Flutter in One-Dimensional Collapsible Tubes

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We seek improved one-dimensional (1-D) models, or long wave approximations, for flutter of physiological vessels conveying inviscid flow; the applications are to wheezing. Two-dimensional analyses predict that walls with coupled axial and normal displacements are unstable to long wave flutter, while walls without such coupling are stable; such coupling has little effect on the short wave behaviour. Qualitatively correct coupled approximations must account for inertia in both the axial and the normal motion of the wall. If the coupling, wall to fluid mass ratio, and stiffness terms are correctly represented, the tube cross-section has little effect. Long wave flutter occurs in simple 1-D models. However, there are qualitative errors at all finite wavelengths. This is because wall normal inertia is neglected. Further analysis suggests that an axisymmetric shell combined with a 1-D flow might provide a 1-D model that is qualitatively correct at all wavelengths and quantitatively useful for wavelengths as small as one-half the tube circumference.