  We develop a theory of fluid--structure interaction (FSI) between an
oscillatory Newtonian fluid flow and a compliant conduit. We consider the
canonical geometries of a 2D channel with a deformable top wall and an
axisymmetric deformable tube. Focusing on the hydrodynamics, we employ a linear
relationship between wall displacement and hydrodynamic pressure, which has
been shown to be suitable for a leading-order-in-slenderness theory. The
slenderness assumption also allows the use of lubrication theory, and the flow
rate is related to the pressure gradient (and the tube/wall deformation) via
the classical solutions for oscillatory flow in a channel and in a tube
(attributed to Womersley). Then, by two-way coupling the oscillatory flow and
the wall deformation via the continuity equation, a one-dimensional nonlinear
partial differential equation (PDE) governing the instantaneous pressure
distribution along the conduit is obtained, without \textit{a priori}
assumptions on the magnitude of the oscillation frequency (\textit{i.e.}, at
arbitrary Womersley number). We find that the cycle-averaged pressure (for
harmonic pressure-controlled conditions) deviates from the expected steady
pressure distribution, suggesting the presence of a streaming flow. An
analytical perturbative solution for a weakly deformable conduit is obtained to
rationalize how FSI induces such streaming. In the case of a compliant tube,
the results obtained from the proposed reduced-order PDE and its perturbative
solutions are validated against three-dimensional, two-way-coupled direct
numerical simulations. We find good agreement between theory and simulations
for a range of dimensionless parameters characterizing the oscillatory flow and
the FSI, demonstrating the validity of the proposed theory of oscillatory flows
in compliant conduits at arbitrary Womersley number.
