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

Published online by Cambridge University Press:  23 May 2018

J. John Soundar Jerome Open the ORCID record for J. John Soundar Jerome [Opens in a new window]  and
B. Di Pierro
J. John Soundar Jerome*
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, Laboratoire de Mécanique des Fluides et d’Acoustique, CNRS UMR–5509, Boulevard 11 novembre, 69622 Villeurbanne CEDEX, Lyon, France
B. Di Pierro
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, Laboratoire de Mécanique des Fluides et d’Acoustique, CNRS UMR–5509, Boulevard 11 novembre, 69622 Villeurbanne CEDEX, Lyon, France
*
†Email address for correspondence: [email protected]
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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 $\unicode[STIX]{x1D70F}_{d}$.

JFM classification

Boundary Layers: Boundary layer structure Granular media: Granular media Non-Newtonian Flows: Non-Newtonian Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 847 , 25 July 2018 , pp. 365 - 385
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Copyright
© 2018 Cambridge University Press 

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