Heil, M. & Waters, S.L. (2008) How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. Journal of Fluid Mechanics 601, 199-227.

We present a combined theoretical and computational analysis of 3D unsteady finite-Reynolds number flows in collapsible tubes whose walls perform prescribed high-frequency oscillations which resemble those typically observed in experiments with a Starling resistor. Following an analysis of the flow fields, we investigate the system's overall energy budget and establish the critical Reynolds number, $Re_{crit}$, at which the wall begins to extract energy from the flow. We conjecture that $Re_{crit}$ corresponds to the Reynolds number beyond which collapsible tubes are capable of performing sustained self-excited oscillations. Our computations suggest a simple functional relationship between $Re_{crit}$ and the system parameters, and we present a scaling argument to explain this observation. Finally, we demonstrate that, within the framework of the instability mechanism analysed here, self-excited oscillations of collapsible tubes are much more likely to develop from steady-state configurations in which tube is buckled non-axisymmetrically, rather than from axisymmetric steady states, which is in pleasing agreement with experimental observations.

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