Absolute and Convective Instabilities of a Collapsible Tube

Abstract

In this paper we extend a numerical method, developed previously by the author, to compute the eigen modes of collapsible viscoelastic duct convening a fowing fuid. A new technique is developed in order to eliminate the need for recurrence formula used in the old method. This gives a more powerful and tractable method to find the eigen modes of the system. The new method has allowed the identification of a new unstable modes in a collapsible tube. It is found that there is a set of standing non axisymmetric waves representing an absolute instabilities and a set of unstable upstream and downstream propagated waves representing a convective instabilities. Two standing waves have equal frequency in their cusp points. The frequency of the other standing waves, in their cusp points, are a multiple of the frequency of the first wave and that in good agreement with experimental finding available in the literature. It is found that the first absolute unstable mode becomes convective at high Reynolds number while the other standing wave remain absolutely unstable modes for Re higher than 100. The frequency ratio of the absolute unstable modes in their cusp point are preserved for all Reynolds number and that with an good agreement with the experience. The absolute unstable mode which becomes convective at high Reynolds number keeps a monochromatic wave with a frequency equal to the frequency of the second absolute unstable mode, in its cusp point, and that for all Reynolds number higher than the limit of absolute instability of the first mode in surprising agreement with the experimental results. It is founded that the viscosity of the solid stabilizes the standing wavesat different degree. The boundary separating the absolute instabilities zone from the convective instabilities zone are found.