Topology transitions and singularities in viscous flows

Topological reconfigurations of the boundaries of thin fluid layers in Hele-Shaw flow are studied. A systematic treatment of the dynamics of the bounding interfaces is developed through an expansion in the aspect ratio of the layer, yielding nonlinear partial differential equations for the local thickness. For both density-stratified fluid layers and gravity-driven jets, numerical study of the dynamics at second order suggests strrongly the collision of the interfaces in finite time. There are associated singularities both in the fluid velocity and in geometric properties of the interfaces.