Deformable roll coating flows: steady state and linear perturbation analysis

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter a continuous liquid layer and to apply it to a continuous flexible substrate. Of the two rolls that make a forward or reverse roll coating gap, one is often covered by a layer of more-or-less deformable elastomer. Liquid carried into the converging side of the nip can develop high enough pressure to deform the resilient cover, which changes the nip profile and thus alters the velocity and pressure field. This elastohydrodynamic coupled action is not yet well understood. Theoretical modelling has to take into account the viscous flow, the roll deformation and the free-surface effects in order to predict the flow behaviour. In this work the flow between a rigid and a deformable counter-rotating roll that shares the same angular speed is described by the lubrication approximation together with a viscocapillary model, based in the film-flow equation, for the film-split region. The deformation of the elastomer layer is modelled by Hookean springs oriented radially, which constitute a one-dimensional model. The stability of the system to transverse perturbation is analysed by examining the time-dependent response to infinitesimal disturbances in order to identify those that grow fastest. The corresponding system of equations is solved by Newton's method with first-order continuation. The relationship between coating thickness, operational parameters (loading force, gap setting, roll velocities, etc.), liquid properties and the properties of the cover is reported, as well as the critical capillary number for onset of ribbing and wavelengths of the ribbing pattern predicted by the mathematical model. The results indicate how a deformable cover may be used in order to delay the onset of ribbing for a desired coating thickness. In order to validate the theoretical predictions of the viscocapillary/Hookean spring model, the symmetric film-split flows between a pair of rigid rolls and a pair consisting of a deformable roll and a rigid one were also analysed experimentally.