Similarity solutions for unsteady shear-stress-driven flow of Newtonian and power-law fluids : slender rivulets and dry patches

Mohd Yatim, Yazariah and Duffy, Brian and Wilson, Stephen (2012) Similarity solutions for unsteady shear-stress-driven flow of Newtonian and power-law fluids : slender rivulets and dry patches. Journal of Engineering Mathematics, 73 (1). pp. 53-69. ISSN 0022-0833 (https://doi.org/10.1007/s10665-011-9499-8)

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Abstract

Unsteady flow of a thin film of a Newtonian fluid or a non-Newtonian power-law fluid with power-law index N driven by a constant shear stress applied at the free surface, on a plane inclined at an angle α to the horizontal, is considered. Unsteady similarity solutions representing flow of slender rivulets and flow around slender dry patches are obtained. Specifically, solutions are obtained for converging sessile rivulets (0 < α < π/2) and converging dry patches in a pendent film (π/2 < α < π), as well as for diverging pendent rivulets and diverging dry patches in a sessile film. These solutions predict that at any time t, the rivulet and dry patch widen or narrow according to |x|3/2, and the film thickens or thins according to |x|, where x denotes distance down the plane, and that at any station x, the rivulet and dry patch widen or narrow like |t|−1, and the film thickens or thins like |t|−1, independent of N.

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