The purpose of this work is to develop a continuum theory that may be used to predict the effects of anisotropic boundary vibrations on loose granular assemblies. In order to do so, we extend statistical averaging techniques employed in the kinetic theory to derive an anisotropic flow theory for rapid, dense flows of identical, inelastic spheres. The theory is anisotropic in the sense that it treats the full second moment of velocity fluctuations, rather than only its isotropic piece, as a mean field to be determined. In this manner, the theory can, for example, predict granular temperatures that are different in different directions. The flow theory consists of balance equations for mass, momentum, and full second moment of velocity fluctuations, as well as constitutive relations for the pressure tensor, the flux of second moment, and the source of second moment. The averaging procedure employed in deriving the constitutive relations is based on a Maxwellian that is perturbed due to the presence of a deviatoric second and full third moment of velocity fluctuations. Because the theory is anisotropic, it can predict the normal stress differences observed in granular shear flows, as well as the evolution to isotropy in an assembly with granular temperatures that are initially highly anisotropic. In order to complement the theory, we employ similar statistical techniques to derive boundary conditions that ensure that the flux of momentum as well as the flux of second moment are balanced at the vibrating boundary. The bumps are hemispheres arranged in regular arrays, and the fluctuating boundary motion is described by an anisotropic Maxwellian distribution function. The bumpiness of the surface may be adjusted by changing the size of the hemispheres, the spacing between the hemispheres in two separate array-directions, and the angle between the two directions. Statistical averaging consistent with the constitutive theory yields the rates at which momentum and full second moment are transferred to the flow. In order to present results in a form that is easy to interpret physically, the statistical parameters that describe the boundary fluctuations are related in a plausible manner to amplitudes and frequencies of sinusoidal vibrations that may differ in three mutually perpendicular directions, and to phase angles that may be adjusted between the three directions of vibration. The focus of the results presented here is on the steady response of unconfined granular assemblies that are thermalized and driven by horizontal bumpy vibrating boundaries. In a first detailed study of the effects of the boundary geometry and boundary motion on the overall response of the assemblies, the anisotropic theory is reduced to a more familiar isotropic form. The resulting theory predicts the manner in which the profiles of isotropic granular temperature and solid volume fraction as well as the uniform velocity and corresponding flow rate vary with spacings between the bumps, angle of the bump-array, energy of vibration, direction of vibration, and phase angles of the vibration. In a second study, we solve the corresponding, but more elaborate, boundary value problem for anisotropic flows induced by anisotropic boundary vibrations. The main focus in presenting these results is on the differences between granular temperatures in three perpendicular directions normal and tangential to the vibrating surface, and how each is affected by the bumpiness of the boundary and the direction of the vibration. In each case, we calculate the corresponding nonuniform velocity profile, solid volume fraction profile, and mass flow rate.

Granular materials exist in many forms in nature ranging from space debris to sand dunes and from breakfast cereals to pharmaceutical tablets. They can behave like a solid or a viscous fluid or a gas. The gas-like nature of granular materials in rapid flows allows the use of models based on kinetic theory thus revealing in depth complex physics and phenomena. However unlike conventional fluids here the energy balance requires additional dissipation terms as a consequence of inelasticity. The complexity of their interaction and diversity in application has led to numerous studies using experimental methods and numerical simulations in order to determine the most appropriate constitutive relationships for granular gases. With large dissipation the form of the constitutive relationship becomes particularly important, especially in the presence of non-equipartition and anisotropy. This thesis is focused on constitutive models of simple granular flows. A vibrated bed is often used as an idealisation of granular flows, providing a convenient approximation to the simplest type of flow: binary and instantaneous collisions with no rotations. Using finite element method (FE) based COMSOL modules we solve conservation of mass, momentum and energy resulting from granular kinetic theory in axi-symmetric form to generate time and spatial resolved solutions of packing fraction, velocity and granular temperature and compare the predictions to numerical simulation and experiment. At first we show the comparison for two closure sets, one based on a simple near elastic approach while the second based on revised Enskog theory for dense inelastic flows. The results for the second approach show good agreement with the results of previously validated near elastic models and experimental results. The observed differences between the two closure sets are small except for the observation of temperature upturn in a dilute region of the cell away from base. One cause of this is the presence of additional constitutive terms in the balance equations and are a consequence of inelasticity. The models also consider time varying effects at low frequency of excitation. These solutions show existence of wave-like effects in the cell with associated temperature upturn within the hydrodynamic applicability region. Presence of instantaneous cyclic rolling is also seen in both approaches. Evidence from MD simulations and experiments qualitatively support the findings of hydrodynamic models in phase resolved as well as time average behaviour. Subsequently, the frequency of vibration was varied to unlink the wave motion from the bulk temperature. Lack of agreement between experiment and the model predictions are shown to be due to lack of separation of time scale between the grain-base interaction and the base frequency. A sharp decrease of heat flux is measured showing that the energy input is frequency dependent. Analysis of the bulk behaviour shows that at high frequency, hard sphere based models are able to capture the steady state behaviour reasonably well. Further investigations that modulate the driving with a low frequency amplitude change revealed the dynamic nature of flow with the low frequency component. No significant influence of high frequency signal is noted except the reduction of base heat flux. Independent analysis of bulk behaviour for modulated wave excitation using MD simulations and hydrodynamic models showed wave motion in a pattern similar to non-modulated low frequency vibration. A one-dimensional inviscid model was used to determine the underlying scaling relationships for near elastic granular flows. A form of non-dimensionalisation predicts scaling behaviour for the granular flow. The predictions show good results for the dilute flows using hard sphere MD simulations. Results from MD simulations confirm dilute limit scaling of base temperature, packing fractions and heat flux coefficients. At higher inelasticity and loading condition the model fails to capture the real physics suggesting the need for a more accurate model. This simplified model does, however, set the basis for describing the main scalings for vibrofluidized granular beds, and in the future we anticipate that effects of further inelasticity and enhanced density could be incorporated.