Flow induced bifurcations to three-dimensional oscillatory motions in continuous tubes

Three-dimensional motions of a cantilever tube carrying an incompressible fluid and having rotational symmetry about the vertical axis are examined for bifurcating oscillatory solutions. The system behavior depends on two parameters, $\rho $, the flow velocity and $\beta$, the mass ratio of the fluid and the tube. As the flow rate is increased past a critical value, the zero solution becomes unstable by a double pair of complex conjugate eigenvalues of the linearized operator crossing the imaginary axis. Using ideas from the center manifold theory and the method of integral averaging, the governing partial differential equations are reduced to a set of four nonlinear ordinary differential equations. It is then shown that the system bifurcates into two qualitatively distinct oscillatory motions. These oscillatory solutions themselves, for some values of the mass ratio $\beta$, bifurcate into more complex amplitude modulated oscillations.