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

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 $μ(I)$--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 $\sqrt{νt}$, where $ν$ 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 $\sqrt{ν_g t}$ analogous to a Newtonian fluid where $ν_g$ is an equivalent granular kinematic viscosity depending not only on the intrinsic properties of the granular media such as grain diameter $d$, density $ρ$ and friction coefficients but also on the applied pressure $p_w$ at the moving wall and the solid fraction $ϕ$ (constant). In addition, the $μ(I)$--rheology indicates that this growth continues until reaching the steady-state boundary layer thickness $δ_s = β_w (p_w/ϕρg )$, independent of the grain size, at about a finite time proportional to $β_w^2 (p_w/ρg d)^{3/2} \sqrt{d/g}$, where $g$ is the acceleration due to gravity and $β_w = (τ_w - τ_s)/τ_s$ is the relative surplus of the steady-state wall shear-stress $τ_w$ over the critical wall shear stress $τ_s$ (yield stress) that is needed to bring the granular media into motion... (see article for a complete abstract).