Scalar conservation laws with nonconstant coefficients with application to particle size segregation in granular flow

TL;DR

This paper models particle size segregation in granular flows using scalar conservation laws with nonconstant coefficients, verified through experiments and tailored to different shear profiles.

Contribution

It introduces a continuum model for segregation with nonconstant velocity profiles and solves complex initial value problems reflecting experimental shear conditions.

Findings

Velocity profiles fitted with exponential and piecewise linear functions

Solutions involve curved characteristics and material interfaces

Model verified against experimental segregation data

Abstract

Granular materials will segregate by particle size when subjected to shear, as occurs, for example, in avalanches. The evolution of a bidisperse mixture of particles can be modeled by a nonlinear first order partial differential equation, provided the shear (or velocity) is a known function of position. While avalanche-driven shear is approximately uniform in depth, boundary-driven shear typically creates a shear band with a nonlinear velocity profile. In this paper, we measure a velocity profile from experimental data and solve initial value problems that mimic the segregation observed in the experiment, thereby verifying the value of the continuum model. To simplify the analysis, we consider only one-dimensional configurations, in which a layer of small particles is placed above a layer of large particles within an annular shear cell and is sheared for arbitrarily long times. We fit the measured velocity profile to both an exponential function of depth and a piecewise linear function which separates the shear band from the rest of the material. Each solution of the initial value problem is non-standard, involving curved characteristics in the exponential case, and a material interface with a jump in characteristic speed in the piecewise linear case.