Consolidation of Basics of Dimensional Analysis

Abstract

The echelon matrix procedure is introduced, as a new base procedure for dimensional analysis, by its application to examples selected by previous authors. It is thereby shown that Buckingham's actual and very perceptive enumeration of the number of nondimensional groups is never in error, despite repeated claims to the contrary. The achievement or the forcing of the echelon matrix formation is shown also to be the natural entry to the stepwise and the proportionality‐stepwise procedures, which, like the basic echelon matrix procedure, offer advantages in straightforward execution, over the traditional, nonintegrated, procedures for dimensional analysis which are ascribed to Rayleigh and to Buckingham. The echelon matrix procedure is also shown to be valuable in examination of various initiatives and claimed improvements to dimensional analysis of the past and present. Vector lengths or “directional analysis” and such concepts are reviewed, and their supposed advantages are shown to be based on misconceptions. It is then shown that the stepwise‐proportionality procedure alone offers an infallible check in the matter of dimensional homogeneity of the variables. An outline is given of the logical sequence of arrangement of variables and execution of analysis by the alternative procedures. Overall, it is shown that dimensional analysis is still a meaningful method for the future provided it is dissociated from past and current misconceptions and some supposed improvements, and is taught and executed in a manner which relates it to the mainstream of basic modern mathematics.