Transverse flows in rapidly oscillating elastic cylindrical shells

We analyse the flows in fluid-conveying tubes whose elastic walls perform small-amplitude high-frequency oscillations. We show that the velocity perturbations induced by the wall motion are dominated by their transverse components and use numerical simulations to analyse the two-dimensional flows that develop in the tube's cross-sections. Asymptotic methods are then employed to derive explicit predictions for the flow fields and for the total viscous dissipation, whose magnitude plays an important role in the development of self-excited oscillations. We show that in cases with fluid-structure interaction, the coupled oscillations are controlled by the ratio or the fluid and wall densities, and by a material parameter that is equivalent to the Womersley number, and indicates the importance Of fluid inertia and wall elasticity relative to the fluid's viscosity. We present numerical simulations of the coupled oscillations and use asymptotic techniques to derive explicit predictions for their period and decay rate. Finally, we discuss the implications of our results for the development of self-excited oscillations in three-dimensional collapsible tubes.