Pattern Formation and Time-Dependence in Flowing Sand

  • G. W. Baxter
  • R. P. Behringer
  • T. Fagert
  • G. A. Johnson
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 26)


We describe three experiments characterizing the dynamics of sand flowing from a hopper. In the first experiment, we have measured time-dependence in the normal stress caused by sand flowing from a hopper in order to test recent predictions by Pitman and Schaeffer. In the second experiment, we have used fast X-ray transmission imaging techniques to follow the evolution of patterns and the propagation of fronts associated with the patterns. The patterns are seen in rough faceted sand but not in smooth sand of the same size. In mixtures of rough and smooth sand, the amplitude of the patterns vanishes smoothly as X, the mass fraction of rough sand, falls below 0.22. In the third experiment we looked for the change between mass and funnel flow. In funnel flow, the material is stagnant near the hopper walls, and the flow is typically complex; in mass flow there is smooth “laminar” flow throughout the hopper. Contrary to expectations, funnel flow was observed over a wide range of hopper opening angles θ, spanning 19° < θ < 80°. Finally, we describe cellular automata models for the flow of sand which reproduce a number of the features seen in the experiments.


Mass Flow Rate Granular Material Cellular Automaton Granular Flow Cellular Automaton Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    FOR A RECENT REVIEW, SEE R. Jackson, Some Mathematical and Physical Aspects of Continuum Models for the Motion of Granular Materials, in The Theory of Dispersed Multiphase Flow, R. Meyer ed., Academic Press 1983.Google Scholar
  2. [2]
    G. Schaeffer, Instability in the Evolution Equations Describing Imcompressible Granular Flow, J. Diff. Eq., 66 (1987), pp. 19–50.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    B. Pitman and D. G. Schaefer, Stability of Time Dependent Compressible Granular Flow in Two Dimensions, Comm. Pure Appl Math., 40 (1987), pp. 421–447.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    A Drescher, T. W. Cousens, AND P. L. Bransby, Kinematics of the mass flow of granular material through a plane hopper, geotechnique 28 (1978), pp. 27–42.CrossRefGoogle Scholar
  5. [5]
    R.L. Mhalowskl, Flow of granular material through a plane hopper, Powder Tech, 39 (1984), pp. 29–40.CrossRefGoogle Scholar
  6. [6]
    J.O. Cutress and R.F. Pulfer, X-ray investigations of Bowing powders, Powder Tech., 1 (1967), pp. 213–220.CrossRefGoogle Scholar
  7. [7]
    A. Jenike, Gravity Flow of bulk Solids, Bulletin No. 108, Utah Eng. Expt, Station, Univ. ofUtah, salt Lake City, 1961.Google Scholar
  8. [8]
    SEE N. Al-Din and D. J. Gunn, The Flow of Non-Cohesive Solids through Orifices, Chemical Engineering Science, 39 (1984), pp. 121–127, and references therein.CrossRefGoogle Scholar
  9. [9]
    M is constant, within a resolution of about 5%, on time-scales which are long compared to t m, but short compared to the time to obtain a time series or to empty the hopper. Measurements of M were carried out on a time scale of about 10s by seeing how long it took to fill small cups. M was then determined by ratio of the mass in the cup to the filling time.Google Scholar
  10. [10]
    R. K. Otnes and L. Enochson, Digital Time Series Analysis, Wiley, N.Y., 1972.zbMATHGoogle Scholar
  11. [11]
    See I. Vardoulakis — elsewhere in these proceedings.Google Scholar
  12. [12]
    TOMMASO TOFFOLI AND NORMAN MARGOLUS, Cellular Automata Machines: A New Environment for Modeling, MIT Press 1987.Google Scholar
  13. [13]
    G.W. Baxter and R.P. Behringer, to be published.Google Scholar
  14. [14]
    We are aware of only one other instance in which automata are being applied to granular flows. Work in progress by Drs. Peter Ha, Gary Gutt and Bradley Werner (see also Gary Gutt, Ph.D. thesis, California Institute of Technology, 1989) uses a considerably different adaptation of cellular automata to granular flow.Google Scholar
  15. [15]
    G. william Baxter and R. P. Behringer, Pattern Formation in Flowing Sand, Phys. Eef. Lett., 62 (1989), p. 2825.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • G. W. Baxter
    • 1
  • R. P. Behringer
    • 1
  • T. Fagert
    • 2
  • G. A. Johnson
    • 2
  1. 1.Department of Physics and Center for Nonlinear StudiesDuke UniversityUSA
  2. 2.Department of RadiologyDuke University Medical CenterUSA

Personalised recommendations