Multiple Scale Analysis

  • Roberto Mauri
Part of the Soft and Biological Matter book series (SOBIMA)


In Chap.  10, we have assumed to know both the structure of the effective equations and the micro-scale morphology of the multiphase systems, and thus we focussed on how to determine the effective properties appearing in the associated constitutive equations. On the other hand, in this chapter we only assume to know the governing equations, describing the transport of momentum, energy and mass at the micro-scale level, and then we intend to average them out, to find the effective equations at the macro- (or meso-) scale. First, in Sect. 11.1, we show how to perform a direct volume averaging, using a multi-pole expansion technique. Clearly, though, any averaging procedure must assume a clear separation of scales, that is the typical length- and time-scales at the micro-level must be much smaller than their macroscopic (or mesoscopic) counterparts. Accordingly, the most natural way to move up from one scale to the other, and thus determine the effective equations of a multiphase system, is by using multiple scale analysis. In Sect. 11.2, first we explain the idea underlying this approach and then show two examples of application to derive the Smoluchowsky equation and study Taylor dispersion. Finally, in Sect. 11.3, this approach is generalized, describing the coarse-graining homogenization procedure and thus show how some results on deterministic chaos can be found. In particular, we see that the transport of colloidal particles in non-homogeneous random velocity fields is described through a convection-diffusion equation that can also be derived from the Stratonovich stochastic process seen in Chap.  5.


Effective Diffusivity Tracer Particle Solvability Condition Brownian Particle Effective Equation 
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.


  1. 1.
    Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978) zbMATHGoogle Scholar
  2. 2.
    Brenner, H.: PhysicoChem. Hydrodyn. 1, 91 (1980) Google Scholar
  3. 3.
    Brenner, H.: Philos. Trans. R. Soc. Lond. A 297, 81 (1980) ADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Brenner, H., Edwards, D.A.: Macrotransport Processes. Butterworth-Heinemann, Stoneham (1993) Google Scholar
  5. 5.
    Brown, J.W., Churchill, R.V.: Fourier Series and Boundary Value Problems. McGraw-Hill, New York (1993) Google Scholar
  6. 6.
    Childress, S.: J. Chem. Phys. 56, 2527 (1972) ADSCrossRefGoogle Scholar
  7. 7.
    Dill, L.H., Brenner, H.: J. Colloid Interface Sci. 93, 343 (1983) CrossRefGoogle Scholar
  8. 8.
    Haber, S., Mauri, R.: J. Fluid Mech. 190, 201 (1988) ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Kim, S., Russel, W.B.: J. Fluid Mech. 154, 269 (1985) ADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Koch, D.L., Brady, J.F.: J. Fluid Mech. 154, 399 (1985) ADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Koch, D.L., Brady, J.F.: J. Fluid Mech. 180, 387 (1987) ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II. Nonequilibrium Statistical Mechanics. Springer, Berlin (1991), Chap. 2.4 zbMATHCrossRefGoogle Scholar
  13. 13.
    Lundgren, T.S.: J. Fluid Mech. 51, 273 (1972) ADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Mauri, R.: J. Eng. Math. 29, 77 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Mauri, R.: Phys. Fluids 15, 1888 (2003) ADSCrossRefGoogle Scholar
  16. 16.
    Mauri, R.: Phys. Rev. E 68, 066306 (2003) ADSCrossRefGoogle Scholar
  17. 17.
    Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1. MIT Press, Cambridge (1971), pp. 540–547, 591–606 Google Scholar
  18. 18.
    Pope, S.B.: Turbulent Flows, pp. 480–482. Cambridge University Press, Cambridge (2000) zbMATHCrossRefGoogle Scholar
  19. 19.
    Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory. Springer, Berlin (1980) zbMATHGoogle Scholar
  20. 20.
    Sneddon, I.N.: Proc. Glasg. Math. Assoc. 4(144), 151 (1960) Google Scholar
  21. 21.
    Taylor, G.I.: Proc. R. Soc. A 219, 186 (1953) ADSCrossRefGoogle Scholar
  22. 22.
    Taylor, G.I.: Proc. R. Soc. A 223, 446 (1953) ADSCrossRefGoogle Scholar
  23. 23.
    Taylor, G.I.: Proc. R. Soc. A 225, 473 (1954) ADSCrossRefGoogle Scholar
  24. 24.
    Vergassola, M., Avellaneda, M.: Physica D 106, 148 (1997) MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Roberto Mauri
    • 1
  1. 1.Department of Chemical Engineering, Industrial Chemistry and Material ScienceUniversity of PisaPisaItaly

Personalised recommendations