Nonlinear oscillations of a collapsible tube subjected to unsteady external pressure

Abstract

The non-linear dynamics of an extremely thin-walled collapsible tube with internal flow subjected to a time-varying external pressure are studied experimentally and theoretically. For the constant chamber pressure case, we observe the existence of a fixed-point attractor, period-1 attractor, and quasiperiodic attractor. The period-1 limit cycle oscillations are essentially relaxation oscillations with up-down asymmetry in the time domain, and as the Reynolds number increases, the asymmetry becomes greater. With the forcing (varying chamber pressure), the system has no fixed points; its response can be period-n, quasiperiodic, or chaotic, depending upon the Reynolds number, driving amplitude, and frequency. For the forced system, at a low Reynolds number, the external forcing dominates the self-excited oscillations and symmetric oscillations are observed; at a higher Reynolds number, the reverse is true. In experiments and theory, aperiodic oscillations for the forced system are always observed in regimes beyond the Hopf bifurcation point of the unforced system. Distended and collapsed cases, under forcing, exhibit only 1:1 synchronous oscillation. These suggest that a natural oscillation timescale of the system must be present for the external forcing to induce aperiodicity. In the experiments, the forced system exhibits signs of quasiperiodic route to chaos at lower driving amplitude, while period-doubling route to chaos at higher driving amplitude. When the system is forced near its natural frequency, an aperiodic response is totally suppressed.