Hopf Bifurcation Phenomena in Tubes Carrying a Fluid

Abstract

Self-induced planar nonlinear oscillations of a cantilever tube carrying an incompressible fluid are investigated. The system behavior is governed by a nonself-adjoint integrodifferential equation and an ordinary differential equation which are coupled nonlinearly. The system behavior depends on three parameters,$\rho_0 $, the flow velocity,$\beta $, the mass ratio of the tube and the fluid and $\alpha $, a parameter related to the pressure loss in the tube. For small $\rho_0 $, the zero solution is asymptotically stable. As $\rho_0$ is increased, for each $\beta $, a critical value of $\rho 0,\rho 0 = \rho _{cr} $, is reached when the zero solution becomes unstable by a pair of complex conjugate eigenvalues of the linearized system crossing the imaginary axis. The system satisfies conditions for Hopf bifurcation and the zero solution bifurcates into periodic orbits. The bifurcated periodic solutions are determined using center-manifold and averaging ideas. In general, the system equations include changes in flow velocity introduced by the tube motions. However, based on the order of $\alpha $, two classes of problems are shown to exist. When $\alpha $ is very large compared to the size of the motion, it is shown that fluid velocity changes do not enter the first order theory and the results show that in this case only supercritical bifurcation occurs. In the second case, either sub or supercritical solutions are found to occur depending on the value of $\alpha $.