# Laboratoire de Mécanique des Fluides et d’Acoustique - UMR 5509

LMFA - UMR 5509
Laboratoire de Mécanique des Fluides et d’Acoustique
Lyon
France

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Article dans J. Fluid Mech. (2018)

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

J. John Soundar Jerome & Bastien Di Pierro

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 viscoplastic 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 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 $\beta_w^2(p_w/\rho gd)$, 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 $\sqrt{t}$ 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 $1/\sqrt{t}$ 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 $u_w\sim(g\delta_s^2/\nu_g)\,f(\beta_w)$ where $f(\beta_w)$ is either $O(1)$ if $\tau_w\sim\tau_s$ or logarithmically large as $\tau_w$ approaches $\tau_d$.

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