A study of the bifurcation behaviour of a model of flow through a collapsible tube

Most of the elastic tubes found in the mammalian body will collapse from a distended circular cross section and when collapsed may undergo flow-induced oscillations. A mathematical model describing fluid flow in a collapsible tube is analysed using the software package AUTO-86. AUTO-86 is used for continuation and bifurcation problems in systems of non-linear ordinary differential equations. The model is a third-order lumped-parameter type and is based on the classical “Starling resistor”; it describes the unsteady flow behaviour and, in particular, the experimentally observed self-excited oscillations, in a way which is simple enough to give physical understanding, yet still firmly based on fluid mechanical principles. Some of the bifurcation types found in this model bear close resemblance to the types suggested by experimental observations of self-excited oscillations in collapsible tubes; they thus shed some light on the various topological changes which occur in practice, particularly in view of the fact that some of the points found numerically are diffcult to achieve experimentally, while the existence of others can only be inferred indirectly and uncertainly from experiment.