# Visco-inertial regimes of immersed granular flows

By means of extensive coupled MD-LBM numerical simulations, allowing for grain dynamics and full sub-particle resolution of the fluid phase, we analyze steady inertial granular flows driven by a viscous fluid. We show that, for a broad range of system parameters (shear rate, confining stress, fluid viscosity and relative fluid-grain density), the frictional strength and packing fraction can be described by a single ‘visco-inertial’ dimensionless parameter combining the inertial and Stokes numbers. Remarkably, we also find that for all combinations of system parameters, including dry granular flows, the friction coefficient is a linear function of the ratio of shear-induced porosity to the overall porosity. This relation leads to a simple expression of the friction coefficient as a function of the visco-inertial number. We also map the frictional behavior under constant confining pressure into effective viscous behavior under volume-controlled conditions. This mapping predicts the divergence of effective normal and shear viscosities in inverse square of the distance to the critical packing fraction. Our results are in excellent agreement with recent available experimental data.

**Forces and pressure fields**

**Velocity field at low and high fluid viscosity**