Self-propelling, soft, and slender structures in fluids: Cosserat rods immersed in the velocity–vorticity formulation of the incompressible Navier–Stokes equations

We present a hybrid Eulerian–Lagrangian method for the direct simulation of three-dimensional, heterogeneous, active, and self-propelling structures made of soft fibers and operating in incompressible viscous flows. Fiber-based organization of matter is pervasive in nature and engineering, from biological architectures made of cilia, hair, muscles or bones to polymers, composite materials or soft robots. In nature, many such structures are adapted to manipulate flows for feeding, swimming or energy harvesting, through mechanisms that are often not fully understood. While simulations can support the analysis (and subsequent translational engineering) of these systems, extreme fibers’ aspect-ratios, large elastic deformations, two-way coupling with three-dimensional flows, and self-propulsion all render the problem numerically challenging. To address this, we couple Cosserat rod theory, where fibers’ dynamics is accurately captured in one-dimensional fashion, with the velocity–vorticity formulation of the Navier–Stokes equations, through a virtual boundary technique. The favorable properties of the resultant hydroelastic solver are demonstrated against a battery of benchmarks, and further showcased in a range of multi-physics scenarios, involving magnetic actuation, viscous streaming, biomechanics, multi-body interaction, and untethered swimming.