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A note on Stokes’ problem in dense granular media using the $\unicode[STIX]{x1D707}(I)$ -rheology

  • J. John Soundar Jerome (a1) and B. Di Pierro (a1)

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 $\unicode[STIX]{x1D707}(I)$ -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 $\sqrt{\unicode[STIX]{x1D708}t}$ , where $\unicode[STIX]{x1D708}$ 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 $\sqrt{\unicode[STIX]{x1D708}_{g}t}$ analogous to a Newtonian fluid where $\unicode[STIX]{x1D708}_{g}$ is an equivalent granular kinematic viscosity depending not only on the intrinsic properties of the granular medium, such as grain diameter $d$ , density $\unicode[STIX]{x1D70C}$ and friction coefficients, but also on the applied pressure $p_{w}$ at the moving wall and the solid fraction $\unicode[STIX]{x1D719}$ (constant). In addition, the $\unicode[STIX]{x1D707}(I)$ -rheology indicates that this growth continues until reaching the steady-state boundary layer thickness $\unicode[STIX]{x1D6FF}_{s}=\unicode[STIX]{x1D6FD}_{w}(p_{w}/\unicode[STIX]{x1D719}\unicode[STIX]{x1D70C}g)$ , independent of the grain size, at approximately a finite time proportional to $\unicode[STIX]{x1D6FD}_{w}^{2}(p_{w}/\unicode[STIX]{x1D70C}gd)^{3/2}\sqrt{d/g}$ , where $g$ is the acceleration due to gravity and $\unicode[STIX]{x1D6FD}_{w}=(\unicode[STIX]{x1D70F}_{w}-\unicode[STIX]{x1D70F}_{s})/\unicode[STIX]{x1D70F}_{s}$ is the relative surplus of the steady-state wall shear stress $\unicode[STIX]{x1D70F}_{w}$ over the critical wall shear stress $\unicode[STIX]{x1D70F}_{s}$ (yield stress) that is needed to bring the granular medium into motion. For the case of Stokes’ first problem when the wall shear stress $\unicode[STIX]{x1D70F}_{w}$ is imposed externally, the $\unicode[STIX]{x1D707}(I)$ -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 $\unicode[STIX]{x1D70F}_{d}$ which is the maximum internal resistance via grain–grain contact friction within the context of the $\unicode[STIX]{x1D707}(I)$ -rheology. Then the wall shear stress $\unicode[STIX]{x1D70F}_{w}$ 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 $u_{w}\sim (g\unicode[STIX]{x1D6FF}_{s}^{2}/\unicode[STIX]{x1D708}_{g})f(\unicode[STIX]{x1D6FD}_{w})$ where $f(\unicode[STIX]{x1D6FD}_{w})$ is either $O(1)$ if $\unicode[STIX]{x1D70F}_{w}\sim \unicode[STIX]{x1D70F}_{s}$ or logarithmically large as $\unicode[STIX]{x1D70F}_{w}$ approaches $\unicode[STIX]{x1D70F}_{d}$ .

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