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We study convective motion in vertically vibrated three-dimensional granular beds by comparing the predictions of a model based on a hydrodynamic description to Navier–Stokes order with experimental results obtained using positron emission particle tracking (PEPT). The three-dimensional conservation equations relating mass, momentum and energy are solved using the finite element (FE) method for a viscous vibrofluidized bed by using only observable system parameters such as particle number, size, mass and coefficients of restitution. The mean velocity profiles from the viscous model show reasonable agreement with the experimental results at relatively low altitudes for the range of experimental values studied, though the velocity fields at higher altitudes were systematically underestimated by the model. We confirm that the convection rolls are influenced by the sidewall coefficient of restitution and demonstrate the scaling relationships that operate, where increasing amplitude of vibration leads to a reduction in the angular velocity of the rolls.

An experimental investigation of an idealized rapidly sheared granular flow was performed to test the predictions of a model based on the kinetic theory of dry granular media. Glass ballotini beads were placed in an annular shear cell and the lower boundary rotated to induce a shearing motion in the bed. A single particle was tracked using the positron emission particle tracking (PEPT) technique, a method that determines the location of a particle through the triangulation of gamma photons emitted by a radioactive tracer particle. The packing fraction and velocity fields within the three-dimensional flow were measured and compared to the predictions of a model developed using the conservation and balance equations applicable to dissipative systems, and solved incorporating constitutive relations derived from kinetic theory. The comparison showed that kinetic theory is able to capture the general features of a rapid shear flow reasonably well over a wide range of shear rates and confining pressures.

Molecular dynamic (MD) simulations are used to probe the ability of Navier–Stokes-order theories to predict each of the constitutive quantities – heat flux, stress tensor and dissipation rate – associated with granular materials. The system under investigation is bounded by two opposite walls of set granular temperature and is characterized by zero mean flow. The comparisons between MD and theory provide evidence of higher-order effects in each of the constitutive quantities. Furthermore, the size of these effects is roughly one order of magnitude greater, on a percentage basis, for heat flux than it is for stress or dissipation rate. For the case of heat flux, these effects are attributed to super-Burnett-order contributions (third order in gradients) or greater, since Burnett-order contributions to the heat flux do not exist. Finally, for the system considered, these higher-order contributions to the heat flux outweigh the first-order contribution arising from a gradient in concentration (i.e. the Dufour effect)

A combination of molecular-dynamics simulations, theoretical predictions, and previous experiments are used in a two-part study to determine the role of the Knudsen layer in rapid granular flows. First, a robust criterion for the identification of the thickness of the Knudsen layer is established: a rapid deterioration in Navier-Stokes-order prediction of the heat flux is found to occur in the Knudsen layer. For (experimental) systems in which heat flux measurements are not easily obtained, a rule-of-thumb for estimating the Knudsen layer thickness follows, namely that such effects are evident within 2.5 (local) mean free paths of a given boundary. Second, comparisons of simulation and experimental data with Navier-Stokes order theory are used to provide a measure as to when Knudsen layer effects become non-negligible. Specifically, predictions that do not account for the presence of a Knudsen layer appear reliable for Knudsen layers collectively composing up to 20% of the domain, whereas deterioration of such predictions becomes apparent when the domain is fully comprised of the Knudsen layer., 9 figures, accepted to Journal of Fluid Mechanics

Equations relating the energy flux, energy dissipation rate, and pressure within a three-dimensional vibrofluidized bed are derived and solved numerically, using only observable system properties, such as particle number, size, mass and coefficient of restitution, to give the granular temperature and packing fraction distributions within the bed. These are compared with results obtained from positron emission particle tracking experiments and the two are found to be in good agreement, without using fitting parameters, except at high altitudes when using a modified heat law including a packing fraction gradient term. Criteria for the onset of the Knudsen regime are proposed and the resulting temperature profiles are found to agree more closely with the experimental distributions. The model is then used to predict the scaling relationship between the height of the centre of mass and mean weighted bed temperature with the number of particles in the system and the excitation level.