The Macroscopic Modelling of Multi-Phase Mixtures

  • D. Lhuillier
Part of the International Centre for Mechanical Sciences book series (CISM, volume 370)


The usual classification of materials into solids, liquids and gases is not entirely satisfying. Those who already used a tooth-paste, made a mayonnaise or wiped muddy shoes have experienced materials with strange properties, neither completely solid nor completely liquid. There also exist liquid-like materials (paint, egg white etc...) which do not behave as ordinary liquids. A careful investigation of those soft solids or odd liquids reveals they are all made of several chemical species and that the basic blocks of material are of a supramolecular size. In fact, these complex fluids are multi-phase mixtures, the simplest example of which is a suspension of particles in a fluid.The present lectures will concern the macroscopic modelling of multi-phase mixtures. Let us insist on the two underlined terms. Macroscopic: we will not focuss on the way to get exact results concerning the flow around particles but we will select among these results, the ones which prove important when considering the mixture as a continuous medium. Modelling: the description to be given will be a simplified one, obtained after a selection of the relevant physical phenomena, and a choice of the best variables to represent them. In the first chapter we will present fluid-mechanical results, while in chapter two we will consider the possible implications of the second law of thermodynamics. In these two chapters, a certain number of (more or less) intuitive statements will be made concerning the averaging procedure. These statements will be justified in chapter three while the last chapter will insist on phenomena linked with the fluctuational kinetic energy.


Internal Energy Dissipation Rate Velocity Fluctuation Momentum Balance Particle Volume Fraction 
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.
    Hinch, E.J.: An averaged equation approach to particles interactions in a fluid suspension, J. Fluid Mech. 83 (1977), 695–720.ADSCrossRefMATHGoogle Scholar
  2. 2.
    Batchelor, G.K.: The stress-tensor in a suspension of force-free particles, J.Fluid Mech. 41 (1970), 545–570.MathSciNetADSCrossRefMATHGoogle Scholar
  3. 3.
    Shaqfeh, E. and G. Fredrickson: The hydrodynamic stress in a suspension of rods, Phys. Fluids A2 (1990), 7–24.MathSciNetADSCrossRefMATHGoogle Scholar
  4. 4.
    Leal, L.G.: Macroscopic transport properties of a sheared suspension, J. Coll. Interface Sc. 58 (1977), 296–311.CrossRefGoogle Scholar
  5. 5.
    Buyevich, Y.A.: Heat and mass transfer in disperse media I. Average field equations II. Constitutive equations, I.t. J. Heat Mass Transfer 35 (1992), 2445–2463.CrossRefMATHGoogle Scholar
  6. 6.
    De Groot, S.R. and P. Mazur: Non-equilibrium Thermodynamics (ch.IIl), North-Holland, Amsterdam, 1969.Google Scholar
  7. 7.
    Maugin, G.A. and W. Muschik: Thermodynamics with internal variables, I. General concepts II. Applications, J. Non-Equilib. Thermo. 19 (1994), 217–289.MATHGoogle Scholar
  8. 8.
    Lhuillier, D.: From molecular mixtures to suspensions of particles, J. Phys.II France 5 (1995), 19–36.CrossRefGoogle Scholar
  9. 9.
    Barthes-Biesel, D. and A. Acrivos: Deformation and burst of a liquid droplet freely suspended in a linear shear field, J. Fluid Mech., 61 (1973), 1–21.ADSCrossRefMATHGoogle Scholar
  10. 10.
    Hand, G.L.: A theory of anisotropic fluids, J. Fluid Mech. 13 (1962), 33–46.MathSciNetADSCrossRefMATHGoogle Scholar
  11. 11.
    Onuki, A.: Dynamic equations of polymers with deformations in the semi-dilute regions, J. Phys. Soc. Japan, 59 (1990), 3423–3426.ADSCrossRefGoogle Scholar
  12. 12.
    Doi, M.: Effects of viscoelasticity on polymer diffusion, in: Dynamics and Patterns in Complex fluids (Ed. A. Onuki and K. Kawasaki) Springer Proceedings in Physics (1990) vol.52.Google Scholar
  13. 13.
    Drew, D.A.: Mathematical modelling of two-phase flows, Ann.Rev.Fluid Mech. 15 (1983), 261–291.ADSCrossRefGoogle Scholar
  14. 14.
    Lhuillier, D.: Ensemble averaging in slightly non-uniform suspensions, Eur.J.Mech., B/Fluids, 11 (1992), 649–661.MathSciNetMATHGoogle Scholar
  15. 15.
    Zhang, D.Z. and A. Prosperetti: Averaged equations for inviscid disperse two-phase flow, J.Fluid Mech. 267 (1994), 185–219MathSciNetADSCrossRefMATHGoogle Scholar
  16. 16.
    Nigmatulin, R.I.: Dynamics of Multiphase media, Hemisphere,New-York, 1990.Google Scholar

Copyright information

© Springer-Verlag Wien 1996

Authors and Affiliations

  • D. Lhuillier
    • 1
  1. 1.Pierre et Marie Curie UniversityParisFrance

Personalised recommendations