Rheology of dense granular suspensions under extensional flow

We study granular suspensions under a variety of extensional deformations and simple shear using numerical simulations. The viscosity and Trouton's ratio (the ratio of extensional to shear viscosity) are computed as functions of solids volume fraction $\phi$ close to the limit of zero inertia. Suspensions of frictionless particles follow a Newtonian Trouton's ratio for $\phi$ all the way up to $\phi_0$, a universal jamming point that is independent of deformation type. In contrast, frictional particles lead to a deformation-type-dependent jamming fraction $\phi_m$, which is largest for shear flows. Trouton's ratio consequently starts off Newtonian but diverges as $\phi\to\phi_m$. We explain this discrepancy in suspensions of frictional particles by considering the particle arrangements at jamming. While frictionless particle suspensions have a nearly isotropic microstructure at jamming, friction permits more anisotropic contact chains that allow jamming at lower $\phi$ but introduce protocol dependence. Finally, we provide evidence that viscous number rheology can be extended from shear to extensional deformations, with a particularly successful collapse for frictionless particles. Extensional deformations are an important class of rheometric flow in suspensions, relevant to paste processing, granulation and high performance materials.