Centre for Differential Equations, Continuum Mechanics and Applications and School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa
Academic Editor: Fazal M. Mahomed
Copyright © 2013 N. Mindu and D. P. Mason. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The migration of melt through the mantle of the Earth is governed by a third-order nonlinear partial differential equation for the voidage or volume fraction
of melt. The partial differential equation depends on the permeability of the
medium which is assumed to be a function of the voidage. It is shown that the
partial differential equation admits, as well as translations in time and space,
other Lie point symmetries provided the permeability is either a power law
or an exponential law of the voidage or is a constant. A rarefactive solitary
wave solution of the partial differential equation is derived in the form of a
quadrature for the exponential law for the permeability.