The stress tensor in a granular flow at high shear rates

Abstract

The stress tensor in a granular shear flow is calculated by supposing that binary collisions between the particles comprising the granular mass are responsible for most of the momentum transport. We assume that the particles are smooth, hard, elastic spheres and express the stress as an integral containing probability distribution functions for the velocities of the particles and for their spatial arrangement. By assuming that the single-particle velocity distribution function is Maxwellian and that the spatial pair distribution function is given by a formula due to Carnahan & Starling, we reduce this integral to one depending upon a single non-dimensional parameter R: the ratio of the characteristic mean shear velocity to the root mean square of the precollisional particle-velocity perturbation. The integral is evaluated asymptotically for R [Gt ] 1 and R [Lt ] 1 and numerically for intermediate values. Good agreement is found between the stresses measured in experiments on dry granular materials and the theoretical predictions when R is given the value 1·7. This case is probably the one for which the present analysis is most appropriate. For moderate and large values of R, the theory predicts both shear and normal stresses that are proportional to the square of the particle diameter and the square of the shear rate, and depend strongly on the solids volume fraction. A provisional comparison is made between the stresses predicted in the limit R → ∞ and the experimental results of Bagnold for shear flow of neutrally buoyant wax spheres suspended in water. The predicted stresses are of the correct order of magnitude and yield the proper variation of stress with concentration. When R [Lt ] 1, the shear stress is linear in the shear rate, and the analysis can be applied to shear flow in a fluidized bed, although such an application is not developed further here.

Distributions of critical flux: modelling, experimental analysis and consequences for cross-flow membrane filtration

“…Particle velocity distributions caused by multi-body hydrodynamic interactions have been studied by numerical simulation [14][15]. As an example, the fluctuation of velocity around a mean value during settling of a concentrated suspension follows a Gaussian distribution [14].…”

“…One can think that shear induced diffusion, colloidal interaction induced diffusion or lateral migration could lead to distributions in particle velocity in the same way. In the domain of granular flow (as for example in powder flow) where a large number of small particles are arranged in a random way, particle velocity fluctuation was defined by Savage and Jeffrey in 1981 [15] by the term "granular temperature" which quantifies the random motion of particles around the mean velocity. The intensity of distribution used later in this paper could then be linked to the concept of granular temperature and then associated to the dense phase kinetic theory used for the description of the granular flow of particles.…”

A Multiphase Model for the Numerical Simulation of Ice-Formation in Sea-Water

“…p s and τ s,ij can be evaluated as function of the granular temperature Θ, according to the kinetic theory, originally applied by Bagnold [2] to this purpose. Starting from [2], several efforts have been devoted over the years to the development of the kinetic theory, see e.g [35,38,30,26].…”

A novel mechanism for bubble formation in fluidized systems: the effects of granular temperature on the stability in fluidization

“…The concept of granular temperature was first applied by a group of researches (Savage and Jeffrey, 1981;Haff, 1983, Jenkins andRichman, 1986) for the modeling of flows where solid particles were present. Its use for the study of the stability of fluidized systems is relatively recent (Buyevich and Kapbasov, 1994;Koch andSangani, 1999, Didwania, 1999;Didwania and Costa, 2000).…”