Thin-Film Drainage Becomes Singular at Saddles

From coating curved parts to creeping lava, thin-film gravitational drainage over complex surfaces can produce abrupt accumulation, usually associated with sharp edges or defects. Here, experiments, simulations, and asymptotics show that a smooth saddle alone generates a logarithmically singular ridge, regularized within a dynamic region where drainage balances hydrostatic pressure and capillarity. In a Newtonian lubrication picture, this mechanism can inform redistribution in manufacturing and geophysics, motivating extensions to complex rheology and interacting geometric features.