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.