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We consider the slow motion of a thin viscous film flowing over a topographical feature (trench or mound) under the action of an external body force. Using the lubrication approximation, the equations of motion simplify to a single nonlinear partial differential equation for the evolution of the free surface in time and space. It is shown that the problem is governed by three dimensionless parameters corresponding to the feature depth, feature width and feature steepness. Quasi-steady solutions for the free surface are reported for a wide range of these parameters. Our computations reveal that the free surface develops a ridge right before the entrance to the trench or exit from the mound and that this ridge can become large for steep substrate features of significant depth. Such capillary ridges have also been observed in the contact line motion over a planar substrate where the buildup of pressure near the contact line is responsible for the ridge. For flow over topography, the ridge formation is a manifestation of the effect of the capillary pressure gradient induced by the substrate curvature. In addition, the minimum film thickness is always found near the concave corner of the feature. Both the height of the ridge and the minimum film thickness are found to be strongly dependent on both the profile depth and steepness. Finally, it is found that either finite feature width or a significant vertical component of gravity can suppress these effects in a way that is made quantitative and which allows the operative physical mechanism to be explained.

When the coating film around a vertical fibre exceeds a critical thickness hc, the interfacial disturbances triggered by Rayleigh instability can undergo accelerated growth such that localized drops much larger in dimension than the film thickness appear. We associate the initial period of this strongly nonlinear drop formation phenomenon with a self-similar intermediate asymptotic blow-up solution to the long-wave evolution equation which describes how static capillary forces drain fluid into the drop. Below hc, we show that strongly nonlinear coupling between the mean flow and axial curvature produces a finite-amplitude solitary wave solution which prevents local finite-time blow up and hence disallows further growth into drops. We thus estimate hc by determining the existence of solitary wave solutions. This is accomplished by a matched asymptotic analysis which joins the capillary outer region of a large solitary wave to the thin-film inner region. Our estimate of hc = 1.68R3H–2, where R is the fibre radius and H is the capillary length H = (σ/ρg)½, is favourably compared to experimental data.

Consider the gravity-driven flow of a thin liquid film down a vertical fibre. A model of two coupled evolution equations for the local film thickness h and the local flow rate q is formulated within the framework of the long-wave and boundary-layer approximations. The model accounts for inertia and streamwise viscous diffusion. Evolution equations obtained by previous authors are recovered in the appropriate limit. Comparisons to experimental results show good agreement in both linear and nonlinear regimes. Viscous diffusion effects are found to have a stabilizing dispersive effect on the linear waves. Time-dependent computations of the spatial evolution of the film reveal a strong influence of streamwise viscous diffusion on the dynamics of the flow and the wave selection process.

The stability of a viscous film flowing down a vertical fiber under the action of gravity is analyzed both experimentally and theoretically. At large or small film thicknesses, the instability is convective, whereas an absolute instability mode is observed in an intermediate range of film thicknesses for fibers of small enough radius. The onset of the experimental irregular wavy regime corresponds precisely to the theoretical prediction of the threshold of the convective instability.