Journal of Nonlinear Science

, Volume 20, Issue 6, pp 689–707 | Cite as

Scalar Conservation Laws with Nonconstant Coefficients with Application to Particle Size Segregation in Granular Flow

  • Lindsay B. H. May
  • Michael Shearer
  • Karen E. Daniels
Article

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 nonstandard, involving curved characteristics in the exponential case, and a material interface with a jump in characteristic speed in the piecewise linear case.

Keywords

Conservation laws First order partial differential equations Granular flow Particle segregation 

Mathematics Subject Classification (2000)

35L60 74L10 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Lindsay B. H. May
    • 1
  • Michael Shearer
    • 1
  • Karen E. Daniels
    • 2
  1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  2. 2.Department of PhysicsNorth Carolina State UniversityRaleighUSA

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