  In this article we address flow problems that carry a multiscale character in
time. In particular we consider the Navier-Stokes flow in a channel on a fast
scale that influences the movement of the boundary which undergoes a
deformation on a slow scale in time. We derive an averaging scheme that is of
first order with respect to the ratio of time-scales $\epsilon$. In order to
cope with the problem of unknown initial data for the fast scale problem, we
assume near-periodicity in time. Moreover, we construct a second-order accurate
time discretisation scheme and derive a complete error analysis for a
corresponding simplified ODE system. The resulting multiscale scheme does not
ask for the continuous simulation of the fast scale variable and shows powerful
speed-ups up to 1:10000 compared to a resolved simulation. Finally, we present
some numerical examples for the full Navier-Stokes system to illustrate the
convergence and performance of the approach.
