The role of boundaries for displacements and motion in two-dimensional fluid or elastic films and membranes

TL;DR

This paper investigates how boundaries influence flow and displacement fields in two-dimensional fluid and elastic films, showing that boundaries can stabilize systems with net forces, unlike unbounded three-dimensional media.

Contribution

It demonstrates that boundaries, especially no-slip ones, are crucial for stabilizing flows and displacements in 2D films with net forces, providing quantitative analysis of mobility and interactions.

Findings

Boundaries cause flow and displacement fields to decay inversely with distance.

No-slip boundaries stabilize the system by regulating flow and displacement.

Free-slip boundaries require kinked boundaries for stabilization.

Abstract

Thin fluid or elastic films and membranes are found in nature and technology, for instance, as confinements of living cells or in loudspeakers. When applying a net force, resulting flows in an unbounded two-dimensional incompressible low-Reynolds-number fluid or displacements in a two-dimensional linearly elastic solid seem to diverge logarithmically with the distance from the force center, which has led to some debate. Recently, we have demonstrated that such divergences cancel when the total (net) force vanishes. Here, we illustrate that, if a net force is present, the boundaries play a prominent role. Already a single no-slip boundary regulates the flow and displacement fields and leads to their decay to leading order inversely in distance from a force center and the boundary. In other words, it is the boundary that stabilizes the system in this situation, unlike the three-dimensional case, where an unbounded medium by itself is able to absorb a net force. We quantify the mobility and displaceability of an inclusion as a function of the distance from the boundary, as well as interactions between different inclusions. In the case of free-slip boundary conditions, a kinked boundary is necessary to achieve stabilization.