Saddle connections near degenerate critical points in Stokes flow within cavities

摘要

Streamline patterns and their bifurcations in two-dimensional incompressible flow near a simple degenerate critical point away from boundaries are investigated by using the normal form theory for the streamfunction obtained by [M. Brøns, J.N. Hartnack, Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids 11 (1999) 314–324]. For the normal form of order six, a bifurcation diagram is constructed with two bifurcation parameters. The theory is applied to the patterns and bifurcations found numerically in the studies of Stokes flow in a double-lid-driven rectangular cavity with two control parameters (the cavity aspect ratio A (height to width), and the speed ratio S). Bifurcations in the cavity are obtained using an analytic solution for the streamfunction developed for any value of S and A. Using this solution for special values (S, A) a global bifurcation is identified with a single heteroclinic connection which connects three saddle points in a triangle and does not appear in the unfolding of the simple linear degenerate critical points lying in a line y = constant.