Rheology of suspensions of viscoelastic spheres: deformability as an effective volume fraction

We study suspensions of deformable (viscoelastic) spheres in a Newtonian solvent in plane Couette geometry, by means of direct numerical simulations. We find that in the limit of vanishing inertia the effective viscosity $\mu$ of the suspension increases as the volume-fraction occupied by the spheres $\Phi$ increases and decreases as the elastic modulus of the spheres $G$ decreases; the function $\mu(\Phi,G)$ collapses to an universal function, $\mu(\Phi_e)$, with a reduced effective volume fraction $\Phi_e(\Phi,G)$. Remarkably, the function $\mu(\Phi_e)$ is the well-known Eilers fit that describes the rheology of suspension of rigid spheres at all $\Phi$. Our results suggest new ways to interpret macro-rheology of blood.