Dip coating on a planar non-vertical substrate in the limit of negligible surface tension

The problem of dip coating of a planar non-vertical substrate is considered for negligible surface tension effects. As in the problem of vertical withdrawal (Cerro and Scriven, J. Fluid Mech. 208 (1980) 40), a singularity arises in the approximate steady-state equation governing the shape of the air-liquid interface; the ultimate thickness of the entrained film on the substrate follows directly from the elimination of this singularity. The removable singularity corresponds to a critical point that divides regions where waves propagate upstream and downstream (subcritical flow) and waves only propagate downstream (supercritical flow). These waves move at the speed of characteristics of the linearized time-dependent hyperbolic equation governing the film flow. The range of flows and film thicknesses that may be achieved by dip coating in an inclined substrate configuration are examined; these film thicknesses are self-metered, in that it they are determined by the substrate speed, substrate angle, and fluid properties. We also examine the parameter space to find flow conditions where the final uniform thickness of the film is not determined by a critical point. In these cases, the film thickness is set by some other means, such as in premetered die coating where both the flow rate and substrate speed are imposed. Examination of the asymptotic behavior of the film downstream shows that some of the possible uniform film thicknesses are not attainable. While there may be two possible uniform film thickness solutions for an imposed volumetric flow rate, there can be at most one supercritical and one subcritical flow solution. (C) 2001 Elsevier Science Ltd. All rights reserved.