A note on Stokes’ problem in dense granular media using the -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$\unicode[STIX]{x1D707}(I)$-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$\unicode[STIX]{x1D70F}_{d}$.