Accuracy of slender body theory in approximating force exerted by thin fiber on viscous fluid

We consider the mapping properties of integral operator arising in nonlocal slender body theory (SBT) for model geometry a straight, periodic filament. It is well known that classical singular SBT suffers from high wavenumber instabilities, making it unsuitable approximating inverse problem, where fiber velocity prescribed and must be inverted to find force density along fiber. Regularizations therefore used instead. Here, we two regularization methods: spectral truncation -regularization Tornberg Shelley (2004). compare these approximations underlying partial differential equation (PDE) solution, which problem simply Stokes Dirichlet with data constrained constant on cross sections. For straight-but-periodic radius , explicitly calculate spectrum both PDE approximations. prove original agrees closely at low wavenumbers but differs frequencies, allowing us define truncated approximation cutoff . -regularized approximations, obtain rigorous -based convergence solution as : A regularity gives convergence, while least yields convergence. Moreover, determine dependence error estimate parameter