Streamline Topologies in Stokes Flow Within Lid-Driven Cavities

Abstract

Stokes flow in a rectangular cavity with two moving lids (with equal speed but in opposite directions) and aspect ratio A (height to width) is considered. An analytic solution for the streamfunction, ψ, expressed as an infinite series of Papkovich–Fadle eigenfunctions is used to reveal changes in flow structures as A is varied. Reducing A from A=0.9 produces a sequence of flow transformations at which a saddle stagnation point changes to a centre (or vice versa) with the generation of two additional stagnation points. To obtain the local flow topology as A→0, we expand the velocity field about the centre of the cavity and then use topological methods. Expansion coefficients depend on the cavity aspect ratio which is considered as a separation parameter. The normal-form transformations result in a much simplified system of differential equations for the streamlines encapsulating all features of the original system. Using the simplified system, streamline patterns and their bifurcations are obtained, as A→0.