Solitary waves in fluid -filled elastic tubes

The propagation of solitary waves in fluid-filled elastic tubes was investigated by direct analysis of the governing field equations. The primary advantage of this approach over the widely used asymptotic techniques is that, for a specified wave speed, the solution of the ‘exact’ amplitude of the solitary waves only requires the roots of an algebraic equation. The shape of the wave can be found to any required degree of accuracy numerically. This approach can be applied beyond the long-wave approximation for any amplitude of wave.

The presented direct approach was used to consider a fluid-filled elastic thin-walled tube where axial displacements were neglected and the velocity of the fluid was averaged over the tube radius. It was shown that errors can become as large as 20% for displacements up to 25% of the tube radius when the reductive perturbation technique is used. The direct approach was also used to investigate a problem in plasma physics, specifically ion-acoustic waves, to illustrate a broader application of the proposed technique.

The kinematically exact shell equations for the tube, including both axial and radial displacements were considered and it was shown that, by casting the problem in a variational framework, it becomes possible to find explicit first integrals of the governing equations. The first integrals then allowed the speed, amplitude and shape of the resulting solitary wave to be determined ‘exactly’ using the proposed direct approach. The results showed that the wave amplitude calculated using the exact displacements were an order of magnitude greater than found when axial displacements were neglected. It was subsequently shown that the axial strain was of the same order as the magnitude of the radial strain and that in the prestressed reference configuration their relationship was approximately linear.

Exploiting this approximate linear relationship, a linear function was found from the axial first integral, permitting the reduction of the governing equations to a problem of one equation in one dependent variable, while still retaining a contribution for the axial displacement. The amplitude predicted using this approximate approach was found to differ from the exact value by as little as 3%.

Finally, the tube wall pressure predicted from our inviscid, incompressible one-dimensional fluid model was compared to a two-dimensional flow, simulated using a modified discrete-vortex method. The tube geometries examined corresponded to the solitary wave profiles for four representative wave speeds. It was shown that the predicted pressures from the existing one-dimensional model compared well with the two-dimensional flow. Based upon this, it should be expected that solitary waves predicted using a two-dimensional fluid model will be in close agreement with the results presented in this dissertation.