Mediated interactions between rigid inclusions in two-dimensional elastic or fluid films

TL;DR

This paper investigates the interactions between rigid disk-like inclusions in two-dimensional elastic or viscous films, revealing that force-free conditions eliminate logarithmic divergences in their Green's functions, which aids understanding of membrane and microswimmer dynamics.

Contribution

It provides a detailed analysis of the interaction behavior of rigid inclusions in 2D films, highlighting the absence of divergence under force-free conditions, a less-explored aspect compared to 3D systems.

Findings

Logarithmic divergence is removed when inclusions are force-free.

Interactions are dominated by pairwise mutual effects.

Results support studies of membrane-inclusion systems and microswimmer dynamics.

Abstract

Interactions between rigid inclusions in continuous three-dimensional linearly elastic solids and low-Reynolds-number viscous fluids have largely been quantified during the past. Prime example systems are given by functionalized elastic composite materials or fluid colloidal suspensions. Here, we address the significantly less frequently studied situation of rigid inclusions in two-dimensional elastic or low-Reynolds-number fluid films. We concentrate on the situation in which disk-like inclusions remain well separated from each other and do not get into contact. Specifically, we demonstrate and explain that the logarithmic divergence of the associated Green's function is removed in the absence of net external forces on the inclusions, in line with physical intuition. For instance, this situation applies when only pairwise mutual interactions between the inclusions prevail. Our results will support, for example, investigations on membranes functionalized by appropriate inclusions, both of technical or biological origin, or the dynamics of active microswimmers in appropriately prepared thin films.