  Filaments are ubiquitous within the microscopic world. They occur frequently
in both biological and industrial environments and display varied and rich
dynamics. Their wide range of applications has spurred the development of a
special branch of asymptotics focused on the behaviour of filaments, called
slender-body theory. Slender-body theories are typically computationally
efficient and focus on the mechanics of an isolated fibre that is not too
curved. However, slender-body theories that work beyond these standard limits
are needed to explore more complex systems. Recently, we developed tubular-body
theory for slow viscous flows, an approach similar to slender-body theory that
allows the hydrodynamic traction on any isolated cable-like body in a highly
viscous fluid to be determined exactly. In this paper, we extend tubular-body
theory to model filaments near plane interfaces by performing an similar
expansion on the single-layer boundary integral equations for bodies by a plane
interface. In the derivation of the new theory, called tubular-body theory for
interfaces, we established a criteria for the convergence of the tubular-body
theory series representation, before comparing the result to boundary integral
simulations for a prolate spheroid by a wall. The tubular-body theory for
interfaces simulations are found to capture the lubrication effects when close
to the plane wall. Finally we simulate the hydrodynamics of a helix beneath a
free interface and a plane wall to demonstrate the broad applicability of the
technique.
