen--Leslie equations governing the fluid velocity and pressure and the director orientation in cases when both the aspect ratio of the channel and the distortion of the director field are small. We demonstrate a variety of flow and director-orientation patterns occurring in different parameter regimes. In the limit of weak flow effects flow alignment does not occur and the appropriate solution of the governing equations is found explicitly. In the limit of strong flow effects flow alignment occurs and orientational boundary layers exist near the substrate and near the blade, and in addition, an orientational internal layer may also exist within which the director orientation changes from +theta_0 to -theta_0 where theta_0 is the flow-alignment angle