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Many engineering and physiological applications deal with situations when afluid is moving in flexible tubes with elastic walls. In the real-lifeapplications like blood flow, there is often an additional complexity ofvorticity being present in the fluid. We present a theory for the dynamics ofinteraction of fluids and structures. The equations are derived using thevariational principle, with the incompressibility constraint of the fluidgiving rise to a pressure-like term. In order to connect this work with theprevious literature, we consider the case of inextensible and unshearable tubewith a straight centerline. In the absence of vorticity, our model reduces toprevious models considered in the literature, yielding the equations ofconservation of fluid momentum, wall momentum and the fluid volume. We showthat even when the vorticity is present, but is kept at a constant value, thecase of an inextensible, unshearable and straight tube with elastics wallscarrying a fluid allows an alternative formulation, reducing to a singlecompact equation for the back-to-labels map instead of three conservationequations. That single equation shows interesting instability in solutions whenthe vorticity exceeds a certain threshold. Furthermore, the equation in stableregime can be reduced to Boussinesq-type, KdV and Monge-Amp\`ere equationsequations in several appropriate limits, namely, the first two in the limit oflong time and length scales and the third one in the additional limit of thesmall cross-sectional area. For the unstable regime, we numerical solutionsdemonstrate the spontaneous appearance of large oscillations in thecross-sectional area.

Swirling fluid flow in flexible, expandable elastic tubes: variational approach, reductions and integrability