Three-dimensional ribbing instability in symmetric forward-roll film-coating processes

We propose a framework for interpreting the formation, evolution, and spatial persistence of ribbing in coating processes, and present companion 'quantifying' parallel spectral element simulations of fully nonlinear unsteady three-dimensional free-surface symmetric forward-roll film-coating fluid flows. The framework couples, by means of a transition region, two well-understood phenomena: the 'viscous fingering' instability of a splitting meniscus; and the levelling of viscous films under the effect of surface tension. The transition-region length, L-t, is on the order of the coating film thickness, while the downstream extent of the levelling region - the distance over which ribs persist - L-l, depends on the fluid properties, the flow conditions, and the wavenumber content of the nonlinear meniscus rib profile. Numerical results are presented for the evolution of the coating flow from perturbed unstable two-dimensional steady states to three-dimensional saturated ribbed states for several representative supercritical capillary numbers, Ca, and spanwise periodicity lengths, b. Nonlinear state selection is briefly discussed.