Time-averaged oscillatory flows in deformable channels

The interaction between deformable surfaces and oscillatory driving is known to yield complex secondary time-averaged flows due to inertial and elastic nonlinearities. Here, we revisit the problem of oscillatory flow in a channel with a deformable top wall, and analyze it under lubrication theory for small deformations but arbitrary Stokes numbers. We find that the oscillatory pressure does not vary linearly along the length of a deformable channel, but instead decays exponentially with spatial oscillations. We show that this decay occurs over an elasto-visco-inertial length scale that depends on the aspect ratio of the channel, its elastic compliance, and the frequency of the oscillatory flow, but is independent of the amplitude of deformation. The inertia of the oscillatory flow along with the deformation of the channel results in a time-averaged secondary flow, which we quantify using perturbation theory. Our theoretical findings are in excellent quantitative agreement with previously reported finite element simulations across a range of dimensionless parameters that characterize the oscillatory flow and the compliance of the channel. We then adapt these ideas to compliant cylindrical tubes in which radial deformation is resisted by elastic hoop stresses.