Integral representation of channel flow with interacting particles

TL;DR

This paper develops a boundary integral method for modeling low-Reynolds-number flow with suspended particles in a channel, deriving analytical and numerical tools to understand particle interactions and their long-term behavior.

Contribution

It introduces a new integral representation for particle-laden channel flow, including a multipole expansion and a numerical approach for hydrodynamic interactions.

Findings

Derived a dipolar multipole expansion for flow around particles.

Confirmed analytically and numerically the 'pair exchange' phenomenon.

Predicted particle separation into singlets and pairs in dilute suspensions.

Abstract

We construct a boundary integral representation for the low-Reynolds-number flow in a channel in the presence of freely-suspended particles (or droplets) of arbitrary size and shape. We demonstrate that lubrication theory holds away from the particles at horizontal distances exceeding the channel height and derive a multipole expansion of the flow which is dipolar to the leading approximation. We show that the dipole moment of an arbitrary particle is a weighted integral of the stress and the flow at the particle surface, which can be determined numerically. We introduce the equation of motion that describes hydrodynamic interactions between arbitrary, possibly different, distant particles, with interactions determined by the product of the mobility matrix and the dipole moment. Further, the problem of three identical interacting spheres initially aligned in the streamwise direction is considered and the experimentally observed "pair exchange" phenomenon is derived analytically and confirmed numerically. For non-aligned particles, we demonstrate the formation of a configuration with one particle separating from a stable pair. Our results suggest that in a dilute initially homogeneous particulate suspension flowing in a channel the particles will eventually separate into singlets and pairs.