A note on Stokes' problem in dense granular media using the $\mu (I)$--rheology

Abstract

The classical Stokes' problem describing the fluid motion due to a steadily moving infinite wall is revisited in the context of dense granular flows of mono-dispersed beads using the recently proposed $\mu (I)$--rheology. In Newtonian fluids, molecular diffusion brings about a self-similar velocity profile and the boundary layer in which the fluid motion takes place increases indefinitely with time $t$ as $\sqrt{\nu t}$, where $\nu$ is the kinematic viscosity. For a dense granular visco-plastic liquid, it is shown that the local shear stress, when properly rescaled, exhibits self-similar behaviour at short-time scales and it then rapidly evolves towards a steady-state solution. The resulting shear layer increases in thickness as $\sqrt{{\nu }_{g}t}$ analogous to a Newtonian fluid where ${\nu }_g$ is an equivalent granular kinematic viscosity depending not only on the intrinsic properties of the granular media such as grain diameter $d$, density $\rho$ and friction coefficients but also on the applied pressure $p_w$ at the moving wall and the solid fraction $\varphi$ (constant). In addition, the $\mu (I)$--rheology indicates that this growth continues until reaching the steady-state boundary layer thickness ${\delta }_s={\beta }_w(p_w/\varphi \rho g)$, independent of the grain size, at about a finite time proportional to ${\beta }_w^2(p_w/\rho gd{)}^{3/2}\sqrt{d/g}$, where $g$ is the acceleration due to gravity and ${\beta }_w=({\tau }_w-{\tau }_s)/{\tau }_s$ is the relative surplus of the steady-state wall shear-stress ${\tau }_w$ over the critical wall shear stress ${\tau }_s$ (yield stress) that is needed to bring the granular media into motion... (see article for a complete abstract).

in press (Journal of Fluid Mechanics)