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

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.