  We propose a finite element method for simulating one-dimensional solid
models moving and experiencing 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 special case of a planar
non-shearable rod. The discretization adopts one-dimensional Hermite elements
for the rod and low-order Lagrange two-dimensional 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.
