Theoretical Foundation of Dimensional Analysis

Abstract

Physical problems are described by relationships, which are dominated by quantities having a certain dimension, such as length, time, mass, force, temperature etc. These relations must be so structured that dependent and independent quantities are combined so as to yield dimensionally correct formulas. For instance, a physically correctly written formula must possess on each of its sides, left and right, the same physical dimension. Similarly, in an equation which describes a physical fact, quantities with different dimensions cannot be added. Such properties are connected with what is called dimensional homogeneity. It holds for all mathematical expressions describing physical facts. In other applied sciences, for instance mathematical economy, dimensional homogeneity is not requested to hold, a fact that allows equations with more general structure.

This and the following chapters are thoroughly revised and extended versions of a chapter on dimensional analysis and model theory of Hutter: “Fluid- and Therrrrodynamik — eine Einführung” which appeared in the German Language by Springer Verlag, Berlin, etc. [109].