Three-dimensional streaming around a cylinder in slender microchannels

When oscillatory flow is driven past an obstruction, its inertia produces a secondary steady flow known as streaming. Although streaming has been utilized in a variety of microfluidic applications, a systematic understanding of streaming in highly confined microfluidic environments remains missing. Here, we study three-dimensional streaming flows by conducting experiments involving a cylindrical obstacle sandwiched in a microchannel with one dimension (channel depth) much shorter than the other two dimensions. Notably, we find that the flow changes direction across the channel depth, distinct from previous observations of three-dimensional microstreaming flows. We understand the observations by applying inertial lubrication theory to solve the incompressible Navier-Stokes equations for small oscillation amplitudes. We show that for slender channels, the streaming is produced by Stokes layers at the confining top and bottom walls of the channel, and reverses direction to maintain zero channel-averaged flux. Finally, we use particle tracking measurements from the experiments of streaming around cylinders with different aspect ratios at different driving frequencies, and find a streaming speed that decays as the inverse cube of distance from the cylinder, in quantitative agreement with our theory. The agreement between our theory and experiments is promising for the control of three-dimensional flows in microfluidic particle trapping and micromixing applications.