The solutions for the unidirectional flow of a thin rivulet with prescribed volume flux down an inclined planar substrate are used to describe the locally unidirectional flow of a rivulet with constant width (i.e. pinned contact lines) but slowly varying contact angle as well as the possible pinning and subsequent de-pinning of a rivulet with constant contact angle and the possible depinning and subsequent re-pinning of a rivulet with constant width as they flow in the azimuthal direction from the top to the bottom of a large horizontal cylinder. Despite being the same locally, the global behaviour of a rivulet with constant width can be very different from that of a rivulet with constant contact angle. In particular, while a rivulet with constant non-zero contact angle can always run from the top to the bottom of the cylinder, the behaviour of a rivulet with constant width depends on the value of the width. Specifically, while a narrow rivulet can run all the way from the top to the bottom of the cylinder, a wide rivulet can run from the top of the cylinder only to a critical azimuthal angle. The scenario in which the hitherto pinned contact lines of the rivulet de-pin at the critical azimuthal angle and the rivulet runs from the critical azimuthal angle to the bottom of the cylinder with zero contact angle but slowly varying width is discussed. The pinning and de-pinning of a rivulet with constant contact angle, and the corresponding situation involving the de-pinning and re-pinning of a rivulet with constant width at a non-zero contact angle which generalises the de-pinning at zero contact angle discussed earlier, are described. In the latter situation, the mass of fluid on the cylinder is found to be a monotonically increasing function of the constant width

We use the lubrication approximation to analyse three closely related problems involving a thin rivulet or ridge (i.e. a two-dimensional droplet) of fluid subject to a prescribed uniform transverse shear stress at its free surface due to an external airflow, namely a rivulet draining under gravity down a vertical substrate, a rivulet driven by a longitudinal shear stress at its free surface, and a ridge on a horizontal substrate, and find qualitatively similar behaviour for all three problems. We show that, in agreement with previous numerical studies, the free surface profile of an equilibrium rivulet/ridge with pinned contact lines is skewed as the shear stress is increased from zero, and that there is a maximum value of the shear stress beyond which no solution with prescribed semi-width is possible. In practice, one or both of the contact lines will de-pin before this maximum value of the shear stress is reached, and so we consider situations in which the rivulet/ridge de-pins at one or both contact lines. In the case of de-pinning only at the advancing contact line, the rivulet/ridge is flattened and widened as the shear stress is increased from its critical value, and there is a second maximum value of the shear stress beyond which no solution with a prescribed advancing contact angle is possible. In contrast, in the case of de-pinning only at the receding contact line, the rivulet/ridge is thickened and narrowed as the shear stress is increased from its critical value, and there is a solution with a prescribed receding contact angle for all values of the shear stress. In general, in the case of de-pinning at both contact lines there is a critical “yield” value of the shear stress beyond which no equilibrium solution is possible and the rivulet/ridge will evolve unsteadily. In an Appendix we show that an equilibrium rivulet/ridge with prescribed flux/area is quasi-statically stable to two-dimensional perturbations

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The behaviour of a steady thin sessile or pendent ridge of fluid on an inclined planar substrate which is strongly coupled to the external pressure gradient a rising from an inviscid airflow parallel to the substrate far from the ridge is described. When the substrate is nearly horizontal a very wide ridge can be supported against gravity by capillary and/or external pressure forces; otherwise only a narrower (but still wide) ridge can be supported. Classical thin-aerofoil theory is adapted to obtain the governing singular integro-differential equation for the profile of the ridge in each case. Attention is focused mainly on the case of a very wide sessile ridge. The effect of strengthening the airflow is to push a pinned ridge down near to its edges but to pull it up near to its middle. At a critical airflow strength the upslope contact angle reaches the receding contact angle at which the upslope contact line de-pins, and continuing to increase the airflow strength beyond this critical value results in the de-pinned ridge becoming narrower, thicker and closer to being symmetric in the limit of a strong airflow. The effect of tilting the substrate is to skew a pinned ridge in the downslope direction. Depending on the values of the advancing and receding contact angles, the ridge may first de-pin at either the upslope or the downslope contact line but, in general, eventually both contact lines de-pin. The special cases in which only one of the contact lines de-pins are also considered. It is also shown that the behaviour of a very wide pendent ridge is qualitatively similar to that of a very wide sessile ridge, while the important qualitative difference between the behaviour of a very wide ridge and a narrower ridge is that, in general, for the latter one or both of the contact lines may never de-pin