A rivulet of a power-law fluid with constant width draining down a slowly varying substrate

Abstract

The flow of a slowly varying rivulet of a power-law fluid with prescribed constant width (i.e. with pinned contact lines) but slowly varying contact angle down a slowly varying substrate, specifically the flow in the azimuthal direction around the outside of a large horizontal circular cylinder, is described. The solution for a rivulet of a perfectly wetting fluid (which can never have constant width) is obtained, and it is shown that, despite having the same local behaviour, the global behaviour of a rivulet of a non-perfectly wetting fluid is qualitatively different from that of a rivulet with prescribed constant contact angle but slowly varying width. Specifically, it is described how the contact lines of a sufficiently narrow rivulet can remain pinned as it drains all the way from the top to the bottom of the cylinder, but how the contact lines of a wider rivulet de-pin at a critical position on the lower half of the cylinder, and how thereafter it drains to the bottom of the cylinder with zero contact angle and slowly varying width. How the shape of the rivulet and the velocity within it depend on the power-law index N is described in detail. In particular, it is shown that whereas neither the shape of the rivulet nor the velocity within it vary monotonically with N, its mass always decreases monotonically with N. Despite the limitations of the power-law model, the present results provide rare analytical insight into non-Newtonian rivulet flow, and, in particular, are a useful benchmark for the study of rivulet flow of more realistic non-Newtonian fluids.