A note on Stokes' problem in dense granular media using the µ(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 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 √ 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 2 w (p w /ρgd) 3/2 √(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. For the case of Stokes' first problem when the wall shear stress τ w is imposed externally, the µ(I)–rheology suggests that the wall velocity simply grows as √ t before saturating to a constant value whereby the internal resistance of the granular media balances out the applied stresses. In contrast, for the case with an externally imposed wall speed u w , the dense granular media near the wall initially maintains a shear stress very close to τ d which is the maximum internal resistance via grain-grain contact friction within the context of the µ(I)–rheology. Then the wall shear stress τ w decreases as 1/ √ 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 ∼ (gδ 2 s /ν g)f (β w) where f (β w) is either O(1) if τ w ∼ τ s or logarithmically large as τ w approaches τ d.
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J John Soundar Jerome, B Di Pierro. A note on Stokes' problem in dense granular media using the µ(I)–rheology. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2018, 847, pp.365-385. ⟨10.1017/jfm.2018.250⟩. ⟨hal-01743924⟩

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