Technical University of Denmark
Lyngby, Denmark
The paper focuses on the classification of flow patterns. That is the topological description of flow patterns in an incompressible viscous two-dimensional flow away from any boundaries. The vector field is expanded in a point in the fluid, and the expansion coefficients are considered as bifurcation parameters. Through tools from modern nonlinear dynamics a series of nonlinear coordinate changes are applied whereby a simplified system (a normal form) is obtained encapsulating all the features of the original system. Bifurcations of the flow patterns occur at degenerate configurations. Using the theory of unfolding the paper gives a description of the most common bifurcations that can occur in an incompressible fluid. Further the case of reflectional symmetry is considered and this theory is applied to the topology of streamlines in Stokes flow in a cavity as studied recently by Gaskell \emph{et al.} [Journal of Fluid Mechanics {\bf 337},pp.263-282]. Finally it is shown that all the general flow patterns classified in the paper can be realized in steady Navier-Stokes flow or in Stokes flow. Whereas a difficulty arises in the case of Navier-Stokes flow with reflectional symmetry.