  Surface roughness significantly modifies the liquid film thickness entrained
when dip coating a solid surface, particularly at low coating velocity. Using a
homogenization approach, we present a predictive model for determining the
liquid film thickness coated on a rough plate. A homogenized boundary condition
at an equivalent flat surface is used to model the rough boundary, accounting
for both flow through the rough texture layer, through an interface
permeability term, and slip at the equivalent surface. While the slip term
accounts for tangential velocity induced by viscous shear stress, accurately
predicting the film thickness requires the interface permeability term to
account for additional tangential flow driven by pressure gradients along the
interface. We find that a greater degree of slip and interface permeability
signifies less viscous stress that would promote deposition, thus reducing the
amount of free film coated above the textures. The model is found to be in good
agreement with experimental measurements and requires no fitting parameters.
Furthermore, our model may be applied to arbitrary periodic roughness patterns,
opening the door to flexible characterization of surfaces found in natural and
industrial coating processes.
