A Granular Mixture Model with Goodman–Cowin-Type Microstructure and its Application to Shearing Flows in Binary Solid–Fluid Bodies

  • Kolumban HutterEmail author
  • Yongqi Wang
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)


A continuum theory of a granular mixture is formulated. In the basic balance laws, we introduce an additional balance of equilibrated forces to describe the microstructural response according to Goodman and Cowin (Arch Rational Mech Anal, 44:249–266,1972, [10]) and Passman et al. (Rational Thermodynamics, Springer, New York, pp. 286–325, 1984, [18]) for each constituent. Based on the müllerLiu form of the second law of thermodynamics, a set of constitutive equations for a viscous solid–fluid mixture with microstructure is derived. These relatively general equations are then reduced to a system of ordinary differential equations describing a steady flow of the solid–fluid mixture between two horizontal plates. The resulting boundary value problem is solved numerically and results are presented for various values of parameters and boundary conditions. It is shown that simple shearing generally does not occur. Typically, for the solid phase, in the vicinity of a boundary, if the solid volume fraction is small, a layer of high shear rate occurs, whose thickness is nearly between 5 and 15 grain diameters, while if the solid volume fraction is high, an interlock phenomenon occurs. The fluid velocity depends largely on the drag force between the constituents. If the drag coefficient is sufficiently large, the fluid flow is nearly the same as that of the solid, while for a small drag coefficient, the fluid shearing flow largely decouples from that of the solid in the entire flow region. Apart from this, there is a tendency for solid particles to accumulate in regions of low shear rate.


Granular materials Solid–fluid mixture GoodmanCowin -type microstructure Viscous fluid Constitutive behavior Shearing flows in binary mixtures 


  1. 1.
    Ahmadi, G.: On mechanics of saturated granular materials. Int. J. Non-linear Mech. 15, 251–262 (1980)CrossRefGoogle Scholar
  2. 2.
    Ahmadi, G.: A continuum theory for two phase media. Acta Mech. 44, 299–317 (1982)CrossRefGoogle Scholar
  3. 3.
    Bagnold, R.A.: Experiments on a gravity free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. (Lond.) A225, 49–63 (1954)CrossRefGoogle Scholar
  4. 4.
    Bailard, J.A.: An experimental study of granular fluid flow. Thesis, University of California, San Diego (1978)Google Scholar
  5. 5.
    Bluhm, J., de Boer, R., Wilmanski, K.: The thermodynamic structure of the two component model of porous incompressible materials with true mass densities. Mech. Res. Commun. 22(2), 171–180 (1995)CrossRefGoogle Scholar
  6. 6.
    Bowen, R.M.: Incompressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 18, 1129–1148 (1980)CrossRefGoogle Scholar
  7. 7.
    Bowen, R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20, 697–735 (1982)CrossRefGoogle Scholar
  8. 8.
    Ehlers, W.: Compressible, incompressible and hybrid two-phase models in porous media theories. In: Angel, Y.C. (ed.) Anisotropy and Inhomogeneity in Elasticity and Plasticity. AMD, vol. 158, pp. 25–38. ASME, New York (1993)Google Scholar
  9. 9.
    Ehlers, W., Kubik, J.: On finite dynamic equations for fluid-saturated porous media. Acta Mech. 105, 101–117 (1994)CrossRefGoogle Scholar
  10. 10.
    Goodman, M.A., Cowin, S.C.: A continuum theory for granular materials. Arch. Rational Mech. Anal. 44, 249–266 (1972)CrossRefGoogle Scholar
  11. 11.
    Hanes, D.M., Inman, D.L.: Observations of rapidly flowing granular-fluid materials. J. Fluid Mech. 150, 357–380 (1985)CrossRefGoogle Scholar
  12. 12.
    Homsy, G., Ei-Kaissay, M.M., Didwania, A.: Instability waves and theorigin of bubbles in fluidized beds, part 2: comparison with theory. Int. J. Multiph. Flow 6, 305–318 (1980)CrossRefGoogle Scholar
  13. 13.
    Johnson, G., Massoudi, M., Rajagopal, K.R.: Flow of a fluid-solid mixture between flat plates. Chem. Eng. Sci. 46(7), 1713–1723 (1991)CrossRefGoogle Scholar
  14. 14.
    Liu, I-Shih: Method of lagrange multipliers for exploitation of the entropy principle. Arch. Rational Mech. Anal. 46, 131–148 (1972)CrossRefGoogle Scholar
  15. 15.
    Liu, I-Shih, Müller, I.: Thermodynamics of mixtures of fluids. In: Truesdell, C. (ed.) Rational Thermodynamics, pp. 264–285. Springer, New York (1984)Google Scholar
  16. 16.
    Massoudi, M.: Stability analysis of fluidized beds. Int. J. Eng. Sci. 26, 765–769 (1988)CrossRefGoogle Scholar
  17. 17.
    Nunziato, J.W., Passman, S.L., Thomas, J.P.: Gravitational flow of granular materials with incompressible grains. J. Rheol. 24, 395–420 (1980)CrossRefGoogle Scholar
  18. 18.
    Passman, S.L., Nunziato, J.W., Walsh, E.K.: A theory of multiphase mixtures. In: Truesdell, C. (ed.) Rational Thermodynamics, pp. 286–325. Springer, New York (1984)CrossRefGoogle Scholar
  19. 19.
    Passman, S.L., Nunziato, J.W., Bailey, P.B.: Shearing motion of a fluid-saturated granular material. J. Rheol. 30, 167–192 (1986)CrossRefGoogle Scholar
  20. 20.
    Savage, S.B.: Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92(1), 53–96 (1979)CrossRefGoogle Scholar
  21. 21.
    Svendsen, B., Hutter, K.: On the thermodymics of a mixture of isotropic materials with constraints. Int. J. Eng. Sci. 33, 2021–2054 (1995)CrossRefGoogle Scholar
  22. 22.
    Svendsen, B.: A thermodynamic model for volume-fraction-based mixtures. Z. Angew. Math. Mech. 77 (1996)Google Scholar
  23. 23.
    Truesdell, C.: A new definition of a fluid, II: the Maxwellian fluid. Journal de Mathématiques Pures et Appliquées 9(30), 111–155 (1951)Google Scholar
  24. 24.
    Truesdell, C.: Rational Thermodynamics. McGraw-Hill, New York (1969)Google Scholar
  25. 25.
    Wang, Y., Hutter, K.: Shearing flows in a Goodman-Cowin type granular material - theory and numerical results. Part. Sci. Technol. 51, 605–632 (1999)Google Scholar
  26. 26.
    Wang, Y., Hutter, K.: A constitutive model of multiphase mixtures and its application in shearing flows of saturated solid-fluid mixtures. Granul. Matter 1, 163–181 (1999)CrossRefGoogle Scholar
  27. 27.
    Wilmanski, K.: The thermodynamical model of compressible porous materials with the balance equation of porosity. Preprint No. 310, Edited by Weierstra"s-Institute für Angewandte Analysis und Stochastik, Berlin (1997)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.c/o Laboratory of Hydraulics, Hydrology, GlaciologyETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringDarmstadt University of TechnologyDarmstadtGermany

Personalised recommendations