A slender rivulet of a power-law fluid driven by either gravity or a constant shear stress at the free surface

Similarity solutions that describe the flow of a slender non-uniform rivulet of non-Newtonian power-law fluid down an inclined plane are obtained. Rivulets driven by either gravity or a constant shear stress at the free surface are investigated, and in both cases solutions are obtained for both weak and strong surface-tension effects. We find that, despite the rather different physical mechanisms driving the flow, the solutions for gravity-driven and shear-stress-driven rivulets are qualitatively similar. When surface-tension effects are weak there is a unique similarity solution in which the transverse rivulet profile has a single global maximum. This solution represents both a diverging and shallowing sessile rivulet and a converging and deepening pendent rivulet. On the other hand, when surface-tension effects are strong there is a one-parameter family of similarity solutions in which the transverse profile of a diverging and shallowing rivulet has one global maximum, while that of a converging and deepening rivulet has either one global maximum or two equal global maxima. We also show how the present similarity solutions can be modified to accommodate a fixed-contact-angle condition at the contact line by incorporating sufficiently strong slip at the solid/fluid interface into the model.