A soft elastohydrodynamic lubrication model is formulated for deformable roll coating involving two contra-rotating rolls, one rigid and the other covered with a compliant layer. Included is a finite-strip model (FSM) for the deformation of the layer and a lubrication model with suitable boundary conditions for the motion of the fluid. The scope of the analysis is restricted to Newtonian fluids, linear elasticity/viscoelasticity and equal roll speeds, with application to the industrially relevant highly loaded or ‘negative gap’ regime. Predictions are presented for coated film thickness, inter-roll thickness, meniscus location, pressure and layer deformation as the control parameters – load (gap), elasticity, layer thickness and capillary number, $\hbox{\it Ca}$ – are varied. There are four main results: Hookean spring models are shown to be unable to model effectively the deformation of a compliant layer when Poisson's ratio $\nu\rightarrow 0.5$. In particular, they fail to predict the swelling of the layer at the edge of the contact region which increases as $\nu\rightarrow 0.5$; they also fail to locate accurately the position of the meniscus, $X_M$, and to identify the presence, close to the meniscus, of a ‘nib’ (constriction in gap thickness) and associated magnification of the sub-ambient pressure loop.Scaling arguments suggest that layer thickness and elasticity may have similar effects on the field variables. It is shown that for positive gaps this is true, whereas for negative gaps they have similar effects on the pressure profile and flow rate yet quite different effects on layer swelling (deformation at the edge of the contact region) and different effects on $X_M$.For negative gaps and $\hbox{\it Ca}\,{\sim}\,O(1)$, the effect of varying either viscosity or speed and hence $\hbox{\it Ca}$ is to significantly alter both the coating thickness and $X_M$. This is contrary to the case of fixed-gap rigid roll coating.Comparison between theoretical predictions and experimental data shows quantitive agreement in the case of $X_M$ and qualitive agreement for flow rate. It is shown that this difference in the latter case may be due to viscoelastic effects in the compliant layer.

Two-dimensional flow is considered in a fluid bead located in the gap between a pair of contra-rotating cylinders and bounded by two curved menisci. The stability of such bead flows with two inlet films, and hence no contact line, are analysed as the roll speed ratio S is increased. One of the inlet films can be regarded as an 'input flux' whilst the other is a 'returning film' whose thickness is specified as a fraction ζ of the outlet film on that roll. The flow is modelled via lubrication theory and for Ca ≪ 1, where Ca represents the capillary number, boundary conditions are formally developed that account for S 1 and the non-constant gap. It is shown that there is a qualitative difference in the results between the single and double inlet film models unless small correction terms to the pressure drops at the interfaces are taken into account. Futhermore, it is shown that the inclusion of these small terms produces an O(1) effect on the prediction of the critical value of S at which bead break occurs. When the limits of the returning film fraction are examined it is found that as ζ → 0 results are in good agreement with those for the single inlet film. Further it is shown for a fixed input flux that as ζ → 1 a transition from bead break to upstream flooding of the nip can occur and multiple two-dimensionally stable solutions exist. For a varying input flux and fixed and 'sufficiently large' values of ζ there is a critical input flux λ(ζ) such that as S is increased from zero: (i) bead break occurs for λ<λ;(ii) upstream flooding occurs for λ>λ;(iii) when λ = λ the flow becomes neutrally stable at a specific value of S beyond which there exist two steady solutions (two-dimensionally stable) leading to bead break and upstream flooding, respectively.