Dimensional analysis: Not a recipe!

Abstract

Dimensional analysis is commonly used to reduce the number of parameters and variables that shall be taken into account in the analysis of a physical problem, by means of the construction of a few non-dimensional products. It is derived from the simple principle that the very concept of a physical law implies that it shall be expressed by mathematical relationships between measures of the involved physical quantities, which must be invariant with respect to any change in the units chosen for measuring these quantities. This principle is expressed mathematically through Vaschy-Buckingham’s theorem, also known as the pi theorem. Inspired from Saint-Guilhem’s papers (Saint-Guilhem R. 1962. Les principes de l’analyse dimensionnelle, invariance des relations vectorielles dans certains groupes d’affinités. Mémorial des sciences mathématiques. Paris: Gauthier-Villars, Vol. 152 Saint-Guilhem R. 1971. Les principes généraux de la similitude physique. Gauthier-Villars: Eyrolles; Saint-Guilhem R. 1985. Sur les fondements de la similitude physique : le théorème de Federman. J Mec Th Appl 4 (3): 337–356; Debongnie JF. 2016. Sur le théorème de Vaschy-Buckingham. [http:// hdl.handle.net/2268/197814] and Barenblatt GI. 1987. Dimensional analysis. New York: Gordon & Breach Sc. Publ) the paper aims at proposing a fairly didactic presentation, where, as strongly advised by Barenblatt, basic concepts such as physical quantities, dimensions, consistent systems of units are first explained. Then, the constitution of dimensionless products is thoroughly developed before specifying the correct number of independent dimensionless products that can be obtained from a given number of physical quantities. Finally, the pi theorem is stated, with a proof in the spirit of Vaschy’s original one. Historical comments evoke Galileo’s analysis, refer to the many contributions by celebrated scholars in the 18th and 19th centuries and conclude with more recent mathematical approaches.