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We consider the accuracy of slender body theory in approximating the forceexerted by a thin fiber on the surrounding viscous fluid when the fibervelocity is prescribed. We term this the slender body inverse problem, as it isknown that slender body theory converges to a well-posed PDE solution when theforce is prescribed and the fiber velocity is unknown. From a PDE perspective,the slender body inverse problem is simply the Dirichlet problem for the Stokesequations, but from an approximation perspective, nonlocal slender body theoryexhibits high wavenumber instabilities which complicate analysis. Here weconsider two methods for regularizing the slender body approximation: spectraltruncation and the $\delta$-regularization of Tornberg and Shelley (2004). Fora straight, periodic fiber with constant radius $\epsilon>0$, we explicitlycalculate the spectrum of the operator mapping fiber velocity to force for boththe PDE and the approximations. We show that the spectrum of the originalslender body approximation agrees closely with the PDE solution at lowwavenumbers but differs at high frequencies, allowing us to define a truncatedapproximation with a wavenumber cutoff $\sim1/\epsilon$. For both the truncatedand $\delta$-regularized approximations, we obtain similar convergence resultsto the PDE solution as $\epsilon\to0$: a fiber velocity with $H^1$ regularitygives $O(\epsilon)$ convergence, while a fiber velocity with at least $H^2$regularity yields $O(\epsilon^2)$ convergence. Moreover, we determine thedependence of the $\delta$-regularized error estimate on the regularizationparameter $\delta$.

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