Flows in forward deformable roll coating gaps: Comparison between spring and plane-strain models of roll cover

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-roll coating gap, one is often covered by a layer of deformable elastomer. Thin films can be obtained without the risk of clashing two hard rolls. Liquid carried into the converging side of the gap can develop high enough pressure to deform the resilient cover, which changes the gap geometry and thus alters the velocity and pressure fields. The complete understanding of the flow in this situation is vital to the optimization of this widely used coating method; however, this elastohydrodynamic action is not well understood. The situation is similar to what is called the Soft-Elastohydrodynamic Lubrication regime (Soft EHL); however, the range of minimum distance between the rotating rolls, roll speed, and therefore flow rate through the gap in the roll coating process is one to three orders of magnitude larger than the typical values reported in previous work on Soft EHL. Earlier works on deformable roll coating analyzed the action with both the lubrication approximation and the full Navier-Stokes solution and different one-dimensional models of roll cover deformation. In order to test the accuracy of the past approaches, and to evaluate the relationship between the empirical constant used in the one-dimensional model to the relevant physical parameters, a complete, two-dimensional formulation has to be employed for both the liquid flow and the solid deformation. In this work, the flow between a rigid and a deformable rotating roll was examined by solving the complete Navier-Stokes system coupled with a non-linear plane-strain model of the roll cover deformation. The approximate and computationally cheaper approach is evaluated in which the compliant wall is represented by an array of radially-oriented Hookean springs. The equation system was solved by the Galerkin/finite element method; the resulting set of non-linear algebraic equations of the fully coupled problem was solved by Newton's method with initialization by pseudo-arc-length continuation as parameters were varied, Results show how roll deformation affects the total flow rate and forces on the rolls and illustrate how a deformable roll can be used to obtain thin coated layers with much less sensitivity to roll runout than those obtained with rigid rolls. (C) 1997 Academic Press.