Analytical solution of the $\mu(I)-$rheology for fully developed granular flows in simple configurations

TL;DR

This paper provides an analytical solution for the $$-rheology in steady, fully developed granular flows, predicting shear band scaling and plug flow behavior in simple configurations, enhancing understanding of dense granular flow dynamics.

Contribution

It offers the first analytical solutions for the $$-rheology in specific flow geometries, linking flow characteristics to parameters like velocity and pressure.

Findings

Shear band height scales with $U_0^{1/4}P_0^{1/2}$.

Plug flow size increases as pressure gradient decreases.

Granular material behaves like a Bingham plastic at small pressure gradients.

Abstract

Using the $\mu(I)$ continuum model recently proposed for dense granular flows, we study theoretically steady and fully developed granular flows in two configurations: a plane shear cell and a channel made of two parallel plates (Poiseuille configuration). In such a description, the granular medium behaves like a fluid whose viscosity is a function of the inertia. In the shear plane geometry our calculation predicts that the height of the shear bands scales with $U_0^{1/4}P_0^{1/2}$, where $U_0$ is the velocity of the moving plate and $P_0$ the pressure applied at its top. In the Poiseuille configuration, the medium is sheared between the lateral boundaries and a plug flow is located in the center of the channel. The size of the plug flow is found to increase for a decreasing pressure gradient. We show that, for small pressure gradient, the granular material behaves like a Bingham plastic fluid.