Streamline topologies near a fixed wall using normal forms

Streamline patterns and their bifurcations in two-dimensional incompressible flow in the vicinity of a fixed wall has been investigated from a topological point of view by Bakker [Bifurcations in Flow Patterns. Kluwer Academic Publishers, 1991]. Bakkers work is revisited in a more general setting allowing curvature of the fixed wall and a time dependence of the streamlines. The velocity field is expanded at a point on the wall, and the expansion coefficients are considered as bifurcation parameters. A series of non-linear coordinate changes results in a much simplified system of differential equations for the streamlines (a normal form) encapsulating all the features of the original system. From this, a complete description of bifurcations up to codimension three close to a simple linear degeneracy are obtained. Further the case of a non-simple degeneracy is considered. Finally the effect of the Navier-Stokes equations on the local topology is considered.