Lowry-Corry, Angela Emily Rosemary2015-09-072015-09-072015-05-27http://hdl.handle.net/10539/18518A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in ful lment of the requirements for the degree of Masters of Science. May 27, 2015.Abstract The unsteady three-dimensional ow of a thin slender rivulet of incompressible Newtonian uid down an inclined porous plane is investigated. The leak-o velocity is not speci ed in the model but is determined in the process of deriving the invariant solution. A second order nonlinear partial di erential equation in two spatial variables and time and containing the leak-o velocity is derived for the height of the thin slender rivulet. Using Lie group analysis it is found that the partial di erential equation can be reduced in two steps to an ordinary di erential equation provided the leak-o velocity satis es a rst order linear partial di erential equation in three variables. An exact analytical solution with a dry patch in the central region is derived for a special leak-o velocity. Two models are considered, one with the leak-o velocity proportional to the height of the rivulet and the other with leak-o velocity proportional to the cube of the height. Numerical solutions are obtained for the height of the rivulet using a shooting method which also determines the two-dimensional boundary of the rivulet on the inclined plane. The e ect of uid leak-o on the height and width of the rivulet is investigated numerically and compared in the two models. The conservation laws for the partial di erential equation with no uid leak-o are investigated. Two conserved vectors are derived, the elementary conserved vector and a new conserved vector. The Lie point symmetry of the partial di erential equation associated with each conserved vector is obtained. Each associated Lie point symmetry is used to perform a double reduction of the partial di erential equation, but the solutions obtained are not physically signi cant.enNewtonian fluids.Partial differential operators.Lie groups.Thesis