A model based on the lubrication approximation is put forward for the general case of asymmetric forward roll coating of Newtonian liquids. Two more-rigorous theories are developed, one based on asymptotic expansions for small ratios of gap-to-roll diameter (H0/R), the second on Galerkin/finite-element solutions of the full Navier–Stokes equations over the relevant flow domain. The lubrication model is useful only as an approximation at high capillary numbers $(Ca\equiv \mu\overline{V}/\sigma) $. The asymptotic analysis is accurate when H0/R < 0.001 and Ca > 0.1. The ratio of the film thicknesses on the two rolls is predicted to equal the speed ratio to the 0.65 power, which is confirmed experimentally. The Galerkin/finite-element solutions give full details of the steady two-dimensional free-surface flows including complex recirculation patterns in the film-splitting region, and show how the film-splitting stagnation line becomes a static contact line in the limit as one roll surface becomes stationary.

The ribbing instability, an extremely common cause of non-uniform liquid films in coating operations, is investigated both theoretically and experimentally. The Navier–Stokes system for the two-dimensional flow in symmetric film-splitting in forward roll coating is solved by finite-element analysis. Stability of the flow with respect to three-dimensional disturbances is examined by applying linear stability theory in a consistent finite-element approach, taking Fourier components in the transverse direction. The resulting generalized asymmetric eigenproblem is solved for the growth rates of disturbances as functions of wavenumber. The theory accurately predicts the critical capillary number and wavenumber at the transition to large-amplitude ribs. A sensitive experimental technique for detecting the ribs was developed that relies on low-angle reflection of a focused strip of white light off the meniscus between the rolls. This allowed detection of much smaller amplitude ribs, and much smaller critical capillary numbers were measured. The results indicate that the transition to ribbing is an imperfect bifurcation due to end effects, and clarify earlier discordances in the literature.

The flow in the metering gap of a reverse roll coater is examined by experiments and finite element solutions of the Navier-Stokes equations. At high speed ratios and capillary numbers, the metered film flow deviates strongly from predictions of lubrication theory: the wetting line moves through the gap center and the metered film thickness passes through a minimum. The two flow instabilities found are ribbing, a sinusoidal cross-web waviness extending smoothly down-web; and cascade, an irregular V-shaped cross-web wave, repeated quasiperiodically down-web. Experimental operability diagrams define parameter ranges where these instabilities and the steady two-dimensional flow are encountered. Ribbing behavior is understood by consideration of the pressure gradient at the free surface. The mechanism of cascade is the intrusion through the gap of the wetting line, which causes the metered film to thicken and eventually reattach to the metering roll in a cyclical manner.