The incompressible Stokes equations can classically be recast in a boundary
integral (BI) representation, which provides a general method to solve
low-Reynolds number problems analytically and computationally. Alternatively,
one can solve the Stokes equations by using an appropriate distribution of flow
singularities of the right strength within the boundary, a method particularly
useful to describe the dynamics of long slender objects for which the numerical
implementation of the BI representation becomes cumbersome. While the BI
approach is a mathematical consequence of the Stokes equations, the singularity
method involves making judicious guesses that can only be justified a
posteriori. In this paper we use matched asymptotic expansions to derive an
algebraically accurate slender-body theory directly from the BI representation
able to handle arbitrary surface velocities and surface tractions. This
expansion procedure leads to sets of uncoupled linear equations and to a single
one-dimensional integral equation identical to that derived by Keller and
Rubinow (1976) and Johnson (1979) using the singularity method. Hence we show
that it is a mathematical consequence of the BI approach that the leading-order
flow around a slender body can be represented using a distribution of
singularities along its centreline. Furthermore when derived from either the
single-layer or double-layer modified BI representation, general slender
solutions are only possible in certain types of flow, in accordance with the
limitations of these representations