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

Wilson, S.K. and Duffy, B.R. and Hunt, R. (2002) A slender rivulet of a power-law fluid driven by either gravity or a constant shear stress at the free surface. Quarterly Journal of Mechanics and Applied Mathematics, 55 (3). pp. 385-408. ISSN 0033-5614

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Abstract

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.