Universal square-root-log scaling for slender-body end effects

Abstract

The charge on a conducting cylinder becomes uniformly distributed at large aspect ratios, albeit very slowly and with nonuniformity persisting near the ends. We show that the charge density and field magnitude are enhanced near the ends by a factor scaling as the square-root of the logarithm of the aspect ratio. This scaling is obtained by locally resumming a perturbation-series solution to the integral equation of slender-body theory. The same scaling applies to a broad class of "truncated" slender bodies -- including cylinders and shapes tapered near the ends on the cross-sectional scale -- as well as other physical setups. We validate this scaling through boundary-element simulations for cylinders with flat and hemispherical caps, and demonstrate its applicability to diffusion, plasmonics and Stokes flow.