Three-Dimensional Stability Analysis of Free Surface Flows: Application to Forward Deformable Roll Coating

Coating flows, with a few exceptions, need to be steady and two-dimensional flows. Moreover, the flow states need to be stable at the operating conditions chosen. The goal of stability analysis of coating flows is to determine the region in the parameter space at which the flow is stable and therefore the coated layer uniform. To determine the stability of liquid flows, a generalized eigenvalue problem has to be solved. This paper describes a formulation for a linear, three-dimensional stability analysis of free surface flows that reduces the size of the eigenproblem, decreasing the computational cost, with no further simplification, when compared with the methods reported in the literature. This formulation is used to study the instability that arises in film-splitting flows between counter-rotating rolls in a deformable gap. This flow instability leads to nonuniform coating characterized by a wavy thickness profile in the transverse direction. This patterning is usually referred to as “ribs.” This type of instability has received a lot of attention in the literature. However, all previous work has addressed the flow between two rigid rolls. Often, in practice, one of the rolls of a pair is covered by a layer of elastomer. The deformation of the roll cover alters the conformation of the gap, the pressure gradient at the film-split meniscus, and, consequently, the critical parameters at the onset of ribbing change. The results indicate how a deformable cover can be used to delay the onset of ribbing in forward-roll coating.