Heil, M. & Waters, S.L. (2006) Transverse flows in rapidly
oscillating, elastic cylindrical shells.
Journal of Fluid Mechanics 547,
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 2D 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 of 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 3D collapsible
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