Multiple Scale Analysis
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.
KeywordsEffective Diffusivity Tracer Particle Solvability Condition Brownian Particle Effective Equation
- 2.Brenner, H.: PhysicoChem. Hydrodyn. 1, 91 (1980) Google Scholar
- 4.Brenner, H., Edwards, D.A.: Macrotransport Processes. Butterworth-Heinemann, Stoneham (1993) Google Scholar
- 5.Brown, J.W., Churchill, R.V.: Fourier Series and Boundary Value Problems. McGraw-Hill, New York (1993) Google Scholar
- 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
- 20.Sneddon, I.N.: Proc. Glasg. Math. Assoc. 4(144), 151 (1960) Google Scholar