Suspensions of deformable particles in a Couette flow

We consider suspensions of deformable particles in a Newtonian fluid by means of fully Eulerian numerical simulations with a one-continuum formulation. We study the rheology of the visco-elastic suspension in plane Couette flow in the limit of vanishing inertia and examine the dependency of the effective viscosity $\mu$ on the solid volume-fraction $\Phi$, the capillary number $\mbox{Ca}$, and the solid to fluid viscosity ratio $\mbox{K}$. The suspension viscosity decreases with deformation and applied shear (shear-thinning) while still increasing with volume fraction. We show that $\mu$ collapses to an universal function, $\mu \left( \Phi^{\rm e} \right)$, with an effective volume fraction $\Phi^{\rm e}$, lower than the nominal one owing to the particle deformation. This universal function is well described by the Eilers fit, which well approximate the rheology of suspension of rigid spheres at all $\Phi$. We provide a closure for the effective volume fraction $\Phi^{\rm e}$ as function of volume fraction $\Phi$ and capillary number $\mbox{Ca}$ and demonstrate it also applies to data in literature for suspensions of capsules and red-blood cells. In addition, we show that the normal stress differences exhibit a non-linear behavior, with a similar trend as in polymer and filament suspensions. The total stress budgets reveals that the particle-induced stress contribution increases with the volume fraction $\Phi$ and decreases with deformability.