A finite element method for simulating soft active non-shearable rods immersed in generalized Newtonian fluids

Abstract

We propose a finite element method for simulating one-dimensional solid models with finite thickness and finite length that move and experience large deformations while immersed in generalized Newtonian fluids. The method is oriented towards applications involving microscopic devices or organisms in the soft-bio-matter realm. By considering that the strain energy of the solid may explicitly depend on time, we incorporate a mechanism for active response. The solids are modeled as Cosserat rods, a detailed formulation being provided for the planar non-shearable case. The discretization adopts one-dimensional Hermite elements for the rod and two-dimensional low-order Lagrange elements for the fluid’s velocity and pressure. The fluid mesh is boundary-fitted, with remeshing at each time step. Several time marching schemes are studied, of which a semi-implicit scheme emerges as most effective. The method is demonstrated in very challenging examples: the roll-up of a rod to circular shape and later sudden release, the interaction of a soft rod with a fluid jet and the active self-locomotion of a sperm-like rod. The article includes a detailed description of a code that implements the method in the Firedrake library.