The timeless tale of sand in an hourglass : mathematical modelling of granular flow in a silo : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
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Date
2024
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Massey University
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Abstract
Granular materials are an integral part of many processes vital to both nature and modern life. Improving our understanding of the behaviour of flowing granular materials is important for a wide variety of applications. Many models have attempted to capture the dynamics of granular flow, including discrete element models which model individual particles, models which analyse the stress in granular material, and models using the Navier-Stokes equation. Each of these models have various advantages and disadvantages, and while they each can be used to give unique insights into granular flows, no currently available model fully captures the multitude of phenomena that granular materials exhibit. This thesis focuses on the μ(I) model within silos, extending the model to capture more of the complexities of granular flow.
In this thesis, the μ(I) model is extended in a variety of ways. While the base μ(I) model is incompressible, granular material dilates while flowing. This work extends the μ(I) model to be pseudo-compressible. Another consideration is that the μ(I) model is also a purely local model. In contrast, experiments done with flows down slopes and in Couette cells have shown that granular materials exhibits nonlocal behaviour. To account for this, this work applies nonlocal fluidity to the μ(I) model. Finally, granular materials are often composed of many components, each with potentially different material parameters, so a model is developed to allow for multiple material types mixing within a silo.
The interaction of these effects are tested within complex domains including the double opening silo. The double opening silo is analysed, with the μ(I) model capable of replicating the `flow rate dip' behaviour seen in DEM and physical experiments. The extensions to the μ(I) model also improve the dipping behaviour, allowing for greater replication of physical behaviour. These extensions are also applied to silos with inserts, where the model shows strong improvements in material mobility are possible within a silo with even small inserts.
The gestalt μ(I) model and the conclusions drawn within this thesis provide valuable insights for a variety of essential industrial and scientific fields, and pushes forward our collective understanding of continuum modelling for granular flow.
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Bulk solids flow, Granular flow, Mathematical models