A nonlinear model for an extensible slender flexible cylinder subjected to axial flow

In this paper, the weakly nonlinear equations of motion are derived for a slender flexible cylinder subjected to axial flow. The cylinder centreline is considered to be extensible, and hence two coupled nonlinear equations describe its motions, involving both longitudinal and transverse displacements. The fluid forces are formulated in terms of several components, for convenience. For high Reynolds number flows, the dominant, inviscid component is modelled by an extension of Lighthill's slender-body work; frictional, hydrostatic and pressure-loss forces are then added to the inviscid component. The derivation of the equations of motion is carried out in a Lagrangian framework, and the resultant equations are correct to third-order of magnitude, O(ε3), where the transverse displacement of the cylinder is of O(ε). This is the main contribution of this paper; however, the equations have been solved and some interesting results are presented also. Bifurcation diagrams with flow velocity as the independent variable, supported by phase-plane plots, show that the system loses stability via a supercritical pitchfork bifurcation and develops divergence, and at higher flow flutter, which is what has been observed experimentally and predicted by linear theory in the past. It is shown that post-divergence flutter does exist, not as an instability of the trivial equilibrium (as predicted by linear theory), but as a Hopf bifurcation emanating from the nonlinear static equilibrium. For high enough flow, interesting dynamics follow, including quasiperiodicity and chaos.