Rods in flows: the PDE theory of immersed elastic filaments

Abstract: We investigate a family of curve evolution equations approximating the motion of a Kirchhoff rod immersed in a low Reynolds number fluid. The rod is modeled as a framed curve whose energy consists of the bending energy of the curve and the twisting energy of the frame. The equations we consider may be realized as gradient flows of the rod energy under a certain anisotropic metric coming from resistive force theory. Ultimately, our goal is to provide a comprehensive treatment of the PDE theory of immersed rod dynamics. We begin by analyzing the problem without the frame, in which case the evolution is globally well-posed and solutions asymptotically converge to Euler elasticae. Next, in the planar setting, we demonstrate the existence of large time-periodic solutions forced by intrinsic curvature relevant to undulatory swimming. Finally, the majority of the paper is devoted to the evolution of an immersed Kirchhoff rod in 3D, which involves a strong coupling between the curve and frame. Under a physically reasonable assumption on the curvature, solutions exist globally-in-time and asymptotically converge to rod equilibria.