  We present a numerical study on the rheology of semi-dilute and concentrated
filament suspensions of different bending stiffness and Reynolds number, with
the immersed boundary method used to couple the fluid and solid. The filaments
are considered as one-dimensional inextensible slender bodies with fixed aspect
ratio, obeying the Euler-Bernoulli beam equation. To understand the global
suspension behavior we relate it to the filament microstructure, deformation
and elastic energy and examine the stress budget to quantify the effect of the
elastic contribution. At fixed volume fraction, the viscosity of the suspension
reduces when decreasing the bending rigidity and grows when increasing the
Reynolds number. The change in the relative viscosity is stronger at finite
inertia, although still in the laminar flow regime as considered here.
Moreover, we find the first normal stress difference to be positive as in
polymeric fluids, and to increase with the Reynolds number; its value has a
peak for an intermediate value of the filament bending stiffness. The peak
value is found to be proportional to the Reynolds number, moving towards more
rigid suspensions at larger inertia. Moreover, the viscosity increases when
increasing the filament volume fraction, and the rate of increase of the
filament stress with the bending rigidity is stronger at higher Reynolds
numbers and reduces with the volume fraction. We show that this behaviour is
associated with the formation of a more ordered structure in the flow, where
filaments tend to be more aligned and move as a compact aggregate, thus
reducing the filament-filament interactions despite their volume fraction
increases.
