Three mathematical models are developed for meniscus roll coating in which there is steady flow of a Newtonian fluid in the narrow gap, or nip, between two contrarotating rolls in the absence of body forces.The zero flux model predicts a constant pressure gradient within the central core and two eddies, each with an inner structure, in qualitative agreement with observation. The small flux model takes account of a small inlet flux and employs the lubrication approximation to represent fluid velocity as a combination of Couette and Poiseuille flows. Results show that the meniscus coating regime is characterized by small flow rates (λ [Lt ] 1) and a sub-ambient pressure field generated by capillary action at the upstream meniscus. Such flows are found to exist for small modified capillary number, Ca(R/H0)1/2 [lsim ] 0.15, where Ca and R/H0 represent capillary number and the radius to semi-gap ratio, respectively.A third model incorporates the full effects of curved menisci and nonlinear free surface boundary conditions. The presence of a dynamic contact line, adjacent to the web on the upper roll, requires the imposition of an apparent contact angle and slip length. Numerical solutions for the velocity and pressure fields over the entire domain are obtained using the finite element method. Results are in accord with experimental observations that the flow domain consists of two large eddies and fluid transfer jets or ‘snakes’. Furthermore, the numerical results show that the sub-structure of each large eddy consists of a separatrix with one saddle point, two sub-eddies with centres, and an outer recirculation.Finally finite element solutions in tandem with lubrication analysis establish the existence of three critical flow rates corresponding to a transformation of the pressure field, the emergence of a ‘secondary snake’ (another fluid transfer jet) and the disappearance of a primary snake.

A two-roll apparatus is used to explore experimentally the detailed fluid mechanics of meniscus roll coating in which inlets are starved and flow rates are small. Both forward and reverse modes of operation (with contra-and co-rotating rolls) are investigated using optical sectioning combined with dye injection and particle imaging techniques. That part of parameter space where meniscus coating occurs is identified by varying the roll separation and roll speeds and hence flow rate and capillary number.Key features of the flow structures identified in the forward mode include two large eddies (each with saddle point, separatrix and sub-eddies), a primary fluid transfer jet and the existence of two critical flow rates associated with the switching-on of a second fluid transfer jet and the switching-off of the primary transfer jet followed by a change in the flow structure. In the reverse mode, the key features are a single large eddy consisting of two sub-eddies, a saddle point and separatrix, a primary fluid transfer jet and once again two critical flow rates. These correspond to (i) the switching-on of a secondary transfer jet and (ii) the disappearance of a saddle point at the nip resulting in the merger of the primary and secondary transfer jets.Measurements of film thickness and meniscus location made over a range of speed ratios and capillary numbers are compared with theoretical predictions. A plate-roll apparatus is used to confirm the presence, for very small flow rates, of a sub-ambient, almost linear, pressure profile across the bead. Investigated also is the transition from inlet-starved to fully flooded roll coating as flow rate is increased and the changes in flow structure and pressure profile are observed.

A finite element formulation of the Navier—Stokes equations, written in terms of the stream function, ψ, and vorticity, ω, for a Newtonian fluid in the absence of body forces, is used to solve the problem of flow in a triangular cavity, driven by the uniform motion of one of its side walls. A key feature of the numerical method is that the difficulties associated with specifying ω at the corners are addressed and overcome by applying analytical boundary conditions on ω near these singularities. The computational results are found to agree well with previously published data and, for small stagnant corner angles, reveal the existence of a sequence of secondary recirculations whose relative sizes and strengths are in accord with Moffatt's classical theory. It is shown that, as the stagnant corner angle is increased beyond approximately 40°, the secondary recirculations diminish in size rapidly.

Stokes flow is considered in a rectangular driven cavity of depth 2 H and width 2 L, with two stationary side walls and two lids moving in opposite directions with speeds U1 and U2. The flow is governed by two control parameters: the cavity aspect ratio, A = H/L, and the speed ratio, S = U1/ U2. The solution for the streamfuntion is expressed as an infinite series of Papkovich-Faddle eigenfunctions, which is then expanded about any stagnation point to reveal changes in the local flow structure as A and S are varied. An (S, A) control space diagram is constructed, which exhibits an intricate structure due to the intersection and confluence of several critical curves representing flow bifurcations at degenerate critical points. There are eight points where two critical curves intersect and the flow bifurcations are described and interpreted with reference to the theoretical work of Bakker ( Bifurcations in Flow Patterns, Kluwer Academic, 1991) and Brøns and Hartnack ( Phys. Fluids, 1999, 11, 314). For a speed ratio in the range -1 ≤ S < O the various flow trnsformations are tracked as A increases in the range O < A < 3.2, and hence the means is identified by which new eddies appear and become fully developed. It is shown that for S ≠ O, the number of eddies increases from 1 to 3 via several key flow transformations, which become more complicated as | S| is reduced.

The flow of viscous lubricant in narrow gaps is considered for those geometries in which cavitation arises. A detailed review is presented of those boundary conditions which have been proposed for terminating the lubrication regime (i.e. those valid where the cavity forms). Finally it is shown that a uniform cavity-fluid interface remains stable to small disturbances provided that \[ \frac{d}{dx}\left(P+\frac{T}{r}\right) < 0, \] in which T and r represent the surface tension of the fluid and the radius of curvature of the interface respectively whilst dP/dx is the gradient of fluid pressure immediately upstream of the interface.