Hydrodynamics of Fractal Media

Abstract

In the general case, real media are characterized by an extremely complex and irregular geometry. Because methods of Euclidean geometry, which ordinarily deals with regular sets, are used to describe real media, stochastic models in hydrodynamics are taken into account (Monin et al., 2007a,b; Ostoja-Starzewski, 2007a; Vishik et al., 1979; Vishik and Fursikov, 1988; Shwidler, 1985). Another possible way of describing a complex structure of the media is to use fractal theory of sets of non-integer-dimensionality (Mandelbrot, 1983; Frame et al., 2006; Feder, 1988). Although, the non-integer-dimension does not reflect completely the geometric and dynamic properties of the fractal media, it however permits some important conclusions about the behavior of the media. For example, the mass of the fractal media enclosed in a volume of characteristic size R satisfies the scaling law M(R) ∼ R D, whereas for a regular n-dimensional Euclidean object M(R) ∼ R n. We define a fractal medium as a medium with non-integer mass dimension. In general, fractal medium cannot be defined as a medium that is distributed over a fractal. Naturally, in real media the fractal structure cannot be observed on all scales but only those for which R 1 < R < R 2, where R 1 is the characteristic scale of the particles (molecules), and R 2 is the macroscopic scale for uniformity of the investigated structure and processes.