  This paper presents a theory to obtain the force per unit length acting on a
slender filament with a non-circular cross-section moving in a fluid at low
Reynolds number. Using a regular perturbation of the inner solution, we show
that the force per unit length has $O(1/ln(2A))$ + $O(\alpha/ln^2(2A))$
contributions driven by the relative motion of the particle and the local fluid
velocity and an $O(\alpha/(ln(2A)A))$ contribution driven by the gradient in
the imposed fluid velocity. Here, the aspect ratio ($A=l/a_0$) is defined as
the ratio of the size of the particle ($l$) and the cross-sectional dimension
($a_0$); and $\alpha$ is the amplitude of the non-circular perturbation. Using
thought experiments, we show that two-lobed and three-lobed cross-sections
affect the response to relative motion and velocity gradients, respectively. A
two-dimensional Stokes flow calculation is used to extend the perturbation
analysis to cross-sections that deviate significantly from a circle (i.e.,
$\alpha \sim O(1)$). We demonstrate the ability of our method to accurately
compute the resistance to translation and rotation of a slender triaxial
ellipsoid. Furthermore, we illustrate novel dynamics of straight rods in a
simple shear flow that translate and rotate quasi-periodically if they have
two-lobed cross-section; and rotate chaotically and translate diffusively if
they have a combination of two- and three-lobed cross-sections. Finally, we
show the remarkable ability of our theory to accurately predict the motion of
rings, retaining great accuracy for moderate aspect ratios ($\sim 10$) and
cross-sections that deviate significantly from a circle, thereby making our
theory a computationally inexpensive alternative to other Stokes flow solvers.
