A Case Study of Elliptic Functions in a CAS: Jeffery-Hamel Flow with Maple

Elliptic functions and integrals are not as widely known in the scientific populace as they have been in the past, even though they can still be used to solve an impressive array of applied problems. Computer Algebra Systems (CAS) have stepped into this knowledge gap and claim to provide access to many of the formulae that have proven so useful. This paper takes a classical problem in two-dimensional fluid flow— namely, flow into or out of a wedge-shaped channel with a sink or source at the vertex, which flow is known as JefferyHamel flow and has ‘well-known’ solutions containing elliptic functions—and tries to duplicate, or even extend, the classical solutions by using a CAS, in this instance Maple. The purposes of this case study include examining just how good CAS can be at elliptic functions; and, more importantly, identifying needs for improvement. Another purpose is to compare the analytical solution with modern numerical solutions. Finally, we believe that this work will motivate improvements to CAS facilities for automatic case analysis. As an aside, we present some simple methods for integration of elliptic functions that seem not to be widely known.