Unifying viscous and inertial regimes of discontinuous shear thickening suspensions

Abstract

The rheology of dense suspensions in the viscous regime can be characterized by the viscous number J and granular flows by the inertia number I, both relating the shear rate γ ˙ to the confining pressure P. Furthermore, several works have shown that the rheology of suspensions and granular flows can be unified, and this unification is used to characterize suspensions with non-negligible particle inertia from the viscous to the inertial regime (i.e., granular flows). There have also been recent works that apply this unification to suspensions where the number of frictional contacts increases upon the shear rate with constant packing fraction, a model that has successfully described discontinuous shear thickening in suspension flows. We and other researchers have previously shown that the fraction of frictional contacts χ f is a key control parameter for friction-driven shear thickening in the viscous regime. It is, however, difficult to control and study the effect of the number of frictional contacts on the rheology of dense suspensions at constant packing fraction as this quantity increases sharply around a threshold shear rate. In this work, we extend our previous work and use numerical simulations to study particle flows in both the viscous and inertial regimes as well as for suspensions in the crossover between these regimes with varying χ f. By having pressure-controlled simulations, we are able to keep χ f at values between 0 and 1, hence critically testing the validity of a unification at intermediate values. With the help of a binary model composed of nonfrictional and frictional particles, where χ f is well-controlled, we manage to obtain expressions for constitutive laws in both limits. For the critical load model, we find a simple relationship between χ f and the pressure-rescaled threshold force f ^ applicable for both the viscous and inertial regimes. Combining these expressions, we then show that suspensions in the crossover between these regimes can be characterized by four dimensionless numbers, f ^, K, K μ, and K z, where the last three are simple combinations of J and I and verified against numerical simulations. These expressions can be further simplified by approximating K μ = K, which we show is a good approximation close to shear jamming or at low Stokes numbers. In the end, we construct a dimensionless parameter encoding various shear protocols and show predictions of behaviors of suspensions under different shear conditions. Having derived constitutive relations from constant pressure simulations, we finally test our relations under constant volume assumption which indeed well captures the various discontinuous shear thickening behaviors seen at different Stokes numbers. Finally, we discuss the role of having a varying microscopic friction coefficient μ p as a function of the normal force.