Many-particle hydrodynamic interactions in parallel-wall geometry: Cartesian-representation method

TL;DR

This paper introduces a new computational method combining spherical and Cartesian basis sets to accurately evaluate hydrodynamic interactions among particles confined between parallel walls, surpassing previous approximation techniques.

Contribution

We developed a novel algorithm that accurately computes the many-particle friction matrix in wall-confined suspensions using basis transformations between spherical and Cartesian flows.

Findings

The superposition approximation is adequate for single-particle dominated quantities.

The superposition approximation fails for collective phenomena like transverse resistance.

Our method enables precise analysis of hydrodynamic interactions in confined geometries.

Abstract

This paper describes the results of our theoretical and numerical studies of hydrodynamic interactions in a suspension of spherical particles confined between two parallel planar walls, under creeping-flow conditions. We propose a novel algorithm for accurate evaluation of the many-particle friction matrix in this system--no such algorithm has been available so far. Our approach involves expanding the fluid velocity field into spherical and Cartesian fundamental sets of Stokes flows. The interaction of the fluid with the particles is described using the spherical basis fields; the flow scattered with the walls is expressed in terms of the Cartesian fundamental solutions. At the core of our method are transformation relations between the spherical and Cartesian basis sets. These transformations allow us to describe the flow field in a system that involves both the walls and particles. We used our accurate numerical results to test the single-wall superposition approximation for the hydrodynamic friction matrix. The approximation yields fair results for quantities dominated by single particle contributions, but it fails to describe collective phenomena, such as a large transverse resistance coefficient for linear arrays of spheres.