S. K. Wilson, B. R. Duffy, R. Hunt; A Slender Rivulet of a Power‐Law Fluid Driven by Either Gravity or a Constant Shear Stress at the Free Surface. Q J Mechanics Appl Math 2002; 55 (3): 385-408. doi: 10.1093/qjmam/55.3.385
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.