On the Integrability of Subsonic Compressible Potential Flow Around Cylinders*

This work analyzes the local integrability of the potential equation for steady, compressible flow in the plane. The classical problem of flow past a circular cylinder is analyzed for free-stream Mach numbers that yield subsonic flow fields. Theorems from differential geometry are applied to study the local integrability of the governing potential equation. The linearized potential equation is analyzed using both the vector field form and the dual formulation in terms of differential ideals. Differential ideals are applied in the analysis by casting the compressible potential equation as an exterior differential system with an independence condition. The resulting exterior differential system is a Pfaffian system which converts the difficult analysis question of local solvability about non-singular points to a simpler problem in linear algebra. A global version of the integrability result is also applied which gives the geometric conditions required for the solution of the potential equation for the subsonic flow around a circular cylinder to decompose into a foliation. The relationship of the leaves of the foliation to the streamlines around the cylinder is considered.