This thesis is concerned with developing suitable models for the investigation of instabilities in rigid and deformable roll coating flows. Consideration is restricted to Newtonian, incompressible fluids in the absence of inertia. In each coating flow analysed the steady two dimensional base state solutions of the system axe explored before the stability of the system is considered.\ud \ud The particular rigid roll coating flow that is studied in this thesis is the flow formed between an intermediate pair of contra-rotating rolls within a multiple roll coater. The\ud base state is modelled using lubrication theory and the flow domain is terminated with simple boundary conditions. It is found that the form of the meniscus location\ud solutions differ from those found in similax flows by previous workers and this motivates a detailed re-examination of the boundaxy conditions. A matched asymptotic analysis applicable for small capillaxy numbers Ca and small inter-roll gaps gives the leading order correction to the boundary conditions and it is found that the inclusion of the correction terms alter the results significantly. It is found that a critical roll speed ratio,\ud Sc exists beyond which no steady two dimensional solutions exist. A simple stability hypothesis predicts the upstream meniscus to be neutrally stable at Sc and this point is\ud associated with bead break. Under certain operating conditions the stability hypothesis predicts multiple steady states.\ud \ud The lineax stability of this multiple roll coating flow is investigated. Special attention is given to the relationship between the wavelength of the disturbance and\ud the form of the boundary conditions. The main predictions of the stability hypothesis axe confirmed and an additional ribbing instability on the downstream meniscus is predicted.\ud \ud The base flow of a deformable roll coater is investigated using lubrication theory for the fluid and a linear elastic plain strain model for the compliant layer. The boundary\ud conditions developed for the multiple roll problem are extended to higher values of Ca. A finite element method is developed to numerically solve the governing equations.\ud The effects of Youngs modulus E and layer thickness L on the steady state operation of the coater are investigated. Previous workers have used spring models to describe\ud the compliant layer with the implicit assumption that E and L-1 have the same affect. Here it is shown that E and L-1 have a different effect upon the meniscus location in the negative gap regime and hence the underlying assumption of all spring models is shown to be incorrect. The plain strain model is extended to include viscoelastic terms\ud and it is shown that these terms can account for the discrepancy between experimental results and previous steady state elastic theories.\ud \ud The lineax stability of the deformable roll coater is investigated with the plain strain model being extended to account for lateral disturbances. The effect of E and L-1 on\ud the stability of the system is investigated and it is found that in keeping with the steady state results, they have a different effect on the stability of the system in the\ud negative gap regime. It is demonstrated that perturbations to the compliant layer play a negligible role in the stability analysis and it is shown how the viscoelastic extension to the base state can be incorporated
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