Multiple Scale Analysis

In this Chapter, starting from the governing equations, describing the transport of momentum, energy and mass at the micro-scale level, we average them out, to find the effective equations at the macro- (or meso-) scale. First, 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. After explaining the idea underlying this approach, we show two examples of application to derive the Smoluchowsky equation and study Taylor dispersion. Finally, 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.