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

Abstract

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Strongly coupled interaction between a ridge of fluid and an inviscid airflow

“…Comparing (40) with the corresponding equation for a very wide ridge (12) reveals that, as might have been expected, the normal component of gravity is negligible for a narrower ridge, and hence that the leading order solutions for narrower sessile and pendent ridges are identical. Moreover, comparing the definitions of Λ for very wide and narrower ridges (given by (13) and (41), respectively) reveals that the airflow required to support even a narrower ridge on a substrate which is not nearly horizontal is stronger than that required to support a very wide ridge on a nearly horizontal substrate (specifically, U ∞ must be larger by a factor of…”

“…There have, of course, also been many other studies in which the pressure gradient and/or the shear stress on a thin film of fluid due to an airflow is prescribed rather than being coupled to the unknown free surface profile (see, for example, [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] ), but these are less directly relevant to the present strongly coupled problem.…”

Travelling-wave similarity solutions for a steadily translating slender dry patch in a thin fluid film

“…with U ∞ again as in (18). Curves of constant Ψ in the X-Y plane may be regarded as effective streamlines of this depth-integrated flow, with the streamline Ψ = 0 corresponding to the axis…”

Surface-tension- and injection-driven spreading of a thin viscous film

“…They find power-law dependencies on the distance down the plane for the width and the height of the resulting film. Further analysis of gravity-driven rivulet flow of Newtonian and power-law fluids down an inclined plane is given in Yatim, Wilson & Duffy (2010), whereas shear-stress-driven flow is considered in Yatim, Duffy & Wilson (2012). Corresponding travelling-wave similarity solutions are found in Yatim, Duffy & Wilson (2013).…”