Steady Streaming in a Hele-Shaw Cell

This study addresses pulsating flow about a cylindrical object in a Hele-Shaw cell in the distinguished double limit where the characteristic viscous time is comparable to the oscillation time and the stroke length is comparable to the cell height. The ratio of the stroke length to the cylinder radius is used as a small parameter to analyze the flow by matched asymptotic expansions, with consideration given to two distinct flow regions, namely, a thin shear layer surrounding the cylinder whence axial vorticity is shed periodically, and a slender outer region scaling with the cylinder radius where streamlines are nearly parallel to the bounding walls. The incompressible Navier-Stokes equations are solved in both regions to describe the leading-order pulsating flow and the first-order corrections, the latter including a steady-streaming component. The steady outer flow field in the center-plane of the cell, whose streamlines are parameter-independent, is shown to reverse direction for a critical value of the fluid viscosity. Said flow is also shown to possess four vortices that perpetually entrap the majority of fluid surrounding the cylinder. The results of this study are of significant interest in micro-fluidic applications such as particle sorting and valveless pumping.