The theory of scaling

Abstract

Scaled experimentation is an important approach for the investigation of complex systems but for centuries has been impeded by the want of a scaling theory that can accommodate scale effects. The present definition of a scale effect is founded on the violation of an invariance principle arising out of dimensional analysis, i.e. dimensionless equations do not change with scale. However, apart from all but the most rudimentary of systems, most dimensionless governing equations invariably do change with scale, thus providing a very severe constraint on the reach of scaled experimentation. This paper introduces the theory of scaling that in principle applies to all physics and quantifies either implicitly or explicitly all scale dependencies. It is shown here how the route offered by dimensional analysis is nothing more than a particular similitude condition among a countable infinite number of alternative possibilities provided by the new theory. The theory of scaling is founded on a metaphysical concept where space is scaled and the mathematical consequences of this are reflected in the governing equations in transport form. The theory is trialled for known problems in continuum mechanics, electromagnetism and heat transfer to illustrate the breath of the approach and additionally demonstrate the advantages offered by additional forms of similitude.