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 "levees" 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

Motivated by the need for a better understanding of the transport of solutes in microfluidic flows with free surfaces, the advection and dispersion of a passive solute in steady unidirectional flow of a thin uniform rivulet on an inclined planar substrate driven by gravity and/or a uniform longitudinal surface shear stress are analysed. Firstly, we describe the short-time advection of both an initially semi-infinite and an initially finite slug of solute of uniform concentration. Secondly, we describe the long-time Taylor-Aris dispersion of an initially finite slug of solute. In particular, we obtain the general expression for the effective diffusivity for Taylor-Aris dispersion in such a rivulet, and discuss in detail its different interpretations in the special case of a rivulet on a vertical substrate.

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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 very 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.