Travelling-wave similarity solutions for an unsteady shear-stress-driven dry patch in a flowing film

We investigate unsteady flow of a thin film of Newtonian fluid around a symmetric slender dry patch moving with constant velocity on an inclined planar substrate, the flow being driven by a prescribed constant shear stress at the free surface of the film (which would be of uniform thickness in the absence of the dry patch). We obtain a novel unsteady travelling-wave similarity solution which predicts that the dry patch has a parabolic shape and that the film thickness increases monotonically away from the dry patch

Travelling-wave similarity solutions for an unsteady gravity-driven dry patch

Travelling-wave similarity solutions are obtained for three-dimensional unsteady gravity-driven flow of a thin film of Newtonian fluid around a moving symmetric slender dry patch on an inclined plane