Exact solutions for steady granular flow in vertical chutes and pipes


Static yield stress is one of the most important features of granular materials. It underpins the stationary stability of various aggregates, especially those with inclined surfaces. Static yield stress plays a critical role in the failure of many materials, and it is important to understand the boundaries between static and yielding material. This has been studied using different models, some of which have inherent limitations. For instance, some models fail to provide an accurate prediction of the destructive potential of snow avalanches and debris flows when a large proportion of the material is involved in a dense fluid-like flow. Therefore, there is a need for more effective numerical and theoretical models for granular flow to provide insights into the features of the transition from solid-like to fluid-like flow.

There is also intensive research motivated by the industrial flow of grains and powders in silos, hoppers and pipes. This has specifically aroused great interest in vertical chutes and pipes that are critical components of industrial apparatus commonly used to process and transport grains and powders. There have been numerous studies investigating the flow in standpipes and vertical chutes based on different configurations. However, certain features that are considered important for vertical chutes and pipes are yet to be fully clarified. This can be attributed to two main reasons. First, the contrasting geometries of experimental set-ups can result in distinct flow regimes. Second, intrinsic material differences and related bulk rheological responses further complicate the constitutive modeling. Therefore, there is a need for more effective strategies for modelling granular flow within shear zones, such as those present near the walls of closed pipes.

Herein, Dr. Thomas Barker (currently at Cardiff University), Dr. Chongqiang Zhu (currently at Heriot-Watt University) and Dr. Jin Sun from the University of Edinburgh have derived exact solutions for steady granular flow in vertical chutes and pipes. The researchers considered a typical arrangement configuration where a hopper at the top fed the chute while the mass flux was controlled by a converging outlet at the bottom. Discrete element method (DEM) simulations allowed for accurate measurements of the flow velocity, varying packing fraction and internal force fields, which are very difficult to achieve via physical experiments. Their research work is currently published in the Journal of Fluid Mechanics.

The researchers showed that steady uniform flow could only be observed for intermediate flow rates, while jamming and unsteady waves dominated slow flows and fast flow was characterized by non-uniform detachment from the walls. Based on the linear version of the µ(I),Φ(I)-rheology it was possible to derive novel exact solutions for vertical flow. In this formulation, the inertial number (I), which is a non-dimensional strain rate, coupled to both the steady solids volume fraction (Φ) and bulk friction (µ).

Although the solutions did not capture the full nonlinear complexities, they matched critical elements of the DEM flow fields and revealed scaling laws that link various important quantities. In particular, a linear relationship between the shear zone width at the walls and the chute width was demonstrated. Notably, this result does not agree with the previous findings on purely quasi-static flow, which predicts a nearly constant width of the shear zone. These differences suggest minimal finite-size effects for the studied inertial flows.

In summary, the derivation of exact solutions for steady granular flow in vertical chutes and pipes was presented. The results justified the one-dimensional continuum modeling, thus proving vital test data for modeling the problem. The derived solutions not only matched the scaling behavior of the equivalent DEM simulations but also provided a better approximation of spatial flow field variations in parallel walled and cylindrical pipe geometries, suggesting the effectiveness of the proposed model. In a statement to Advances in Engineering, Dr. Thomas Barker, first author, said the proposed model is simple, provides a basis for universal scaling laws and will help broaden the range of application in studying other important granular flows of interest.

Exact solutions for steady granular flow in vertical chutes and pipes - Advances in Engineering Exact solutions for steady granular flow in vertical chutes and pipes - Advances in Engineering

About the author

Dr. Chongqiang Zhu is currently a research associate of School of Engineering and Physical Sciences at Heriot-Watt University, Edinburgh, United Kingdom. Previously, Chongqiang conducted successful postdoctoral research at both the Universities of Edinburgh and Tongji University. He received the B.S. (2012) in Civil Engineering from China University of Petroleum, and the Ph.D. (2017) in geological Engineering from Tongji University. His main research interests involve mechanism and evolution of geophysical flows (e.g. landslides, flow slides and pyroclastic density currents, etc.), geo-disaster prevention and mitigation using both experimental and numerical methods from both micro to macro perspectives. He has successfully revealed the mechanism of enhanced mobility of earthquake-induced landslides by near-fault seismic motions based on comprehensive large-scale shaking table tests and particle-based simulations. He also developed in-house codes (i.e. MPS and MPS-FEM) to reproduce the flowing process of large deformation flows and investigate its interactions with structures. He has published 20 SCI/EI papers in Journal of Fluid Mechanics, Computers and Geotechnics, etc. In this work, he firstly found the linear relation between the chute width and the size of the shear band at the walls based on discrete element simulation.

About the author

Dr Jin Sun is a Reader in the School of Engineering at the University of Edinburgh. He graduated from Iowa State University in 2007 with a PhD in Mechanical and Chemical Engineering and did postdoctoral research at Princeton University until he joined the University of Edinburgh in 2010.

His main research interests lie in rheology and flow of dense granular materials in industrial and natural environments, such as manufacturing of ceramics, food stuff, battery electrodes and pharmaceuticals, and debris and pyroclastic flows. He has been focusing on the complex rheology of geo-materials occurring during natural disasters in recent years and collaborating with geologists to investigate submarine landslides and pyroclastic flows. He employs a combination of computational and experimental methods to understand the macroscopic behaviour of such materials from interactions at different spatial/temporal scales and to devise mathematical models for the rheology and dynamics.

He has been principal investigator of projects funded by the UK national research councils including EPSRC and NERC; and co-investigator on NERC and EU projects. He has been awarded a Royal Academy of Engineering/ Leverhulme Trust Senior Research Fellowship and is a member to UK research council’s grant review panels, the Particle Technology committee of the Institute of Chemical Engineers, and the editorial board of the Geoenvironmental Disasters journal.

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About the author

Dr Thomas Barker was recently appointed Lecturer in Applied Mathematics at Cardiff University, UK. Previously, Thomas completed successful postdoctoral positions at The University of Edinburgh and at The University of Manchester following his PhD in Applied Mathematics at The University of Manchester, awarded in 2017.

Dr Barker’s research is focused on the rheolgical modelling of granular flows, such as those found in debris avalanches, subsea landslides and in pharmaceutical powder processing. He employs a combination of Continuum Mechanics theory, Numerical Solutions of PDEs and Discrete Particle Simulations to understand and predict these important and widespread physical processes. To date Dr Barker has 8 published papers in world-leading journals which have so far attracted a total of over 390 citations. A highlight of these works is the proof of ill-posed regimes for the mu(I)-rheology which has reignited the debate around appropriate constitutive equations for granular materials.

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Barker, T., Zhu, C., & Sun, J. (2021). Exact solutions for steady granular flow in vertical chutes and pipesJournal of Fluid Mechanics, 930, A21-28.

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