Slender body theory for particles with non-circular cross-sections

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.