Unsteady slender rivulet flow down an inclined porous plane

The unsteady three-dimensional flow of a thin slender rivulet of an incompressible Newtonian fluid down an inclined porous plane is investigated. The leak-off velocity is not specified a priori but is determined in the process of deriving the invariant solution. A second order partial differential equation in time and two spatial variables and containing the leak-off velocity is derived for the height of the thin slender rivulet. By using Lie group analysis the partial differential equation is reduced in two steps to an ordinary differential equation provided the leak-off velocity satisfies a first order linear partial differential equation in three variables. An exact analytical solution with a dry patch in the central region is derived for a special leak-off velocity. Three models are considered with the leak-off velocity proportional to the height, the square of the height and the cube of the height of the rivulet using a shooting method which also determines the two-dimensional boundary of the rivulet on the inclined plane. The effect of fluid leak-off on the height and width of the rivulet is investigated numerically and compared in the three models.