A mathematical model of unsteady collapsible tube behaviour

A simple, third-order lumped parameter model is presented to describe unsteady flow in a short segment of collapsible tube held between two rigid segments and contained in a pressurised chamber. Equilibrium states and their stability are analysed in detail, as is fully non-linear time dependent behaviour, including in particular the excitation and sustenance of limit--cycle oscillations. The model explicitly neglects both wave propagation (and hence the possibility of choking) and the influence on the elastic properties of the tube of longitudinal tension, but it is otherwise firmly based on fluid mechanical principles. The results emphasise the profound importance of (a) the unsteady head loss (but with some pressure recovery) in the separated flow at the oscillating throat, and (b) the mechanical properties of the parts of the system both downstream and upstream of the collapsible segment. The nature of the upstream segment in particular determines whether it is an upstream pressure head or the inflow to the collapsible segment that is held constant during oscillations. The results are discussed in the context both of other models and of experiment.