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

TL;DR

This paper develops a 3D streaming theory for flow around obstacles in Hele-Shaw microchannels, revealing a flow reversal across the depth and a faster decay rate of streaming velocity, validated by experiments.

Contribution

It introduces a novel 3D streaming model for confined microchannels, extending previous 2D theories and experimentally confirming key predictions.

Findings

Flow reverses direction across the channel depth.

Streaming velocity decays as the inverse cube of distance.

Experimental validation of decay rate and flow structure.

Abstract

The application of oscillatory flow around an obstacle drives a steady ``streaming'' due to inertial rectification, which has been used in a host of microfluidic applications. While theory has focused largely on two-dimensional (2D) flows, streaming in many practical microfluidic devices is three-dimensional (3D) 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 changes direction across the depth of the channel, which is a feature not observed in less confined streaming setups. 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 decay rate quantitatively using particle tracking measurements from experiments of streaming around cylinders with different aspect ratios at different driving frequencies.