Many engineering and physiological applications deal with situations when a
fluid is moving in flexible tubes with elastic walls. In the real-life
applications like blood flow, there is often an additional complexity of
vorticity being present in the fluid. We present a theory for the dynamics of
interaction of fluids and structures. The equations are derived using the
variational principle, with the incompressibility constraint of the fluid
giving rise to a pressure-like term. In order to connect this work with the
previous literature, we consider the case of inextensible and unshearable tube
with a straight centerline. In the absence of vorticity, our model reduces to
previous models considered in the literature, yielding the equations of
conservation of fluid momentum, wall momentum and the fluid volume. We show
that even when the vorticity is present, but is kept at a constant value, the
case of an inextensible, unshearable and straight tube with elastics walls
carrying a fluid allows an alternative formulation, reducing to a single
compact equation for the back-to-labels map instead of three conservation
equations. That single equation shows interesting instability in solutions when
the vorticity exceeds a certain threshold. Furthermore, the equation in stable
regime can be reduced to Boussinesq-type, KdV and Monge-Amp\`ere equations
equations in several appropriate limits, namely, the first two in the limit of
long time and length scales and the third one in the additional limit of the
small cross-sectional area. For the unstable regime, we numerical solutions
demonstrate the spontaneous appearance of large oscillations in the
cross-sectional area.Comment: 57 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1805.1102
