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The stress tensor in granular shear flows of uniform, deformable disks at high solids concentrations

Published online by Cambridge University Press:  26 April 2006

Marijan Babić ,
Hayley H. Shen  and
Hung Tao Shen
Marijan Babić
Affiliation:
Department of Civil and Environmental Engineering, Clarkson University, Potsdam, New York 13676, USA
Hayley H. Shen
Affiliation:
Department of Civil and Environmental Engineering, Clarkson University, Potsdam, New York 13676, USA
Hung Tao Shen
Affiliation:
Department of Civil and Environmental Engineering, Clarkson University, Potsdam, New York 13676, USA
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Abstract

Application of the kinetic theory of gases to granular flows has greatly increased our understanding of ‘rapid’ granular flows. One of the underlying assumptions is that particles interact only through binary collisions. For a given set of material and flow parameters, as the concentration increases, the transition from a binary collision mode to other modes of interaction occurs. Kinetic theory can no longer be applied. A numerical model is utilized to simulate the mechanical behaviour of a small assembly of uniform, inelastic, frictional, deformable disks in a simple shear flow. There are two objectives: to obtain the ‘empirical’ constitutive law and to gain insight into the mechanisms that operate in the transitional and quasi-static regimes. In a simple shear flow, spatially and temporally averaged dimensionless stresses $\tau^*_{ij} = \tau_{ij}/(\rho_{\rm s}D^2\dot{\gamma}^2)$ are functions of the concentration C, the dimensionless shear rate $B =\dot{\gamma}/(K_n/m)^{\frac{1}{2}}$, and material parameters ζn, Ks/Kn and μ. Here $\dot{\gamma}$ is the shear rate, Kn is the normal stiffness of an assumed viscoelastic contact force model, Ks/Kn is the ratio of tangential to normal stiffness, ζn is the normal damping coefficient, μ is the friction coefficient, and ρs, D and m are the particle density, diameter and mass, respectively. The range of B from 0.001 to 0.0707 was investigated for C ranging from 0.5 to 0.9, with material constants fixed as ζn = 0.0709 (corresponding to the restitution coefficient e = 0.8 in binary impacts), Ks/Kn = 0.8 and μ = 0.5. It is found that for lower concentrations (C < 0.75) dimensionless stresses τ*ij are nearly independent of B, while for higher concentrations (C > 0.75) τ*ij monotonically decreases as B increases. Moreover, their relationship in this regime is well approximated by power law: τ*ij ∝ B−n(C). The powers nij range from nearly zero for C = 0.775 (corresponding to the familiar square power dependency of dimensional stresses on the shear rate in the rapid flow regime), to nearly two for C = 0.9 (corresponding to shear-rate independence in quasi-static regime). The intermediate concentration range corresponds to transition. Distinct mechanisms that govern transitional and quasi-static regimes are observed and discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 219 , October 1990 , pp. 81 - 118
Copyright
© 1990 Cambridge University Press

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