Forward roll coating flows of viscoelastic liquids

Except at low speed, the film splitting flow that occurs in forward roll coating is three-dimensional and results in more or less regular stripes in the machine direction. For Newtonian liquids, the stability of the film-split flow is determined by the competition of capillary forces and viscous forces: the onset of meniscus nonuniformity is marked by a critical value of the capillary number. Although most of the liquids coated industrially are polymeric solutions and dispersions that are not Newtonian, most of previous theoretical analyses of film splitting flows dealt only with Newtonian liquids. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case. This free surface coating flow is analyzed here with differential constitutive models, the Oldroyld-B and the FENE-P equations. The continuity and momentum equations coupled with the constitutive models, and the non-linear mapping equations that transform the free boundary problem into a fixed boundary problem are solved with the DEVSS-G/SUPG method with finite element basis functions. The resulting set of non linear equations is solved by Newton's method with pseudo-arc-length continuation. The results show how the flow field changes with Weissenberg number, leading to changes in the forces at a free surface that may explain why the ribbing instability sets in at smaller Capillary number when the liquid is viscoelastic.