A 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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/18518 |
| Date | 27 May 2015 |
| Creators | Lowry-Corry, Angela Emily Rosemary |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf, application/pdf |
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