Granular packing in complex flow geometries

James A. Robinson, Daniel J. Holland, and Luke Fullard

Phys. Rev. Fluids 7, 074304 – Published 25 July 2022

Abstract

Granular packing is often characterized by a linear relationship between packing fraction $ϕ$ and dimensionless shear rate. We examine this relationship in simple and complex heterogeneous flow geometries. At high inertial number $(I)$ , we observe that the packing fraction depends approximately linearly on $I$ and is seemingly independent of the geometry. However, at low $I$ the packing fraction varies nonlinearly with $I$ and this variation is dependent on the specific geometry and conditions. This response is analogous to the behavior of the stress ratio $(μ)$ , where, at low $I$ , $μ$ depends on the geometry of the system, an effect that is often attributed to nonlocal effects of the granular rheology. We demonstrate that, in steady isochoric flow geometries, the $ϕ$ and $μ$ responses are equivalent and $ϕ$ may be determined locally from $μ$ , even at low $I$ . However, in a more complex, nonisochoric geometry, such as a pseudo-two-dimensional hopper, this $ϕ (μ)$ relationship is not recovered in dense regions. We show that during transient startup in a shear cell $ϕ$ and $μ$ follow a similar response to that seen in these nonisochoric flows. Hence we conclude that the observed discrepancy arises because the hopper is transient in a Lagrangian sense. The simple $ϕ (μ)$ relationship seen in isochoric flows is insufficient in these more complex systems because of differences in the temporal evolution of the stress and packing fraction.

Received 28 April 2021
Accepted 24 June 2022

DOI:https://doi.org/10.1103/PhysRevFluids.7.074304

Physics Subject Headings (PhySH)

Granular packing

Dry granular materials Granular flows

Molecular dynamics

Fluid DynamicsGeneral PhysicsInterdisciplinary PhysicsNonlinear DynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

James A. Robinson and Daniel J. Holland ^*

Chemical and Process Engineering Department, University of Canterbury, 8140 Christchurch, New Zealand

Luke Fullard

School of Fundamental Sciences, Massey University, 4474 Palmerston North, New Zealand

^*daniel.holland@canterbury.ac.nz

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 7, Iss. 7 — July 2022

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Two-dimensional diagrams of the three-dimensional geometries simulated, showing the $x$ and $z$ axes. (a) Shear cell. (b) Shear cell with gravity. (c) Vertical chute. (d) Pseudo-2D hopper. These are not to scale. The width of the first three geometries along the $y$ axis was 12 $d_{m}$ , while the hopper was 10 $d_{m}$ . The height of the shear cell and shear cell with gravity was approximately 50 $d_{m}$ , though it was varied to maintain a constant wall pressure ( $P_{w}$ ).
Reuse & Permissions
Figure 2
$ϕ$ vs $I$ for different geometries with the $x$ axis on a log (a) and linear (b) scale. Specifically the shear cell with gravity (SCG), the vertical chute (VC), and the pseudo-2D hopper (PH). For SCG, $▿$ corresponds to $P_{f} = 2$ and $V_{f} = 4$ , $▿$ corresponds to $P_{f} = 2$ and $V_{f} = 2$ , and $▿$ corresponds to $P_{f} = 4$ and $V_{f} = 4$ . For VC, $□$ corresponds to $W = 25 d_{m}$ , $□$ corresponds to $W = 50 d_{m}$ , and $□$ corresponds to $W = 75 d_{m}$ . For PH, $◊$ corresponds to $W = 22.5 d_{m}$ and $W_{outlet} = 10 d_{m}$ , $◊$ corresponds to $W = 17.5 d_{m}$ and $W_{outlet} = 10 d_{m}$ , and $◊$ corresponds to $W = 17.5 d_{m}$ and $W_{outlet} = 8 d_{m}$ . The solid line corresponds to Eq. (1) and the dashed line corresponds to Eq. (2), fitted to the zero gravity shear cell data (SC = $○$ [see Supplemental Material [39] for details on individual shear cell simulations]). Solid symbols denote $ϕ > ϕ_{s}$ (using $ϕ_{s}$ from the power law).
Reuse & Permissions
Figure 3
$μ$ vs $I$ for simulations in different geometries with the $x$ axis on a log (a) and linear (b) scale. Solid symbols denote $μ < μ_{s}$ . The solid line denotes Eq. (3), as fitted to the zero gravity shear cell data, which gave $μ_{s} = 0.374$ , $μ_{2} = 1.08$ , and $I_{0} = 0.776$ . Colors are as in Fig. 2.
Reuse & Permissions
Figure 4
$ϕ$ vs $μ$ for simulations in different geometries. The solid line denotes $μ_{s}$ and the dashed line denotes $ϕ_{s}$ [taken from the fitted Eq. (2)]. Colors are as in Fig. 2. Solid symbols are used for all points (and do not denote anything).
Reuse & Permissions
Figure 5
Early stress ( $σ$ ), packing ( $ϕ$ ), and velocity ( $U$ ) response during startup in a shear cell with gravity and a vertical chute. [(a)–(d)] Response for the shear cell with gravity, looking at the primary stresses ( $σ_{z z}$ and $σ_{x z}$ ) alongside $ϕ$ and the major velocity component ( $U_{x}$ ), all normalized by their steady state value. These are shown at the $z$ positions of 8.62 $d_{m}$ ( $⧫$ ) and 40.95 $d_{m}$ ( $•$ ) (with the bottom wall being $z = 0$ ) which are in the dense and dilute regions of flow, respectively. [(e)–(h)] Equivalent response for the vertical chute (with the primary stresses being $σ_{x x}$ and $σ_{z x}$ and the major velocity component being $U_{z}$ ) analyzed at the $x$ positions 27.13 $d_{m}$ ( $⧫$ ) and 8.66 $d_{m}$ ( $•$ ) (with the left wall being $x = 0$ ), again corresponding to the dense and dilute regions of flow. A black dashed line shows the steady state case.
Reuse & Permissions
Figure 6
$ϕ$ vs $μ$ for simulations of transient startup in a shear cell with gravity [(a)–(d)] and in a vertical chute [(e)–(f)] for different analyzed time ranges. Transient data are denoted by ( $▴$ ), and the steady state case is shown in solid green, with steady state other geometries using hollow blue symbols (using $▿$ for the shear cell with gravity and $□$ for the chute). A single hopper case ( $◊$ , $W = 17.5 d_{m}$ and $W_{outlet} = 8 d_{m}$ ) is included for reference.
Reuse & Permissions

Physical Review Fluids