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

Abstract

Locally unidirectional steady gravity-driven flow of a thin rivulet of a power-law fluid with prescribed volume flux down a locally planar substrate is considered. First, the solution for unidirectional flow of a uniform rivulet down a planar substrate is obtained, and then it is used to obtain the solution for a slowly varying rivulet with prescribed constant (nonzero) contact angle down a slowly varying substrate, specifically flow in the azimuthal direction around the outside of a large horizontal circular cylinder. The solution is shown to depend strongly on the value of the power-law index of the fluid. For example, a rivulet of strongly shear-thinning fluid "self-channels" its flow down a narrow central channel between two "levées" of slowly moving fluid that form at its sides, and in the central channel there is a "plug-like" flow except in a boundary layer near the substrate. On the other hand, in a rivulet of a strongly shear-thickening fluid the velocity profile is linear except in a boundary layer near the free surface. Another notable qualitative departure from Newtonian behaviour is that, whereas the mass of a rivulet of a Newtonian or a shear-thinning fluid is theoretically infinite, the mass of a rivulet of a shear-thickening fluid is finite.