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The interaction between deformable surfaces and oscillatory driving is known to produce complex secondary time-averaged flows due to inertial and elastic nonlinearities. Here, we revisit the problem of oscillatory flow in a cylindrical tube with a deformable wall, and analyse it under a long-wave theory for small deformations, but for arbitrary Womersley 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 material properties of the fluid and the elastic walls, the geometry of the system, and the frequency of the oscillatory flow, but is independent of the amplitude of deformation. Inertial and geometric nonlinearities associated with the elastic deformation of the channel drive a time-averaged secondary flow. We quantify the flow using numerical solutions of the perturbation theory, and gain insight by developing analytic approximations. The theory identifies a complex non-monotonic dependence of the time-averaged flux on the elastic compliance and inertia, including a reversal of the flow. Finally, we show that our analytic theory is in excellent quantitative agreement with the three-dimensional direct numerical simulations of Pandeet al.(Phys. Rev. Fluids, vol. 8, no. 12, 2023, 124102). 

Driving oscillatory flow around an obstacle generates, due to inertial rectification, a steady ‘streaming’ flow that is useful in a host of microfluidic applications. While theory has focused largely on two-dimensional flows, streaming in many practical microfluidic devices is three-dimensional due to confinement. We develop a three-dimensional streaming theory around an obstacle in a microchannel with a Hele-Shaw-like geometry, where one dimension (depth) is much shorter than the other two dimensions. Utilizing inertial lubrication theory, we demonstrate that the time-averaged streaming flow has a three-dimensional structure. Notably, the flow reverses direction across the depth of the channel, which is a feature not observed in less confined streaming set-ups. This feature is confirmed by our experiments of streaming around a cylinder sandwiched in a microchannel. Our theory also predicts that the streaming velocity decays as the inverse cube of the distance from the cylinder, faster than that expected from previous two-dimensional approaches. We verify this velocity decay quantitatively using particle tracking measurements from experiments of streaming around cylinders with different aspect ratios at different driving frequencies.