Shallow free-surface Stokes flow around a corner

Abstract

The steady lateral spreading of a free-surface viscous flow down an inclined plane around a vertex from which the channel width increases linearly with downstream distance is investigated analytically, numerically and experimentally. From the vertex the channel wall opens by an angle α to the downslope direction and the viscous fluid spreads laterally along it before detaching. The motion is modelled using lubrication theory and the distance at which the flow detaches is computed as a function of α using analytical and numerical methods. Far downslope after detachment, it is shown that the motion is accurately modelled in terms of a similarity solution. Moreover, the detachment point is well approximated by a simple expression for a broad range of opening angles. The results are corroborated through a series of laboratory experiments and the implication for the design of barriers to divert lava flows are discussed. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.

Viscous free-surface flows past cylinders

“…agrees with the small F expansion (61) to leading order in the regime F 1 and agrees with the large F expansion (34) to order F −1 in the regime F 1. Indeed, the simple expression (62) quite accurately captures the behaviour of the maximum flow depth for all F (see figure 8).…”

“…Furthermore, the square obstruction must also be able to withstand a much greater force. The results illustrate that the dependence of the extent and the shape of the dry region on the shape of the downstream side of the barrier is an important problem, and results are reported in [61].…”

Using a squeegee on a layer of viscous or viscoplastic fluid

“…There is also an interesting question of how the ridge of fluid diverted around the ends of the squeegee starts to spread to fill the gap left behind the squeegee. (See [13] for the downslope case. )…”

Depth-integrated models for three-dimensional flow over topography

“…Some early studies include the case of thin-film flow over a one-dimensional trench (Kalliadasis, Bielarz & Homsy 2000;Mazouchi & Homsy 2001;Bielarz & Kalliadasis 2003), while Gaskell et al (2004) considered both one-and two-dimensional topographies. More recently Hinton, Hogg & Huppert (2019, 2020a, 2020b used lubrication models, asymptotic analyses and laboratory experiments to describe free-surface viscous flows over two-dimensional mounds, around a corner and past cylinders of various cross-sections. D'Alessio (2023a) also used a lubrication model to compute flows past cylinders having circular and elliptical cross-sections.…”