  Fluid flows containing dilute or dense suspensions of thin fibers are
widespread in biological and industrial processes. To describe the motion of a
thin immersed fiber, or to describe the forces acting on it, it is convenient
to work with one-dimensional fiber centerlines and force densities rather than
two-dimensional surfaces and surface tractions. Slender body theories offer
ways to model and simulate the motion of immersed fibers using only
one-dimensional data. However, standard formulations can break down when the
fiber surface comes close to intersecting itself or other fibers. In this paper
we introduce a numerical method for a recently derived three-dimensional
slender body boundary value problem that can be stated entirely in terms of a
one-dimensional distribution of forces on the centerline. The method is based
on a new completed single-layer potential formulation of fluid velocity which
circumvents some of the traditional conditioning issues associated with the
unmodified single layer potential. We give numerical results demonstrating the
good conditioning and improved performance of the method in the presence of
near-intersections.
