Solitary waves travelling along an unsmooth boundary

Any physical laws are scale-dependent, the same phenomenon might lead to debating theories if observed using different scales. The two-scale thermodynamics observes the same phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption, and fractal calculus has to be adopted to establish governing equations. Here basic properties of fractal calculus are elucidated, and the relationship between the fractal calculus and traditional calculus is revealed using the two-scale transform, fractal variational principles are discussed for 1-D fluid mechanics. Additionally planet distribution in the fractal solar system, dark energy in the fractal space, and a fractal ageing model are also discussed. nema

Dimension or scale is everything. When a thing is observed by different scales, different results can be obtained. Two scales are enough for most of practical problems, and a new definition of a two-scale dimension instead of the fractal dimension is given to deal with discontinuous problems. Fractal theory considers a self-similarity pattern, which cannot be found in any a real problem, while the two-scale theory observes each problem with two scales, the large scale is for an approximate continuous problem, where the classic calculus can be fully applied, and on the smaller scale, the effect of the porous structure on the properties can be easily elucidated. This paper sheds a new light on applications of fractal theory to real problems. nema