Well-posed continuum modelling of granular flows

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Abstract

Inertial granular flows lie in a region of parameter space between quasi-static and collisional regimes. In each of these phases the mechanisms of energy dissipation are often taken to be the defining features. Frictional contacts between grains and the transmission of energy through co-operative force chains dominate slowly sheared flows. In the opposite extreme infrequent high-energy collisions are responsible for dissipation in so-called gaseous granular flows. Borrowing from each of these extremes, it is postulated that during liquid-like flow, grain energy is transferred through frequent frictional interactions as the particles rearrange. This thesis focuses on the μ(I)-rheology which generalises the simple Coulomb picture, where greater normal forces lead to greater tangential friction, by including dependence on the inertial number I, which reflects the frequency of grain rearrangements. The equations resulting from this rheology, assuming that the material is incompressible, are first examined with a maximal-order linear stability analysis. It is found that the equations are linearly well-posed when the inertial number is not too high or too low. For inertial numbers in which the equations are instead ill-posed numerical solutions are found to be grid-dependent with perturbations growing unboundedly as their wavelength is decreased. Interestingly, experimental results also diverge away from the original μ(I) curve in the ill-posed regions. A generalised well-posedness analysis is used alongside the experimental findings to suggest a new functional form for the curve. This is shown to regularise numerical computations for a selection of inclined plane flows. As the incompressibility assumption is known to break down more drastically in the high-I and low-I limits, compressible μ(I) equations are also considered. When the closure of these equations takes the form suggested by critical state soil mechanics, it is found that the resultant system is well-posed regardless of the details of the deformation. Well-posed equations can also be formed by depth-averaging the μ(I)-rheology. For three-dimensional chute flows experimental measurements are captured well by the depth-averaged model when the flows are shallow. Furthermore, numerical computations are much less expensive than those with the full μ(I) system.

Layman's Abstract

Armed with a host of modern mathematical, numerical and experimental tools the aim: `model the collective motion of rigid, macroscopic spherical particles' sounds like a trivial task. However, due in part to many novel features of this system, an accurate, reliable and holistic framework seems ever further from reach. The work presented here restricts the premise by focusing on the so-called `liquid phase' of granular flow. This regime can be thought of as the motions when grains move sufficiently, but not excessively, quickly relative to one another. Here `sufficiently quickly' is to imply that the grains don't remain in contact with the same neighbours for the duration of interest. Conversely for `excessively' fast relative motion the grains travel freely for much more time than they interact with one another. These other regimes are suitably named as the solid and gaseous phases respectively. Motivation for studying the liquid-like behaviour of granular flows mainly stems from the abundance of important examples in natural and industrial settings. Landslides, where significant volumes of earth and rock are suddenly displaced, and volcanic eruptions often lead to flows of dry granulated material which can have devastating effects on nearby human settlements. The deposits from historical flows of this nature can also be used by geophysicists to infer specific details of the event in order to construct and tune their models. For the mining and pharmaceutical industries, the efficient transport and storage granular material is vital. Silos holding large volumes of grains have been known to collapse due to unforeseen internal forces and pills of medication must be pressed from powders with identical distributions of grain sizes in order to deliver consistent amounts of the active ingredient. When modelling and predicting these important flows many techniques are available. Discrete models track each particle and often employ Newton's equations of motion, with force laws for particle interactions, to predict the future locations and velocities of grains. With these methods, the more grains present in the system the longer the calculations take to complete. As an alternate perspective, continuum models aim to capture the motions of grains en masse by assuming that collections of many particles can be approximated by a single, smoothly varying material. Continuum mechanics has a long history of application to fluid flows and the deformation of solids. The equations which form the basis of continuum models describe the evolution of the velocity, density and forces across the space occupied by the material as time progresses. This general framework is made specific to granular materials with models that link the variables and reflect physical observations. Often the resulting equations are too complex for pen and paper solution so numerical integration on computers is employed. This allows the predictions of the specific model to be compared with the results of small-scale experiments and real world flows. Mathematical analysis of the equations can reveal if they are well-posed or ill-posed. Ill-posed equations predict that small imperfections in the variables grow and come to dominate the solutions. Such equations have no unique numerical solution and are physically unjustified. This thesis aims to develop and test equations for granular flow in order to ensure that they are well-posed and thus able to accurately and reliably model flows.

Keyword(s)

Granular flow; Instability; Rheology

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