Motion of several slender rigid filaments in a Stokes flow

Abstract: We investigate the dynamics of several slender rigid bodies moving in a flow driven by the three-dimensional steady Stokes system in presence of a smooth background flow. More precisely we consider the limit where the thickness of these slender rigid bodies tends to zero with a common rate $\varepsilon$, while their volumetric mass density is held fixed, so that the bodies shrink into separated massless curves. While for each positive $\varepsilon$, the bodies' dynamics are given by the Newton equations and correspond to some coupled second-order ODEs for the positions of the bodies, we prove that the limit equations are decoupled first-order ODEs whose coefficients only depend on the limit curves and on the background flow. These coefficients appear through appropriate renormalized Stokes resistance tensors associated with each limit curve, and through renormalized Fax\'en-type force and torque associated with the limit curves and the background flow. We establish a rate of convergence of the curves of order $O( \vert \log \varepsilon \vert^{-1/2})$. We also determine the limit effect due to the limit curves on the fluid, in the spirit of the immersed boundary method. Both for the convergence of the filament velocities and the fluid velocities we identify an initial exponential relaxation within a $O(\varepsilon^2 \vert \log \varepsilon \vert)$ time.