  Slender objects are commonplace in microscale flow problems, from soft
deformable sensors to biological filaments such as flagella and cilia. Whilst
much research has focussed on the local translational motion of these slender
bodies, relatively little attention has been given to local rotation, even
though it can be the dominant component of motion. In this study, we explore a
classically motivated ansatz for the Stokes flow around a rotating slender body
via superposed rotlet singularities, which leads us to pose an alternative
ansatz that accounts for both translation and rotation. Through an asymptotic
analysis that is supported by numerical examples, we determine the suitability
of these flow ansatzes for capturing the fluid velocity at the surface of a
slender body, assuming local axisymmetry of the object but allowing the
cross-sectional radius to vary with arclength. In addition to formally
justifying the presented slender-body ansatzes, this analysis reveals a
markedly simple relation between the local angular velocity and the torque
exerted on the body, which we term resistive torque theory. Though reminiscent
of classical resistive force theories, this local relation is found to be
algebraically accurate in the slender-body aspect ratio, even when translation
is present, and is valid and required whenever local rotation contributes to
the surface velocity at leading asymptotic order.
