Abstract
Vertical chutes and pipes are a common component of many industrial apparatus used in the transport and processing of powders and grains. Here, a typical arrangement is considered first in which a hopper at the top feeds the chute and a converging outlet at the bottom controls the mass flux. Discrete element method (DEM) simulations reveal that steady uniform flow is only observed for intermediate flow rates, with jamming and unsteady waves dominating slow flows and non-uniform wall detachment in fast flow. Focusing on the steady uniform regimes, a progressive idealisation is carried out by matching with equivalent DEM simulations in periodic cells. These investigations justify a one-dimensional continuum modelling of the problem and provide key test data. Novel exact solutions are derived here for vertical flow using a linear version of the ‘ $\mu(I),\varPhi(I)$ -rheology’, for which the bulk friction $\mu$ and steady solid volume fraction $\varPhi$ depend on the inertial number I. Despite not capturing the full nonlinear complexities, the solutions match important aspects of the DEM flow fields and reveal simple scaling laws linking many quantities of interest. In particular, this study clearly demonstrates a linear relation between the chute width and the size of the shear zones at the walls. This finding contrasts with previous works on purely quasi-static flow, which instead predict a roughly constant shear zone width, a difference which implies that finite-size effects are minimal for the inertial flows studied here.
Highlights
One of the defining features of granular materials is their so-called static yield stress, which means that sufficient forcing is required for irreversible plastic flow to occur
Exact solutions for steady granular flow in vertical chutes μ(I),Φ(I)-rheology are found for both planar parallel walls and for cylindrical pipe walls under the assumption of no slip
The exact solutions derived here match the scaling behaviour of equivalent Discrete element method (DEM) simulations very well and even provide a good approximation to the spatial variation of the flow fields in both the cylindrical pipe and parallel-walled geometries. This success is despite the simplified nature of the linearised μ(I),Φ(I)-rheology, which has enabled the exact solutions, and which means that certain complexities of the real flow are neglected
Summary
One of the defining features of granular materials is their so-called static yield stress, which means that sufficient forcing is required for irreversible plastic flow to occur. These novel continuum solutions are based solely on mass and linear momentum balances using the μ(I),Φ(I)-rheology of Pouliquen et al (2006) In this theory, both the steady solid volume fraction Φ and the ratio of shear to normal stress μ are taken to be functions of the inertial number I, which is a non-dimensional strain rate designed to reflect the frequency of grain rearrangements. Both the steady solid volume fraction Φ and the ratio of shear to normal stress μ are taken to be functions of the inertial number I, which is a non-dimensional strain rate designed to reflect the frequency of grain rearrangements These relations are well verified for many important flows (see GDR MiDi 2004), but, to date, more attention has been applied to the incompressible μ(I)-rheology of Jop, Forterre & Pouliquen (2006) in which Φ = φ∗ is a constant. The duality in this regard, as well as parallels between cylindrical and rectangular pipes, suggests a universality of the relations and inspires direct application in practical scenarios
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