# The Einstein viscosity with fluid elasticity

## TL;DR

This paper develops a first-order correction to the viscosity of dilute suspensions in viscoelastic fluids, revealing shear-thickening behavior due to fluid elasticity and emphasizing the importance of extensional rheology modeling.

## Contribution

It introduces a perturbation theory for suspension viscosity considering fluid elasticity effects up to second order in Weissenberg number, validated by numerical simulations.

## Key findings

- Suspension shear-thickens near particles due to elastic stretching.
- Extensional rheology significantly influences shear rheology predictions.
- Strain-rate thickening observed at higher order in Weissenberg number.

## Abstract

We give the first correction to the suspension viscosity due to fluid elasticity for a dilute suspension of spheres in a viscoelastic medium. Our perturbation theory is valid to $O(\phi\mathrm{Wi}^2)$ in the Weissenberg number $\mathrm{Wi}=\dot\gamma \lambda$, where $\dot\gamma$ is the typical magnitude of the suspension velocity gradient, and $\lambda$ is the relaxation time of the viscoelastic fluid. For shear flow we find that the suspension shear-thickens due to elastic stretching in strain hot spots near the particle, despite the fact that the stress inside the particles decreases relative to the Newtonian case. We thus argue that it is crucial to correctly model the extensional rheology of the suspending medium to predict the shear rheology of the suspension. For uniaxial extensional flow we correct existing results at $O(\phi\mathrm{Wi})$, and find dramatic strain-rate thickening at $O(\phi\mathrm{Wi}^2)$. We validate our theory with fully resolved numerical simulations.