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We consider suspensions of deformable particles in a Newtonian fluid by meansof fully Eulerian numerical simulations with a one-continuum formulation. Westudy the rheology of the visco-elastic suspension in plane Couette flow in thelimit of vanishing inertia and examine the dependency of the effectiveviscosity $\mu$ on the solid volume-fraction $\Phi$, the capillary number$\mbox{Ca}$, and the solid to fluid viscosity ratio $\mbox{K}$. The suspensionviscosity decreases with deformation and applied shear (shear-thinning) whilestill increasing with volume fraction. We show that $\mu$ collapses to anuniversal function, $\mu \left( \Phi^{\rm e} \right)$, with an effective volumefraction $\Phi^{\rm e}$, lower than the nominal one owing to the particledeformation. This universal function is well described by the Eilers fit, whichwell approximate the rheology of suspension of rigid spheres at all $\Phi$. Weprovide a closure for the effective volume fraction $\Phi^{\rm e}$ as functionof volume fraction $\Phi$ and capillary number $\mbox{Ca}$ and demonstrate italso applies to data in literature for suspensions of capsules and red-bloodcells. In addition, we show that the normal stress differences exhibit anon-linear behavior, with a similar trend as in polymer and filamentsuspensions. The total stress budgets reveals that the particle-induced stresscontribution increases with the volume fraction $\Phi$ and decreases withdeformability.

Suspensions of deformable particles in a Couette flow