Multiscale Mechanics group MSM/TS/CTW

University of Twente, Enschede, The Netherlands

**To: **STW and industrial partners of STW-VICI 10828

**From:** T. Weinhart, Postdoctoral researcher

**Subject:** Bridging the gap between particulate systems and continuum theory – The Macro-Microscale transition for shallow granular flows

Start Date: 1. August 2011

Report Date: 31. January 2012

**1. Introduction:**

This report briefly describes the ongoing activities related to the aforementioned project and provides a road map of the same for the users’ committee. The project is mainly concerned with the micro-macro transition required to obtain the *bulk properties* of a system of granular particles directly from the particle properties. This requires a description of a discrete granular system (the positions and velocities of, and forces on, each particle) in terms of density, velocity, and stress. In particular, we focus on the definition of the macroscopic stress near external boundaries as they appear in most granular systems but still lack a proper definition. Shallow steady granular flow down an incline over a rough bottom is chosen as an industry-relevant reference system.

**2. Summary of Ongoing Efforts and Accomplishments:**

*A) An expression for the stress near an external boundary: *

To obtain bulk density, velocity, and stress, we choose the coarse-graining method described by Isaac Goldhirsch [5] as it has several advantages over other methods, including: *i)* the fields automatically satisfy the conservation equations of continuum mechanics; *ii)* it is not assumed that the particles are rigid or spherical, and *iii)* the results are valid for single particles (no averaging over ensembles of particles is required). In order to obtain a micro-macro transition of systems that include external boundaries such as walls (and most systems do), we extended the stress definition to stress fields near external boundaries, allowing for more accurate statistics near system boundaries [1].

*B) Closure rules for shallow granular flow down an incline over a rough bottom.*

Shallow free-surface granular flow down an incline over a rough bottom is chosen as a reference system. We previously showed that the closure relations can be obtained as functions of the micro-parameters such as the geometric roughness of the base. As the flow behaviour depends strongly on the roughness of the bottom, the roughness was varied by changing the diameter of the basal particles, , where is the diameter of the flowing particles. For steady flows, a linear relationship could be found between the Froude number and the flow height scaled by the stopping height , which is the height below which the flow arrests at a given inclination for a base with , see Figure. This relationship yields a closure relation for the friction parameter of the continuum shallow water equations and thus a micro-macro transition has been established [2]. Recently, we established a closure for the friction parameter as a function of microscopic contact friction of flowing particles with the base [3], and for the segregation rate of polydispersed steady flows as a function of polydispersity [4]

**3. Outlook**

A) The stress definition [1] can be extended to measure partial stresses in polydispersed flows. We plan to use this new tool to study the macroscopic effect of particle size segregation in chute flows.

B) The study of shallow granular chute flows allows us to obtain the quantitative bulk behavior from the material properties of the particles. The next step will be to validate the measured closure rules by comparing continuum simulations of medium scale experiments such as the flow through a contraction to Discrete Particle simulations and experiments.

References:

1.Weinhart, T. and Thornton, A.R. and Luding, S. and Bokhove, O. *From discrete particles to continuum ﬁelds near a boundary.* accepted in Granular Matter, special volume for Goldhirsch, (attached)

2.Weinhart, T. and Thornton, A.R. and Luding, S. and Bokhove, O. *Closure Relations for Shallow Granular Flows from Particle Simulations.* Granular Matter, submitted August, 2011. https://research.utwente.nl/en/publications/closure-relations-for-shallow-granular-flows-from-particle-simula-3

3.Thornton, A.R. and Weinhart, T. and Luding, S. and Bokhove, O. *Friction dependence of shallow granular flows from discrete particle simulations, *submitted to EPL (attached)

4.Thornton, A.R. and Weinhart, T. and Luding, S. and Bokhove, O. Modeling of particle size segregation: Calibration using the discrete particle method, submitted to Modern Physics, https://research.utwente.nl/en/publications/modeling-of-particle-size-segregation-calibration-using-the-discr

5.Goldhirsch, I.: Stress, stress asymmetry and couple stress: from discrete particles to continuous fields. Granul. Mat. 12(3), 239-252 (2010)