Bifurcation Analysis of a Pipe Containing Pulsatile Flow

Abstract

In this paper the dynamic behaviour of a continuum inextensible pipe containing fluid flow is investigated. The fluid is considered to be incompressible, frictionless and its velocity relative to the pipe has the same but time-periodic magnitude along the pipe at a certain time instant. The equations of motion are derived via Lagrangian equations and Hamilton´s principle. The system is non-conservative, and the amount of energy carried in and out by the flow appears in the model. It is well-known, that intricate stability problems arise when the flow pulsates and the corresponding mathematical model, a system of ordinary or partial differential equations, becomes time-periodic. The method which constructs the state transition matrix used in Floquet theory in terms of Chebyshev polynomials is especially effective for stability analysis of systems with multi-degree-of-freedom. The implementation of this method using computer algebra enables us to obtain the boundary curves of the stable domains semi-analytically. The bifurcation analysis was performed with respect to three important parameters: the forcing frequency $\omega$, the perturbation amplitude $\nu$ and the average flow velocity U.