Slow linear shear flow past discrete particles adhered to a plane wall

TL;DR

This paper revisits and simplifies the exact solution for flow disturbance caused by particles adhered to a wall in shear flow, analyzing the flow's far-field behavior and its implications for microfluidic applications.

Contribution

It provides a detailed, simplified derivation of O'Neill's solution and extends the analysis to the flow's universal far-field behavior for particles on a surface.

Findings

Flow disturbance is governed by the particle's stress moment.

The flow due to adhered particles does not decay away from the surface.

The superposition of flow fields estimates the net flow from particle layers.

Abstract

Linear shear flow bounded by a plane wall is an idealization that occurs in microfluidic devices and many other applications. Perfect plane approximation neglects surface irregularities and discrete particles adsorbed at the surface. Here we study the disturbance to the linear shear flow due to the particle(s) rigidly attached to the surface. We first revisit the exact solution of O'Neill for a spherical particle in contact with an infinite plane boundary. While the original paper contains multiple typos and provides very few details of the derivation, we present detailed solution accompanied by an alternative and simpler derivation of the viscous force and the torque exerted on the particle. We further study the universal far-field behavior of the flow due to an arbitrary particle adhered to the surface, and demonstrate that it is controlled by the stress moment of magnitude depending on particle's volume and shape. Using the revised O'Neill solution, we compute the stress moment for a spherical particle. Using the far-field asymptotic form of the flow we estimate the net flow due to uniform and sparse layer of discrete adsorbates by superposition and demonstrate that it does not decay away from the plane.