Two Perturbation Calculations in Fluid Mechanics Using Large-Expression Management

Two uid-ow problems are solved using perturbation expansions, with special emphasis on the reduction of intermediate expression swell. This is done by developing tools in Maple that contribute to the eecient representation and manipulation of large expressions. The tools share a common basis, which is the creation of a hierarchy of representation levels such that expressions located at higher levels are expressed using entries from lower levels. The evaluation of higher-level expressions by the algebra system does not proceed recursively to the lowest level, as would ordinarily be the case, but instead can be directly controlled by the user. The rst uid-ow problem, arising in lubrication theory, is solved by implementing a technique of switch-controlled evaluation. The processes of simpliication and evaluation are controlled at each level by user-manipulated switches. A perturbation solution is derived semi-interactively with the switch-controlled evaluation being used to reduce the size of intermediate expressions. The second uid problem, in convection, is solved by extending a perturbation series in several variables to high order by implementing techniques for the automatic generation of hierarchical expression sequences.