Doctor of Philosophy
Department of Mathematics
Katoanga, Tu'ipulotu Lofitu, Group invariant methods for the velocity equations arising in granular flow, Doctor of Philosophy thesis, Department of Mathematics, University of Wollongong, 1997. http://ro.uow.edu.au/theses/1547
For axially symmetric flows of granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming either a plastic potential or dilatant double shearing, a set of three first order partial differential equations are obtained. These equations turn out to be deceptive, because although they have simple mathematical forms, the determination of simple exact solutions is non-trivial. In this thesis, we apply Lie symmetry methods and other recent developments, namely potential and non-classical symmetry methods to these velocity equations. All the known functional forms of existing solutions are shown to arise systematically by consideration of the "optimal system" of the one-parameter symmetries which are admitted. For one particular family of solutions we obtain a simple and straight forward asymptotic expansion for the stress angle, and show that under certain circumstances, these expansions provide accurate solutions of the highly nonlinear ordinary differential equation. In addition, for the same family of solutions we show that a special case permits one integration which enables the velocity components to be given explicitly in terms of the stress angle. Another special case admits a full integration and a number of new exact simple solutions are determined. A detailed numerical comparison is made for various parameter values of the asymptotic expansion, the full numerical solution and with the new exact analytical solutions. The apphcation of potential symmetry methods to the velocity equations leads to some "equivalent potential systems" and these systems are investigated. Finally, the non-classical symmetry method is applied to the velocity equation, which has not been done previously. A number of new nonclassical symmetries are determined, which will be fully investigated for future studies on the velocity equations.