The Tension Determination Problem for an Inextensible Interface in 2D Stokes Flow

TL;DR

This paper investigates the mathematical problem of determining tension in an inextensible filament in 2D Stokes flow, analyzing existence, regularity, and singularity behavior, with implications for biological and engineering membrane models.

Contribution

It provides a rigorous analysis of the tension determination problem, establishing conditions for unique solutions and regularity properties, especially near circular configurations.

Findings

Unique tension solution exists if and only if the filament is not a circle.

Tension gains one derivative relative to the force density.

Regularity of tension depends differently on tangential and normal force components.

Abstract

Consider an inextensible closed filament immersed in a 2D Stokes fluid. Given a force density $\mathbf{F}$ defined on this filament, we consider the problem of determining the tension $\sigma$ on this filament that ensures the filament is inextensible. This is a subproblem of dynamic inextensible vesicle and membrane problems, which appear in engineering and biological applications. We study the well-posedness and regularity properties of this problem in H\"older spaces. We find that the tension determination problem admits a unique solution if and only if the closed filament is {\em not} a circle. Furthermore, we show that the tension $\sigma$ gains one derivative with respect to the imposed line force density $\mathbf{F}$, and show that the tangential and normal components of $\mathbf{F}$ affect the regularity of $\sigma$ in different ways. We also study the near singularity of the tension determination problem as the interface approaches a circle, and verify our analytical results against numerical experiment.