# Numerical study of filament suspensions at finite inertia

## TL;DR

This study numerically investigates how filament flexibility and inertia influence suspension rheology, revealing that increased inertia raises viscosity and normal stress differences, with microstructure changes affecting overall flow behavior.

## Contribution

It provides new insights into the effects of filament bending stiffness and inertia on suspension rheology using detailed numerical simulations.

## Key findings

- Viscosity decreases with filament flexibility and increases with Reynolds number.
- Normal stress difference is positive and peaks at intermediate filament stiffness.
- Filament microstructure becomes more ordered at higher inertia, reducing interactions.

## Abstract

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.