(172d) Effective Wall-Boundary Conditions for Dense Flows of Granular Materials

A longstanding question in continuum modeling of wall-bounded granular flows and gas-particle flows is the choice of appropriate boundary conditions (BCs) for the solid phase. Among the most common formulations are the simple no-slip condition and the Johnson-Jackson model [1], the latter of which prescribes a slip velocity dependent on the fluctuation energy of the flowing material and on particle properties. Despite their common use, however, very little work has been done to validate these BCs at the micro- or mesoscale, either by experiment or by discrete particle simulations. Furthermore, even if the slip velocity can be predicted accurately, this accuracy may be lost when performing coarse-grid continuum-model simulations, as the boundary layer in a wall-bounded granular flow is typically very small compared to the simulation grid size. In the present work, we aim to address these problems by 1) defining an 'effective' slip velocity that subsumes the boundary layer into the BC and 2) constituting this effective slip velocity in terms of flow and particle properties.

To this end, we perform discrete element method (DEM) simulations of frictional particles in simple-shear flow between two flat, frictional walls. Two distinct regions are observed in the flow: a dense 'core' region, near the center and away from the walls, exhibiting a linear velocity profile; and the boundary layers, which exhibit exponential velocity profiles and lower particle concentrations. The pressure and shear stress in the core region closely follow the local kinetic-theory (KT) model of Chialvo and Sundaresan [2], while the near-wall regions do not because of nonlocal conduction of fluctuation energy. To characterize the slip behavior, we can ignore these nonlocal effects by extrapolating the core velocity profile to the wall and defining the effective slip velocity as the difference between this value and the wall velocity. A linearized version of the KT model, with near-wall behavior denoted as a perturbation from the core behavior, is then combined with Johnson-Jackson-type expressions for the energy and momentum balances at the wall to produce a suggested functional form for the dimensionless slip velocity in terms of the bulk friction coefficient. The DEM data support a model of this form, the coefficients of which we constitute in terms of interparticle and particle-wall friction coefficients. The resulting BC expression is advantageous for its applicability to coarse-grid simulations, as it eliminates the need to resolve the boundary layer. Additionally, its pairing with an algebraic version of the KT model removes the need to track near-wall fluctuation energy or its spatial gradients.

[1] P.C. Johnson and R. Jackson, J. Fluid Mech. 176, 67 (1987).
[2] S. Chialvo and S. Sundaresan, Phys. Fluids, (accepted).