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 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 root vt, where v is the kinematic viscosity. For a dense granular viscoplastic liquid, it is shown that the local shear stress, when properly resealed, 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 root v(g)t analogous to a Newtonian fluid where v(g) is an equivalent granular kinematic viscosity depending not only on the intrinsic properties of the granular medium, 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 phi (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)/phi rho g), independent of the grain size, at approximately a finite time proportional to beta(2)(w)(p(w)/rho gd)3/2 root 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 medium into motion. For the case of Stokes' first problem when the wall shear stress tau(w) is imposed externally, the mu(I)-rheology suggests that the wall velocity simply grows as root t before saturating to a constant value whereby the internal resistance of the granular medium balances out the applied stresses. In contrast, for the case with an externally imposed wall speed u(w) the dense granular medium near the wall initially maintains a shear stress very close to tau(d) which is the maximum internal resistance via grain-grain contact friction within the context of the mu(I)-rheology. Then the wall shear stress tau(w) decreases as 1/root t until ultimately saturating to a constant value so that it gives precisely the same steady-state solution as for the imposed shear-stress case. Thereby, the steady-state wall velocity, wall shear stress and the applied wall pressure are related as u(w) similar to(g delta(2)(s)/v(g))f(beta(w)) where f(beta(w)) is either O(1) if tau(w)similar to tau(s) or logarithmically large as tau(w) approaches tau(d).