Unstable equilibrium behaviour in collapsible tubes

Abstract

Thick-walled silicone rubber tube connected to rigid pipes upstream and downstream was externally pressurised ( p e ) to cause collapse while aqueous fluid flowed through propelled by a constant upstream head. Three types of equilibrium were found: stable equilibria (steady flow) at high downstream flow resistance R 2, self-excited oscillations at low R 2, and ‘unattainable’ (by varying external pressure) or exponentially unstable equilibria at intermediate R 2. The self-excited oscillations were highly non-linear and appeared in four, apparently discrete, frequency bands: 2.7 Hz, 3.8–5.0 Hz, 12–16 Hz and 60–63 Hz, suggesting that the possible oscillation modes may be harmonically related. Stable, intermediate ‘two-in-every-three-beats’ oscillation was also observed, with a repetition frequency in the 3.8–5.0 Hz band. As p e was increased, self-excited oscillations were eventually suppressed, leaving internal fluid pressure varying with no single dominant frequency as a result of turbulent jet dissipation at the downstream rigid pipe connection. Comparison of pressure-wave velocity calculated from the local pressure-area relation for the tube with fluid velocity indicated that supercritical velocities were attained in the course of the self-excited oscillations.