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 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 νgt analogous
to a Newtonian fluid where νg 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 pw 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 δs=βw(pw/ϕρg), independent of the grain size, at about a finite
time proportional to βw2(pw/ρgd)3/2d/g, where g is
the acceleration due to gravity and βw=(τw−τs)/τs is the
relative surplus of the steady-state wall shear-stress τw over the
critical wall shear stress τs (yield stress) that is needed to bring the
granular media into motion... (see article for a complete abstract).Comment: in press (Journal of Fluid Mechanics