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Consider an inextensible closed filament immersed in a 2D Stokes fluid. Givena force density $\mathbf{F}$ defined on this filament, we consider the problemof determining the tension $\sigma$ on this filament that ensures the filamentis inextensible. This is a subproblem of dynamic inextensible vesicle andmembrane problems, which appear in engineering and biological applications. Westudy the well-posedness and regularity properties of this problem in H\"olderspaces. We find that the tension determination problem admits a unique solutionif and only if the closed filament is {\em not} a circle. Furthermore, we showthat the tension $\sigma$ gains one derivative with respect to the imposed lineforce density $\mathbf{F}$, and show that the tangential and normal componentsof $\mathbf{F}$ affect the regularity of $\sigma$ in different ways. We alsostudy the near singularity of the tension determination problem as theinterface approaches a circle, and verify our analytical results againstnumerical experiment.

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