Three-dimensional streaming around an obstacle in a Hele-Shaw cell

The inertial rectification of oscillatory flow driven past an obstacle produces a secondary steady flow called "streaming", which has been used in a variety of microfluidic applications. Understanding of such flows is limited largely to two-dimensional configurations, and typically neglects the high degree of vertical confinement provided by boundary walls in practical microfluidic devices. We develop a three-dimensional streaming theory around an obstruction sandwiched in a microchannel with a Hele-Shaw geometry, in which one dimension (depth) is much shorter than the other two. Applying inertial lubrication theory to solve the incompressible Naiver-Stokes equations for small oscillation amplitudes, we show that the streaming flow has a three-dimensional structure. Notably, the flow changes direction across the depth of channel, distinct from previous observations of three-dimensional microscreaming flows. We show that this vertical flow reversal is supported by our experiments of streaming around short cylinders in microchannels. Our theory also predicts that flow velocity decays as the inverse cube of distance from the cylinder, more rapidly than that expected from two-dimensional approaches. We verify this decay rate quantitatively using particle tracking measurements from experiments of streaming around cylinders with different aspect ratios at different driving frequencies. The ability to generate recirculating three-dimensional flows in microconfined geometries is promising for particle trapping and micromixing applications.