Instabilities in Roll and Slot Coating Flows

This thesis is concerned primarily with the investigation, both analytically and numerically, of instabilities in reverse- and forward-roll and slot coating flows. Consideration is restricted to Newtonian, incompressible fluids in the absence of inertial forces. The onset of the ribbing instability in inlet flooded, reverse roll coating is examined first by applying linear stability theory to a base flow formulated using lubrication theory. Regions of instability are established and found to be in accord with experimental observation. These results are then compared with predictions from a stability hypothesis based on a two-dimensional force balance argument. This simple criterion shows that the effect of various parameters on the stability of the downstream free surface can be ascertained by examining their influence on the pressure gradient and meniscus location. The stability hypothesis is shown to underpredict the critical capillary number and so is only sufficient for predicting stability. Results are also compared to ones obtained numerically by applying linear stability theory to finite element solutions for the entire flow field, the principal difference being that the analytical approach overpredicts the critical capillary number for the onset of instability. A variation of inlet flooded, reverse roll coating is then studied in which the nip is fed from above by a large reservoir of fluid (i.e. a hydrostatic head). The influence of this head on the base flow (obtained using lubrication theory) and its stability is then investigated. An improved model of the dynamic contact line, developed by Shikhmurzaev [1993a] is described in which the dynamic contact angle is no longer kept constant, but is a function of various fluid and geometrical parameters. The limit of this theory for small capillary number is incorporated into the analytical model from which its effect on the base flow and stability is examined. Instabilities in forward roll coating are then investigated. The inlet flooded case is studied using linear stability analysis, a stability hypothesis and the finite element method. As in reverse roll coating, the stability hypothesis at the downstream free surface is sufficient for predicting stability only. The finite element method, on the other hand, leads to solutions that are in close agreement with linear stability theory, unlike the reverse roll case. Inlet starved forward roll coating is examined next and, as with the inlet flooded case, the ribbing instability can still manifest at the downstream free surface. The presence of an instability known as bead break, noted experimentally by Malone [1992] and Gaskell et al [1998], is verified analytically using linear stability theory. It is then shown that a stability hypothesis applied at the upstream free surface gives an accurate description of stability (unlike at the downstream free surface). Finally, the slot coating geometry is explored. A geometrically flexible finite element code is described for which it is possible to use various lip shapes and a roll of variable radius and location (with respect to the slot). Initially, as has always been the case in previous work reported in the literature, the numerical mesh incorporates a downstream wetting line pinned at the lip edge and the effect of the various fluid and geometrical parameters on the resultant pressure profile and upstream meniscus location is examined. These numerical results are seen to compare favourably with predictions obtained analytically using a model based on lubrication theory. However, Sartor [1990] showed that the downstream wetting line does not always remain pinned, but can climb up the die shoulder or regress into the coating gap. This has been confirmed experimentally by Kapur [1998] who also notes that ribs only appear when operating in the unpinned regime. Hence, the case in which the downstream, static contact line has retreated from the lip edge towards the inlet is studied. Pressure profiles and meniscus locations are compared with those for the case of a pinned downstream wetting line and the numerical linear stability analysis used to determine the effect of the fluid and geometrical parameters on the stability of the downstream free surface.